diff --git "a/29FAT4oBgHgl3EQflB1V/content/tmp_files/load_file.txt" "b/29FAT4oBgHgl3EQflB1V/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/29FAT4oBgHgl3EQflB1V/content/tmp_files/load_file.txt" @@ -0,0 +1,1679 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf,len=1678 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='08614v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='AP] 20 Jan 2023 Plane wave stability analysis of Hartree and quantum dissipative systems Thierry Goudon∗1 and Simona Rota Nodari†1 1Université Côte d’Azur, Inria, CNRS, LJAD, Parc Valrose, F-06108 Nice, France Abstract We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' such a coupling lead to challenging issues compared to the usual non linear Schrödinger equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hartree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Particles interacting with a vibrational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Schrödinger-Wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Orbital stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 35Q40 35Q51 35Q55 1 Introduction This work is concerned with the stability analysis of certain solutions of the following Hartree-type equation iBtU ` 1 2∆xU “ γ ˆ σ1 ‹x ˆ Rn σ2Ψ dz ˙ U, (1a) ´ ∆zΨ “ ´γσ2pzq ` σ1 ‹x |U|2˘ pxq (1b) ∗thierry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='goudon@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='fr †simona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='rotanodari@univ-cotedazur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='fr 1 endowed with the initial condition U ˇˇ t“0 “ UInit, (2) and of the following Schrödinger-Wave system: iBtU ` 1 2∆xU “ γΦU, (3a) 1 c2 B2 ttΨ ´ ∆zΨ “ ´γσ2pzqσ1 ‹ |U|2pt, xq, (3b) Φpt, xq “ ¨ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy, (3c) where γ, c ą 0 are given positive parameters, completed with U ˇˇ t“0 “ UInit, Ψ ˇˇ t“0 “ ΨInit, BtΨ ˇˇ t“0 “ ΠInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (4) The variable x lies in the torus Td, meaning that the equations are understood with p2πq´periodicity in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In (3b), the additional variable z lies in Rn and, as explained below, it is crucial to assume n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For reader’s convenience, the scaling of the equation is fully detailed in Appendix A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for our purposes the God-given form functions σ1, σ2 are fixed once for all and the features of the coupling are embodied in the parameters γ, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The system (1a)-(1b) can be obtained, at least formally, from (3a)-(3c) by letting the parameter c run to `8, while γ is kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By the way, system (1a)-(1b) can be cast in the more usual form iBtU ` 1 2∆xU “ ´γ2κ ` Σ ‹x |U|2˘ U, t P R, x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (5) where1 κ “ ˆ Rn σ2pzqp´∆zq´1σ2pzq dz “ ˆ Rn |pσ2pξq|2 |ξ|2 dξ p2πqn ą 0 and Σ “ σ1 ‹ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (6) Letting now Σ resemble the delta-Dirac mass, the asymptotic leads to the standard cubic non linear Schrödinger equation iBtU ` 1 2∆xU “ ´γ2κ|U|2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (7) in the focusing case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These asymptotic connections can be expected to shed some light on the dynamics of (3a)-(3c) and to be helpful to guide the intuition about the behavior of the solutions, see [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The motivation for investigating these systems takes its roots in the general landscape of the analysis of “open systems”, describing the dynamics of particles driven by momentum and energy exchanges with a complex environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Such problems are modeled as Hamiltonian systems, and it is expected that the interaction mechanisms ultimately produce the dissipation of the particles’ energy, an idea which dates back to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Caldeira and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Leggett [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These issues have been investigated for various classical and quantum couplings, and with many different mathematical viewpoints, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' [2, 3, 24, 25, 28, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The case in which the environment is described as a vibrational field, like in the definition of the potential by (3b)-(3c), is particularly appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In 1The Fourier transform of an integrable function ϕ : Rn Ñ C is defined by pϕpξq “ ´ Rn ϕpzqe´iξ¨z dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2 fact, (3a)-(3c) is a quantum version of a model introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' De Bièvre and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Bruneau, dealing with a single classical particle [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Intuitively, the model of [6] can be thought of as if in each space position x P Rd there is a membrane oscillating in a direction z P Rn, transverse to the motion of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When a particle hits a membrane, its kinetic energy activates vibrations and the energy is evacuated at infinity in the z´direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These energy transfer mechanisms eventually act as a sort of friction force on the particle, an intuition rigorously justified in [6, Theorem 2 and Theorem 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We refer the reader to [1, 12, 13, 30, 46] for further theoretical and numerical insight about this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The model of [6] has been revisited by considering many interacting particles, which leads to Vlasov-type equations, still coupled to a wave equation for defining the potential [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Unexpectedly, asymptotic arguments indicate a connection with the attractive Vlasov-Poisson dynamic [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, the particles-environment interaction can be interpreted in terms of Lan- dau damping [19, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The quantum version (3a)-(3c) of the De Bièvre-Bruneau model has been discussed in [21, 20], with a connection to the kinetic model by means of a semi-classical analysis inspired from [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that in (3a)-(3c), the vibrational field remains of classical nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' a fully quantum framework is dealt with in [3] for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A remarkable feature of these systems is the presence of conserved quantities, here inherited from the framework designed in [6] for a classical particle, and the study of these models brings out the critical role of the wave speed c ą 0 and the dimension n of the space for the wave equation (we can already notice that n ě 3 is necessary for (6) to be meaningful), see [6, 18, 19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Schrödinger-Wave system (3a)-(3c) the energy HSWpU, Ψ, Πq “ 1 4 ˆ Td |∇U|2 dx ` 1 4 ¨ TdˆRn ˆΠ2 c2 ` |∇zΨ|2 ˙ dx dz ` γ 2 ˆ Td Φ|U|2 dx, (8) is conserved since we can readily check that d dtHSWpU, Ψ, BtΨq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, for the Hartree system (1a)-(1b), we get d dtHHapUq “ 0 where we have set HHapUq “ 1 4 ˆ Td |∇U|2 dx ´ γ2 κ 4 ˆ Td Σpx ´ yq|Upt, xq|2|Upt, yq|2 dy dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Furthermore, for both model, the L2 norm is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Of course, these conservation properties play a central role for the analysis of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, (1a)-(1b) has further fundamental properties which occur only for the asymptotic model: firstly, (1a)-(1b) is Galilean invariant, which means that, given a solution pt, xq ÞÑ upt, xq and for any p0 P Td, the function pt, xq ÞÑ upt, x ´ tp0qeipx´tp0{2q is a solution too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' secondly, the momentum pptq “ Im ´ ¯upt, xq∇xupt, xq dx is conserved and, accordingly, the center of mass follows a straight line at constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' That these properties are not satisfied by the more complex system (3a)-(3c) makes its analysis more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, we point out that, in contrast to the usual nonlinear Schrödinger equation or Hartree-Newton system, where Σ is the Newtonian potential, the equations (1a)-(1b) or (3a)-(3c) do not fulfil a 3 scale invariance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This also leads to specific mathematical difficulties: despite the possible regularity of Σ, many results and approaches of the Newton case do not extend to a general kernel, due to the lack of scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When the problem is set on the whole space Rd, one is interested in the stability of solitary waves, which are solutions of the equation with the specific form upt, xq “ eiωtQpxq, and, for (3a)-(3c), ψpt, x, zq “ Ψpx, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The details of the solitary wave are embodied into the Choquard equation, satisfied by the profile Q, [32, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It turns out that the Choquard equation have infinitely many solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' among these solutions, it is relevant to select the solitary wave which minimizes the energy functional under a mass constraint, [32, 37] and to study the orbital stability of this minimal energy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This program has been investigated for (7) and (1a)-(1b) in the specific case where Σpxq “ 1 |x| in dimension d “ 3, by various approaches [8, 31, 33, 34, 39, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Quite surprisingly, the specific form of the potential plays a critical role in the analysis (either through explicit formula or through scale invariance properties), and dealing with a general convolution kernel, as smooth as it is, leads to new difficulties, that can be treated by a perturbative argument, see [27, 51] for the case of the Yukawa potential, and [21] for (1a)-(1b) and (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here, we adopt a different viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We consider the case where the problem holds on the torus Td, and we are specifically interested in the stability of plane wave solutions of (3a)-(3c) and (1a)-(1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We refer the reader to [4, 5, 14, 40] for results on the nonlinear Schrödinger equation (7) in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The discussion on the stability of these plane wave solutions will make the following smallness condition 4γ2κ}σ1}2 L1 ă 1 (9) (assuming the plane wave has an amplitude unity) appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Despite its restriction to the periodic framework, the interest of this study is two-fold: on the one hand, it points out some difficulties specific to the coupling and provides useful hints for future works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' on the other hand, it clarify the role of the parameters, by making stability conditions explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In Section 2, we clarify the positioning of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we further discuss some mathematical features of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We also introduce the main assumptions on the parameters that will be used throughout the paper and we provide an overview of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Section 3 is concerned with the stability analysis of the Hartree equation (1a)-(1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Section 4 deals with the Schrödinger-Wave system at the price of restricting to the case where the wave vector of the plane wave solution vanishes: k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For reasons explained in details below, the general case is much more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Section 5 justifies that in general the mode k � 0 is linearly unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, in Appendix A, we provide a physical interpretation of the parameters involved, and for the sake of completeness, in Appendices B and C, we discuss the well-posedness of the Schrödinger-Wave system (3a)-(3c) and its link with the Hartree equation (1a)-(1b) in the regime of large c’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4 2 Set up of the framework 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Plane wave solutions and dispersion relation For any k P Zd, we start by seeking solutions to (3a)-(3c) of the form Upt, xq “ Ukpt, xq :“ exp ` ipωt ` k ¨ xq ˘ , Ψpt, x, zq “ Ψ˚pzq, BtΨpt, x, zq “ Π˚pzq “ 0, (10) with ω ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that the L2 norm of Uk is p2πqd{2 and Ψ˚ actually does not depend on the time variable, nor on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since |Ukpt, xq| “ 1 is constant, the wave equation simplifies to 1 c2 B2 ttΨ ´ ∆zΨ “ ´γσ2pzq @ σ1 D Td, where @ ¨ D Td stands for the average over Td: @ f D Td “ ´ Td fpxq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, z ÞÑ Ψ˚pzq is a solution to (3b) if Ψ˚pzq “ ´γΓpzq @ σ1 D Td, with Γ the solution of ´∆zΓpzq “ σ2pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This auxiliary function Γ is thus defined by the convolution of σ2 with the elementary solution of the Laplace operator in dimension n, or equivalently by means of Fourier transform: Γpzq “ ˆ Rn Cn |z ´ z1|n´2 σ2pz1q dz1 “ F ´1 ξÑz ´pσ2pξq |ξ|2 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (11) The corresponding potential (3c) is actually a constant which reads ´γ ¨ TdˆRn σ1px ´ yqσ2pzqΓpzq @ σ1 D Td dz dy “ ´κγ @ σ1 D2 Td with κ “ ˆ Rn σ2pzqΓpzq dz “ ˆ Rn |∇zΓpzq|2 dz ą 0 (we remind the reader that this formula coincides with (6) and makes sense only when n ě 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It remains to identify the condition on the coefficients so that Uk satisfies the Schrödinger equation (3a): this leads to the following dispersion relation ω ` k2 2 ´ Υ˚ “ 0, Υ˚ “ γ2κ @ σ1 D2 Td ą 0 (12) with k2 “ řd j“1 k2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can compute explicitly the associated energy: HSWpUk, Ψ˚, Π˚q “ p2πqd 2 ˆk2 2 ´ γ2κ 2 @ σ1 D2 Td ˙ “ p2πqd 4 pk2 ´ Υ˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Of course, among these solutions, the constant mode U0pt, xq “ eiωt1pxq has minimal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It turns out that the plane wave Ukpt, xq “ eiωteik¨x equally satisfies (1a)-(1b) provided the dispersion relation (12) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Incidentally, we can check that HHapUkq “ p2πqd 2 ˆk2 2 ´ γ2κ 2 @ Σ D Td ˙ “ p2πqd 4 pk2 ´ Υ˚q is made minimal when k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Hamiltonian structure and symmetries of the problem The conservation properties play a central role in the stability analysis, for instance in the reasonings that use concentration-compactness arguments [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Based on the conserved quantities, one can try to construct a Lyapounov functional, intended to evaluate how far a solution is from an equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the stability analysis relies on the ability to prove a coercivity estimate on the variations of the Lyapounov functional, see [47, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This viewpoint can be further extended by identifying analogies with finite dimensional Hamiltonian systems with symmetries, which has permitted to set up a quite general framework [22, 23], revisited recently in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The strategy relies on the ability in exhibiting a Hamiltonian formulation of the problem BtX “ JBXH pXq, where the symplectic structure is given by the skew-symmetric operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence of Noether’s Theorem, this formulation encodes the conservation properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In partic- ular, it implies that t ÞÑ H pXptqq is a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the problem under consideration, as it will be detailed below, X is a vectorial unknown with components possibly depending on different variables (x P Td and z P Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This induces specific difficulties, in particular because the nature of the coupling is non local and delicate spectral issues arise related to the essential spectrum of the wave equation in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we can easily observe that the systems (1a)-(1b) and (3a)-(3c) are invariant under multiplications by a phase factor of U, the “Schödinger unknown”, and under translations in the x variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to the conservation of the L2 norm of U and of the total momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, the systems (1a)-(1b) and (3a)-(3c) cannot be handled by a direct applica- tion of the results in [4, 22, 23]: the basic assumptions are simply not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Nevertheless, our approach is strongly inspired from [4, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As we will see later, for the Hartree system, a decisive advantage comes from the conservation of the total momentum and the Galilean invariance of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Schrödinger-Wave problem, since the expression of the total momentum mixes up contribution from the “Schrödinger unknown” U and the “wave unknown” Ψ, the information on its conservation does not seem readily useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2 In what follows, we find advantages in changing the unknown by writing Upt, xq “ eik¨xupt, xq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' in turn the Schrödinger equation iBtU ` 1 2∆U “ ΦU becomes iBtU ` 1 2∆u ´ k2 2 u ` ik ¨ ∇u “ Φu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, the parameter k will appear in the definition the energy functional H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This explains a major difference between (1a)-(1b) and (3a)-(3c): for the former, a coercivity estimate can be obtained for the energy functional H , for the latter, when k � 0 there are terms which cannot be controlled easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is reminiscent of the momentum conservation in (1a)-(1b) and the lack of Galilean invariance for (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The detailed analysis of the linearized operators sheds more light on the different behaviors of the systems (1a)-(1b) and (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2For the problem set on Rd, it is still possible, in the spirit of results obtained in [14] for NLS, to justify that orbital stability holds on a finite time interval: the solution remains at a distance ǫ from the orbit of the ground state over time interval of order Op1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ǫq, see [48, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='11 & Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The argument relies on the dispersive properties of the wave equation through Strichartz’ estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Outline of the main results Let us collect the assumptions on the form functions σ1 and σ2 that govern the coupling: (H1) σ1 : Td Ñ r0, 8q is C8 smooth, radially symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' @ σ1 D Td � 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (H2) σ2 : Rn Ñ r0, 8q is C8 smooth, radially symmetric and compactly supported;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (H3) p´∆q´1{2σ2 P L2pRnq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (H4) for any ξ P Rn, pσ2pξq � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Assumptions (H1)-(H2) are natural in the framework introduced in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hypothesis (H3) can equivalently be rephrased as p´∆q´1σ2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it appears in many places of the analysis of such coupled systems and, at least, it makes the constant κ in (6) meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This constant is a component of the stability constraint (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hypothesis (H4) equally appeared in [6, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (W)] when discussing large time asymptotic issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Assumptions (H1)-(H4) are assumed throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Our results can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We assume (9) and consider k P Zd and ω ą 0 satisfying (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Hartree equation, the analysis is quite complete: the plane wave eipωt`k¨xq is spectrally stable (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for any initial perturbation with zero mean, the solutions of the linearized Hartree equation are L2-bounded, uniformly over t ě 0 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the plane wave eipωt`k¨xq is orbitally stable (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Schrödinger-Wave system, only the case k “ 0 is fully addressed: the plane wave peiωt1pxq, ´γΓpzq @ σ D Td, 0q is spectrally stable (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for any initial perturbation of peiωt1pxq, ´γΓpzq @ σ D Td, 0q with zero mean, the solutions of the linearized Schrödinger-Wave system are L2-bounded, uniformly over t ě 0 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the plane wave peiωt1pxq, ´γΓpzq @ σ D Td, 0q is orbitally stable (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When k � 0, the situation is much more involved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' at least we prove that in general the plane wave solution peipωt`k¨xq, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable, see Section 5 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3 Stability analysis of the Hartree system (1a)-(1b) To study the stability of the plane wave solutions of the Hartree system, it is useful to write the solutions of (1a)-(1b) in the form Upt, xq “ eik¨xupt, xq with upt, xq solution to iBtu ` 1 2∆u ´ k2 2 u ` ik ¨ ∇u “ ´γ2κpΣ ‹ |u|2qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (13) 7 If k P Zd and ω ą 0 satisfy the dispersion relation (12), uωpt, xq “ eiωt1pxq is a solution to (13) with initial condition uωp0, tq “ 1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, studying the stability properties of Ukpt, xq “ eiωteik¨x as a solution to (1a)-(1b) amounts to studying the stability of uωpt, xq “ eiωt1pxq as a solution to (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The problem (13) has an Hamiltonian symplectic structure when considered on the real Banach space H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq ˆ H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, if we write u “ q ` ip, with p, q real-valued, we obtain Bt ˆ q p ˙ “ J∇pq,pqH pq, pq with J “ ˆ 0 1 ´1 0 ˙ and H pq, pq “ 1 2 ˆ1 2 ˆ Td |∇q|2 ` |∇p|2 dx ` k2 2 ˆ Tdpp2 ` q2q dx ´ ˆ Td pk ¨ ∇q dx ` ˆ Td qk ¨ ∇p dx ˙ ´ γ2κ 4 ˆ Td Σ ‹ pp2 ` q2qpp2 ` q2q dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to u “ q ` ip, we can write H puq “ 1 2 ˆ1 2 ˆ Td |∇u|2 dx ` k2 2 ˆ Td |upxq|2 dx ` ˆ Td k ¨ p´i∇uqu dx ˙ ´ γ2κ 4 ˆ TdpΣ ‹ |u|2qpxq|upxq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (14) As observed above, H is a constant of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, it is clear that (13) is invariant under multiplications by a phase factor so that Fpuq “ 1 2}u}2 L2 is conserved by the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The quantities Gjpuq “ 1 2 ˆ Td ˆ1 i Bxju ˙ u dx are constants of the motion too, that correspond to the invariance under translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, a direct verification leads to d dtGjpuqptq “ κγ2 2 ˆ Td ˆ Td BxjΣpx ´ yq ‹ |u|2pt, yq|u|2pt, xq dy dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, we shall endow the Banach space H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq ˆ H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq with the inner product Bˆ q p ˙ ˇˇˇ ˆ q1 p1 ˙F “ ˆ Td ` pp1 ` qq1q dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' that can be also interpreted as an inner product for complex-valued functions: xu|u1y “ Re ˆ Td uu1 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (15) 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Linearized problem and spectral stability Let us expand the solution of (13) around uω as upt, xq “ uωpt, xqp1 ` wpt, xqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The linearized equation for the fluctuation reads iBtw ` 1 2∆xw ` ik ¨ ∇xw “ ´2γ2κpΣ ‹ Repwqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (16) We split w “ q ` ip, q “ Repwq, p “ Impwq so that (16) recasts as Bt ˆq p ˙ “ Lk ˆq p ˙ (17) with the linear operator Lk : ˆ q p ˙ ÞÝÑ ¨ ˝ ´k ¨ ∇xq ´ 1 2∆xp 1 2∆xq ` 2γ2κΣ ‹ q ´ k ¨ ∇xp ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (18) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 (Spectral stability for the Hartree equation) Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the spectrum of Lk, the lineariza- tion of (13) around the plane wave uωpt, xq “ eiωt1pxq, in L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq is contained in iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Consequently, this wave is spectrally stable in L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, we expand q, p and σ1 by means of their Fourier series qpt, xq “ ÿ mPZd Qmptqeim¨x, Qmptq “ 1 p2πqd ˆ Td qpt, xqe´im¨x dx, ppt, xq “ ÿ mPZd Pmptqeim¨x, Pmptq “ 1 p2πqd ˆ Td ppt, xqe´im¨x dx, σ1pxq “ ÿ mPZd σ1,meim¨x, σ1,mptq “ 1 p2πqd ˆ Td σ1pxqe´im¨x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that σ1 being real and radially symmetric, we have σ1,m “ σ1,m “ σ1,´m (19) and, by definition, @ σ1 D Td “ p2πqdσ1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, we obtain Lk ˆq p ˙ “ ¨ ˚ ˚ ˝ ř mPZd ˆm2 2 Pm ´ ik ¨ mQm ˙ eim¨x ř mPZd ˆ ´m2 2 Qm ´ ik ¨ mPm ` 2p2πq2dγ2κ|σ1,m|2Qm ˙ eim¨x ˛ ‹‹‚ “ Lk,0 ˆQ0 P0 ˙ ` ÿ mPZd∖t0u Lk,m ˆQm Pm ˙ eik¨x (20) with Lk,0 “ ˆ 0 0 2p2πq2dγ2κ|σ1,0|2 0 ˙ and Lk,m “ ˜ ´ik ¨ m m2 2 ´ m2 2 ` 2p2πq2dγ2κ|σ1,m|2 ´ik ¨ m ¸ (21) for m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 9 Note that, since the Fourier modes are uncoupled, ˆq p ˙ is a solution to (17) if and only if the Fourier coefficients ˆQm Pm ˙ satisfy Bt ˆ Qmptq Pmptq ˙ “ Lk,m ˆ Qmptq Pmptq ˙ for any m P Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, λ P C is an eigenvalue of the operator Lk if and only if there exists at least one Fourier mode m P Zd such that λ is an eigenvalue of the matrix Lk,m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' there exists pqm, pmq � p0, 0q such that λqm ´ m2 2 pm ` ik ¨ mqm “ 0, λpm ` m2 2 qm ` ik ¨ mpm “ 2p2πq2dγ2κ|σ1,m|2qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (22) A straightforward computation gives that λ0 “ 0 is the unique eigenvalue of the matrix Lk,0 with eigenvector p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This means that KerpLkq contains at least the vector subspace spanned by the constant function x P Td ÞÑ ˆ0 1 ˙ , which corresponds to the constant solution upt, xq “ i of (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, if m P Zd ∖ t0u, λm is an eigenvalue of Lk,m if it is a solution to pλ ` ik ¨ mq2 ´ m2 2 ˆ ´m2 2 ` 2p2πq2dγ2κ|σ1,m|2 ˙ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is a second order polynomial equation for λ and the roots are given by λm,˘ “ ´ik ¨ m ˘ |m| 2 b ´m2 ` 4γ2κp2πq2d|σ1,m|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If the smallness condition (9) holds, the argument of the square root is negative for any m P Zd∖t0u, and thus the roots λ are all purely imaginary (and we note that λ´m,˘ “ λm,¯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' More precisely, we have the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 (Spectral stability for the Hartree equation) Let k, m P Zd and Lk,m defined as in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' λ0 “ 0 is the unique eigenvalue of Lk,0 and KerpLk,0q “ span "ˆ 0 1 ˙* ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for any m P Zd ∖ t0u, the eigenvalue of Lk,m are λm,˘ “ ´ik ¨ m ˘ |m| 2 b ´m2 ` 4γ2κp2πq2d|σ1,m|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (a) if 4γ2κp2πq2d |σ1,m|2 m2 ď 1, then λm,˘ P iR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (b) if 4γ2κp2πq2d |σ1,m|2 m2 ą 1, then λm,˘ P C ∖ iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, Repλm,`q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Now, (9) implies 4γ2κp2πq2d |σ1,m|2 m2 ă 1 for all m P Zd ∖ t0u, so that σpLkq Ă iR and uωpt, xq “ eiωt1pxq is spectrally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Conversely, if σ1, σ2 and γ are such that there exists m˚ P Zd ∖ t0u 10 verifying 4γ2κp2πq2d |σ1,m˚|2 m2 ˚ ą 1, then the plane wave uω is spectrally unstable for any k P Zd and ω ą 0 that satisfy the dispersion relation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This proves Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We observe that this result is consistent with the linear stability analysis of (7), see [40, The- orem 1], when replacing formally Σ by the delta-Dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The analogy should be considered with caution, though, since the functional difficulties are substantially different: here u ÞÑ ´ 1 2∆Tdu ´ 2γ2κΣ‹Repuq is a compact perturbation of ´ 1 2∆Td, which has a compact resolvent hence a spectral decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It is important to remark that the analysis of eigenproblems for Lk has consequences on the behavior of solutions to (17) of the particular form Qpt, xq “ eλtqpxq, Ppt, xq “ eλtppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We warn the reader that spectral stability excludes the exponential growth of the solutions of the linearized problem when the smallness condition (9) holds, but a slower growth is still possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be seen by direct inspection for the mode m “ 0: we have BtQ0 “ 0, so that Q0ptq “ Q0p0q and BtP0 “ 2p2πq2dκ @ σ1 D2 TdQ0p0q which shows that the solution can grow linearly in time P0ptq “ P0p0q ` 2p2πq2dγ2κ @ σ1 D2 TdQ0p0qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In fact, excluding the mode m “ 0 suffices to guaranty the linearized stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 (Linearized stability for the Hartree equation) Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let w be the solution of (16) associated to an initial data wInit P H1pTdq such that ´ Td wInit dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists a constant C ą 0 such that suptě0 }wpt, ¨q}H1 ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that if ´ Td wInit dx “ 0 then the corresponding Fourier coefficients Q0p0q and P0p0q are equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, Q0ptq “ P0ptq “ 0 for all t ě 0, so that ´ Td wpt, xq dx “ 0 for all t ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The proof follows from energetic consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, we observe that, on the one hand, 1 2 d dt ˆ Td |∇w|2 dx “ ´γ2κ 2i ˆ Td Σ ‹ pw ` wq∆pw ´ wq dx, and, on the other hand, 1 2 d dt ˆ Td Σ ‹ pw ` wqpw ` wq dx “ ´ 1 2i ˆ Td Σ ‹ pw ` wq∆pw ´ wq dx ´ k ¨ ˆ Td ∇pw ` wqΣ ‹ pw ` wq dx, where we get rid of the last term in the right hand side by assuming k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to the following energy conservation property d dt "1 2 ˆ Td |∇w|2 dx ´ γ2κ 2 ˆ Td Σ ‹ pw ` wqpw ` wq dx “ 0 which holds for k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We denote by E0 the energy of the initial data wInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, we can simply estimate ˇˇˇˇ ˆ Td Σ ‹ pw ` wqpw ` wq dx ˇˇˇˇ ď }Σ ‹ pw ` wq}L2}w ` w}L2 ď }Σ}L1}w ` w}2 L2 ď 4}Σ}L1}w}2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 11 To conclude, we use the Poincaré-Wirtinger estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, since we have already remarked that the condition ´ Td wInit dx “ 0 implies ´ Td wpt, xq dx “ 0 for any t ě 0, we can write }wpt, ¨q}2 L2 “ ›››wpt, ¨q ´ 1 p2πqd ˆ Td wpt, yq dy ››› 2 L2 “ p2πqd ÿ mPZd∖t0u |cmpwpt, ¨qq|2 ď p2πqd ÿ mPZd∖t0u m2|cmpwpt, ¨qq|2 “ }∇wpt, ¨q}2 L2 for any t ě 0, where the cmpwpt, ¨qq’s are the Fourier coefficients of the function x P Td ÞÑ wpt, xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for any solution with zero mean, we infer, for all t ě 0, 2E0 “ ˆ Td |∇w|2pt, xq dx´γ2κ ˆ Td Σ‹pw `wqpw `wqpt, xq dx ě p1´4γ2κ}Σ}L1q ˆ Td |∇wpt, xq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, if (9) is satisfied, we obtain sup tě0 }wpt, ¨q}H1 ď 2 d E0 1 ´ 4γ2κ}Σ}L1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The stability estimate extends to the situation where k � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, from the solution w of (16), we set vpt, xq “ wpt, x ` tkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It satisfies iBtv ` 1 2∆xv “ ´2γ2κΣ ‹ Repvq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, repeating the previous argument, }vpt, ¨q}H1 “ }wpt, ¨q}H1 remains uniformly bounded on p0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This step of the proof relies on the Galilean invari- ance of (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it could have been used from the beginning, but it does not apply for the Schrödinger- Wave system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 The analysis applies mutadis mutandis to any equation of the form (1a), with the potential defined by a kernel Σ and a strength encoded by the constant γ2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, the stability criterion is set on the quantity 4γ2κp2πqd |pΣm| m2 For instance, the elementary solution of pa2´∆xqΣ “ δx“0 with periodic boundary condition has its Fourier coefficients given by pΣm “ 1 p2πqdpa2`m2q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to the physical variable, in the one-dimension case, the function Σ reads Σpxq “ e´a|x| 2a ` coshpaxq ape2aπ ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The linearized stability thus holds provided 4γ2κp2πq2d 1 a2`1 ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Orbital stability In this subsection, we wish to establish the orbital stability of the plane wave uωpt, xq “ eiωt1pxq as a solution to (13) for k P Zd and ω ą 0 that satisfy the dispersion relation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As pointed out before, (13) is invariant under multiplications by a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to define the corresponding orbit through upxq “ 1pxq by O1 “ teiθ, θ P Ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 12 Intuitively, orbital stability means that the solutions of (13) associated to initial data close enough to the constant function x P Td ÞÑ 1 “ 1pxq remain at a close distance to the set O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Stability analysis then amounts to the construction of a suitable Lyapounov functional satisfying a coercivity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This functional should be a constant of the motion and be invariant under the action of the group that generates the orbit O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the construction of such a functional relies on the invariants of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, the plane wave has to be a critical point on the Lyapounov functional so that the coercivity can be deduced from the properties of its second variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The difficulty here is that, in general, the bilinear symmetric form defining the second variation of the Lyapounov function is not positive on the whole space: according to the strategy designed in [22], see also the review [47], it will be enough to prove the coercivity on an appropiate subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here and below, we adopt the framework presented in [4] (see also [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Inspired by the strategy designed in [4, Section 8 & 9], we introduce, for any k P Zd and ω ą 0 satisfying the dispersion relation (12), the set Sω “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Fpuq “ Fp1q “ p2πqd 2 “ p2πqd k2{2 ` ω 2γ2κ @ σ1 D2 Td ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Sω is therefore the level set of the solutions of (13), associated to the plane wave pt, xq ÞÑ uωpt, xq “ eiωt1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we introduce the functional Lωpuq “ H puq ` ωFpuq ´ dÿ j“1 kjGjpuq, (23) which is conserved by the solutions of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have BuLωpuqpvq “ Re ˆ1 2 ˆ Tdp´∆uqv dx ` k2 2 ˆ Td uv dx ´γ2κ ¨ TdˆTd Σpx ´ yq|upyq|2upxqvpxq dy dx`ω ˆ Td uv dx ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a matter of fact, we observe that BuLωp1q “ 0 owing to the dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get B2 uLωpuqpv, wq “ Re ˆ1 2 ˆ Tdp´∆ ` k2qwv dx ´2γ2κ ¨ TdˆTd Σpx ´ yqRe ` upyqwpyq ˘ upxqvpxq dy dx ´γ2κ ¨ TdˆTd Σpx ´ yq|upyq|2wpxqvpxq dy dx ` ω ˆ Td wv dx ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Still by using the dispersion relation, we obtain B2 uLωp1qpv, wq “ Re ¨ ˚ ˚ ˚ ˝ ˆ Td ˆ ´∆w 2 ´ 2γ2κΣ ‹ Repwq ˙ looooooooooooooooomooooooooooooooooon :“Sw vpxq dx ˛ ‹‹‹‚“ xSw|vy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 13 S : H2pTdq Ă L2pTdq Ñ L2pTdq is an unbounded linear operator and its spectral properties will play an important role for the orbital stability of uω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that the operator S is the linearized operator (18), up to the advection term k ¨ ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The main result of this subsection is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 (Orbital stability for the Hartree equation) Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave uωpt, xq “ eiωt1pxq is orbitally stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' @ε ą 0, Dδ ą 0, @vInit P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, }vInit ´ 1}H1 ă δ ñ sup tě0 distpvptq, O1q ă ε (24) where distpv, O1q “ infθPr0,2πr }v ´ eiθ1} and pt, xq ÞÑ vpt, xq P C0pr0, 8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pTdqq stands for the solution of (13) with Cauchy data vInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The key ingredient to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 is the following coercivity estimate on the Lyapounov functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exist η ą 0 and c ą 0 such that @w P Sω, dpw, O1q ă η ñ Lωpwq ´ Lωp1q ě c distpw, O1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (25) Then the the plane wave uωpt, xq “ eiωt1pxq is orbitally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Assume that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 holds and suppose, by contradiction, that uω is not orbitally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, there exists 0 ă ε0 ă 2 3η such that @n P N ∖ t0u, DuInit n P H1pTdq, }uInit n ´ 1}H1 ă 1 n and Dtn P r0, `8r, distpunptnq, O1q “ ε0, pt, xq ÞÑ unpt, xq P C0pr0, 8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pTdqq being the solution of (13) with Cauchy data uInit n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To apply the coercivity estimate of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6, we define zn “ ´ F p1q F punptnqq ¯1{2 unptnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It is clear that zn P Sω since Fpznq “ Fp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, ` unptnq ˘ nPN∖t0u is a bounded sequence in H1pTdq and limnÑ`8 Fpunptnqq “ Fp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, on the one hand, there exists γ P r0, 2πr such that }unptnq}H1 ď }unptnq ´ eiθ1}H1 ` }eiθ1}H1 ď 2dpunptnq, O1q ` }eiθ1}H1 “ 2ε0 ` }1}H1 and, on the other hand, |Fpunptnqq ´ Fp1q| “ 1 2|}unptnq}2 L2 ´ }1}2 L2| ď }unptnq ´ 1}L2pε0 ` }1}H1q ă 1 npε0 ` }1}H1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, limnÑ`8 }zn ´ unptnq}H1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This implies for n P N large enough, ε0 2 ď dpzn, O1q ď 3ε0 2 ă η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6, we obtain LωpuInit n q ´ Lωp1q “ Lωpunptnqq ´ Lωp1q “ Lωpunptnqq ´ Lωpznq ` Lωpznq ´ Lωp1q ě Lωpunptnqq ´ Lωpznq ` cdpzn, O1q2 ě Lωpunptnqq ´ Lωpznq ` c 4ε2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 14 Finally, using the fact that BuLωp1q “ 0 and B2 uLωp1qpw, wq ď C}w}2 H1, we deduce that lim nÑ`8pLωpuInit n q ´ Lωp1qq “ 0, lim nÑ`8pLωpunptnqq ´ Lωpznqq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are thus led to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since BuLωp1q “ 0, the coercivity estimate (25) can be obtained from a similar estimate on the bilinear form B2 uLωp1qpw, wq for any w P H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As pointed out before, the difficulty lies in the fact that, in general, this bilinear form is not positive on the whole space H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The following lemma states that it is enough to have a coercivity estimate on B2 uLωp1qpw, wq for any w P T1SωXpT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Recall that the tangent set to Sω is given by T1Sω “ tu P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, BuFp1q “ 0u “ " pq, pq P H1pTd, Rq ˆ H1pTd, Rq, A ˆ q p ˙ ˇˇˇ ˆ 1 0 ˙ E “ 0 This set is the orthogonal to 1 with respect to the inner product defined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The tangent set to O1 (which is the orbit generated by the phase multiplication) is T1O1 “ spanRti1u so that pT1O1qK “ tu P H1pTd, Cq, xu, i1y “ 0u “ " pq, pq : Td Ñ R, A ˆ q p ˙ ˇˇˇ ˆ 0 1 ˙ E “ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists ˜c ą 0 B2 uLωp1qpu, uq ě ˜c}u}2 H1 (26) for any u P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then there exist η ą 0 and c ą 0 such that (25) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let w P Sω such that distpw, O1q ă η with η ą 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By means of an implicit function theorem argument (see [4, Section 9, Lemma 8]), we obtain that there exists θ P r0, 2πr and v P pT1O1qK such that eiθw “ 1 ` v, distpw, O1q ď }v}H1 ď Cdistpw, O1q for some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we use the fact that H1pTdq “ T1Sω ‘ spanRt1u to write v “ v1 ` v2 with v1 P T1Sω X pT1O1qK and v2 P spanRt1u X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since v “ eiθw ´ 1 and Fpwq “ Fp1q, we obtain 0 “ Fpeiθwq ´ Fp1q “ 1 2 ˆ Td |v|2 dx ` Re ˆ Tdpv1 ` v2q1 dx “ 1 2 ˆ Td |v|2 dx ` Re ˆ Td v21 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since v2 P spanRt1u, it follows that }v2}H1 ď }v}2 H1 }1}L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This implies }v1}H1 “ }v ´ v2}H1 ě }v}H1 ´ 1 }1}L2 }v}2 H1 ě 1 2}v}H1 15 provided }v}H1 ď }1}L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, if }v}H1 is small enough, using that B2 uLωp1qpw, zq ď C}w}H1}z}H1, we obtain B2 uLωp1qpv1, v2q ď C}v}3 H1, B2 uLωp1qpv2, v2q ď C}v}4 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to B2 uLωp1qpv, vq “ B2 uLωp1qpv1, v1q ` op}v}2 H1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, let w P Sω be such that dpw, O1q ă η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have Lωpwq ´ Lωp1q “ Lωpeiθwq ´ Lωp1q “ 1 2B2 uLωp1qpv, vq ` op}v}2 H1q “ 1 2B2 uLωp1qpv1, v1q ` op}v}2 H1q ě ˜c}v1}2 H1 ` op}v}2 H1q ě ˜c 2}v}2 H1 ` op}v}2 H1q ě ˜c 4distpw, O1q2 where we use BuLωp1q “ 0 and v1 P T1Sω X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' At the end of the day, to prove the orbital stability of the plane wave uωpt, xq “ eiωt1pxq it is enough to prove (26) for any u P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be done by studying the spectral properties of the operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, in the simpler case of the Hartree equation, the coercivity of B2 uLωp1q on T1S1 X pT1O1qK can be also obtained directly from the expression B2 uLωp1qpu, uq “ Re ˆˆ Td ˆ ´∆u 2 ´ 2γ2κΣ ‹ Repuq ˙ upxq dx ˙ “ xSu|uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (27) Let u P T1S1 X pT1O1qK and write u “ q ` ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to B2 uLωp1qpu, uq “ 1 2 ˆ Td |∇q|2 dx ´ 2γ2κ ˆ TdpΣ ‹ qqq dx ` 1 2 ˆ Td |∇p|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, since u P T1S1 X pT1O1qK, we have ˆ Td q dx “ 0 and ˆ Td p dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce B2 uLωp1qpu, uq ě 1 2 ˆ Td |∇q|2 dx ´ 2γ2κ ˆ TdpΣ ‹ qqq dx ` 1 4}p}2 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (28) Next, we expand q and Σ in Fourier series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' qpxq “ ÿ mPZd qmeim¨x and Σpxq “ ÿ mPZd Σmeim¨x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that, if Σ “ σ1 ‹ σ1, then Σm “ p2πqdσ2 1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, ´ Td q dx “ 0 implies q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, 1 2 ˆ Td |∇q|2 dx ´ 2γ2κ ˆ TdpΣ ‹ qqq dx “ p2πqd ÿ mPZd∖t0u ˆm2 2 ´ 2γ2κp2πqdΣm ˙ q2 m “ p2πqd ÿ mPZd∖t0u ˆ 1 ´ 4γ2κp2πqd Σm m2 ˙ m2 2 q2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (29) As a consequence, we obtain the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 16 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists δ P p0, 1q such that 4γ2κp2πq2d σ2 1,m m2 ď δ (30) for all m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists ˜c ą 0 such that B2 uLωp1qpu, uq ě ˜c}u}2 H1 (31) for any u P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If (30) holds, then (28)-(29) lead to B2 uLωp1qpu, uq ě 1 ´ δ 2 p2πqd ÿ mPZd∖t0u m2q2 m ` 1 4}p}H1 “ 1 ´ δ 2 }∇q}2 L2 ` 1 4}p}2 H1 ě 1 ´ δ 4 }u}2 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that ´ Td q dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='9 By decomposing the linear operator S into real and imaginary part and by using Fourier series, one can study its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, S has exactly one negative eigenvalue λ´ “ ´2γ2κ @ Σ D Td with eigenspace spanRt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, KerpSq “ spanRti1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, if (30) is satisifed, then infpσpSq X p0, 8qq ě 1´δ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, by applying the same arguments as in [5, Section 6], we can recover the coercivity of B2 uLωp1q on T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6, gives Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 and the orbital stability of the plane wave uω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4 Stability analysis of the Schrödinger-Wave system: the case k “ 0 Like in the case of the Hartree system, to study the stability of the plane wave solutions of the Schrödinger-Wave system (3a)-(3c), it is useful to write its solutions in the form Upt, xq “ eik¨xupt, xq with pt, x, zq ÞÑ pupt, xq, Ψpt, x, zqq solution to iBtu ` 1 2∆xu ´ k2 2 u ` ik ¨ ∇xu “ ´ ˆ γσ1 ‹ ˆ Rn σ2Ψ dz ˙ u, 1 c2 B2 ttΨ ´ ∆zΨ “ ´γσ2σ1 ‹ |u|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (32) If k P Zd and ω ą 0 satisfy the dispersion relation (12), uωpt, xq “ eiωt1pxq, Ψ˚pt, x, zq “ ´γΓpzq @ σ1 D Td, Π˚pt, x, zq “ BtΨ˚pt, x, zq “ 0 17 with Γ the solution of ´∆zΓ “ σ2 (see (11)), is a solution to (32) with initial condition uωp0, tq “ 1pxq, Ψ˚p0, x, zq “ ´γΓpzq @ σ1 D Td, Π˚p0, x, zq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the time being, we stick to the framework identified for the study of the asymptotic Hartree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Problem (32) has a natural Hamiltonian symplectic structure when considered on the real Banach space H1pTdqˆH1pTdqˆL2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqqˆL2pTd ˆRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, if we write u “ q `ip, with p, q real-valued, we obtain Bt ¨ ˚ ˚ ˝ q p Ψ Π ˛ ‹‹‚“ ˆJ 0 0 ´J ˙ ∇pq,p,Ψ,ΠqHSW pq, p, Ψ, Πq with J “ ˆ 0 1 ´1 0 ˙ and HSWpq, p, Ψ, Πq “ 1 2 ˆ1 2 ˆ Td |∇q|2 ` |∇p|2 dx ` k2 2 ˆ Tdpp2 ` q2q dx ´ ˆ Td pk ¨ ∇q dx ` ˆ Td qk ¨ ∇p dx ˙ ` 1 4 ˆ TdˆRn ˆΠ2 c2 ` |∇zΨ|2 ˙ dx dz ` γ 2 ˆ Td ˆˆ TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ pp2 ` q2qpxq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to u “ q ` ip, we can write HSWpu, Ψ, Πq “ 1 2 ˆ1 2 ˆ Td |∇u|2 dx ` k2 2 ˆ Td |upxq|2 dx ` ˆ Td k ¨ p´i∇uqu dx ˙ ` 1 4 ˆ TdˆRn ˆΠ2 c2 ` |∇zΨ|2 ˙ dx dz ` γ 2 ˆ Td ˆˆ TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ |upxq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (33) As a consequence, HSW is a constant of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, it is clear that (32) is invariant under multiplications by a phase factor of u so that Fpuq “ 1 2}u}2 L2 is conserved by the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, now, the quantities Gjpuq “ 1 2 ˆ Td ˆ1 i Bxju ˙ u dx (34) are not constants of the motion: d dtGjpuqptq “ γ 2 ˆ Td ˆ Td Bxjσ1px ´ yq ˆˆ Rn σ2pzqΨpt, y, zq dz ˙ |u|2pt, xq dy dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, they cannot be used in the construction of the Lyapounov functional as we did for the Hartree system (see (23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 18 Finally, we consider the Banach space H1pTdqˆH1pTdqˆL2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqqˆL2pTdˆRnq endowed with the inner product C ¨ ˚ ˚ ˝ q p Ψ Π ˛ ‹‹‚ ˇˇˇ ¨ ˚ ˚ ˝ q1 p1 Ψ1 Π1 ˛ ‹‹‚ G “ ˆ Td ` pp1 ` qq1q dx ` ˆ TdˆRnp∇zΨ∇zΨ1 ` ΠΠ1q dx dz that can be also interpreted as an inner product for complex valued functions: xpu, Ψ, Πq|pu1, Ψ1, Π1qy “ Re ˆ Td uu1 dx ` ˆ TdˆRnp∇zΨ ¨ ∇zΨ1 ` ΠΠ1q dx dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (35) We denote by } ¨ } the norm on H1pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq induced by this inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Preliminary results for the linearized problem: spectral sta- bility when k “ 0 As before, we linearize the system (3a)-(3c) around the plane wave solution obtained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Namely, we expand Upt, xq “ Ukpt, xqp1 ` upt, xqq, Ψpt, x, zq “ ´γ @ σ1 D TdΓpzq ` ψpt, x, zq and, assuming that u, ψ and their derivatives are small, we are led to the following equations for the fluctuation pt, xq ÞÑ upt, xq P C, pt, x, zq ÞÑ ψpt, x, zq P R iBtu ` 1 2∆xu ` ik ¨ ∇xu “ γΦ, ´ 1 c2 B2 ttψ ´ ∆zψ ¯ pt, x, zq “ ´γσ2pzqσ1 ‹ ρpt, xq, ρpt, xq “ 2Re ` upt, xq ˘ , Φpt, xq “ ¨ TdˆRn σ1px ´ yqσ2pzqψpt, y, zq dz dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (36) We split the solution into real and imaginary parts upt, xq “ qpt, xq ` ippt, xq, qpt, xq “ Repupt, xqq, ppt, xq “ Impupt, xqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We obtain pBtq ` 1 2∆xp ` k ¨ ∇xqqpt, xq “ 0, pBtp ´ 1 2∆xq ` k ¨ ∇xpqpt, xq “ ´γ ˆ σ1 ‹ ˆ Rn σ2pzqψpt, ¨, zq dz ˙ pxq, ´ 1 c2 B2 ttψ ´ ∆zψ ¯ pt, x, zq “ ´2γσ2pzqσ1 ‹ qpt, xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (37) 19 It is convenient to set π “ ´ 1 2c2 Btψ, in order to rewrite the wave equation as a first order system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We obtain Bt ¨ ˚ ˚ ˝ q p ψ π ˛ ‹‹‚“ Lk ¨ ˚ ˚ ˝ q p ψ π ˛ ‹‹‚ (38) where Lk is the operator defined by Lk : ¨ ˚ ˚ ˝ q p ψ π ˛ ‹‹‚ÞÝÑ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xp ´ k ¨ ∇xq 1 2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ ˆˆ Rn σ2ψ dz ˙ ´2c2π ´1 2∆zψ ` γσ2σ1 ‹ q ˛ ‹‹‹‹‹‹‚ For the next step, we proceed via Fourier analysis as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We expand q, p, ψ, π and σ1 by means of their Fourier series: ψpt, x, zq “ ÿ mPZd ψmpt, zqeim¨x, ψmpt, zq “ 1 p2πqd ˆ Td ψpt, x, zqe´im¨x dx, πpt, x, zq “ ÿ mPZd πmpt, zqeim¨x, πmpt, zq “ 1 p2πqd ˆ Td πpt, x, zqe´im¨x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, recall that σ1 being real and radially symmetric, (19) holds and, by definition, @ σ1 D Td “ p2πqdσ1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, since the Fourier modes are uncoupled, the Fourier coefficients pQmptq, Pmptq, ψmpt, zq, πmpt, zqq satisfy Bt ¨ ˚ ˚ ˝ Qm Pm ψm πm ˛ ‹‹‚“ Lk,m ¨ ˚ ˚ ˝ Qm Pm ψm πm ˛ ‹‹‚ (39) where Lk,m stands for the operator defined by Lk,m ¨ ˚ ˚ ˝ Qm Pm ψm πm ˛ ‹‹‚“ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´ik ¨ mQm ` m2 2 Pm ´m2 2 Qm ´ ik ¨ mPm ´ γp2πqdσ1,m ˆ Rn σ2pzqψm dz ´2c2πm γp2πqdσ2pzqσ1,mQm ´ 1 2∆zψm ˛ ‹‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Like for the Hartree equation, the behavior of the mode m “ 0 can be analysed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 20 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 (The mode m “ 0) For any k P Zd, the kernel of Lk,0 is spanned by p0, 1, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, equation (39) for m “ 0 admits solutions which grow linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let pQ0, P0, ψ0, π0q P KerpLk,0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It means that $ ’ ’ ’ & ’ ’ ’ % γp2πqdσ1,0 ˆ Rn σ2pzqψ0pzq dz “ 0, π0 “ 0, ∆zψ0 “ 2γp2πqdσ2pzqσ1,0Q0, which yields ψ0pzq “ ´2γ @ σ1 D TdQ0Γpzq with Γpzq “ p´∆q´1σ2pzq so that ´2γ2@ σ1 D2 TdκQ0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It implies that Q0 “ 0, ψ0 “ 0 while P0 is left undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, the first equation in (39) tells us that Q0ptq “ Q0p0q P C is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get Btψ0 “ ´2c2π0 which leads to B2 ttψ0 ´ c2∆zψ0 “ ´σ2pzq 2γc2@ σ1 D TdQ0p0q looooooooomooooooooon :“C1 (40) The solution of (40) with initial condition pψ0pzq, π0pzq “ ´ 1 2c2Btψp0, zqq P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqˆL2pRnq satisfies pψ0pt, ξq “ pψ0p0, ξq cospc|ξ|tq ´ 2c2pπ0pξqsinpc|ξ|tq c|ξ| ´ ˆ t 0 sinpc|ξ|sq c|ξ| pσ2pξqC1 ds where pψ0pt, ξq and pπ0pt, ξq are the Fourier transforms of z ÞÑ ψpt, zq and z ÞÑ πpt, zq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, integrating BtP0 “ ´γ @ σ1 D Td loooomoooon :“C2 ˆ Rn σ2pzqψ0pzq dz we obtain P0ptq “ P0p0q ` C2 ˆ Rn pσ2pξq pψ0p0, ξqsinpc|ξ|tq c|ξ| dξ p2πqn ´ 2c2C2 ˆ Rn pσ2pξqpπ0p0, ξq1 ´ cospc|ξ|tq c2|ξ|2 dξ p2πqn ´ C1C2 ˆ t 0 ˆ s 0 pcpτq dτ ds where pcpτq “ ˆ Rd |pσ2pξq|2 sinpc|ξ|τq c|ξ| dξ p2πqn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This kernel already appears in the analysis performed in [10, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The contribution involving the initial data of the vibrational field can be uniformly bounded by 1 p2πqn ˆˆ Rd |pσ2pξq|2 c2|ξ|2 dξ ˙1{2 #ˆˆ Rd | pψ0p0, ξq|2 dξ ˙1{2 ` 4c2 ˆˆ Rd |pπ0p0, ξq|2 c2|ξ|2 dξ ˙1{2+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, as a consequence of (H2), it turns out that pc is compactly supported, with ´ 8 0 pcpτq dτ “ κ c2, see [10, Lemma 14] and [19, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that ˆ t 0 ˆ s 0 pcpτq dτ ds “ ˆ t 0 pcpτq ˆˆ t τ ds ˙ dτ “ ˆ t 0 pt ´ τqpcpτq dτ „ tÑ8 t κ c2 ´ ˆ 8 0 τpcpτq dτ, 21 which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When k “ 0, basic estimates based on the energy conservation allow us to justify the stability of the solutions with zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, in contrast to what has been established for the Hartree system, this analysis does not extend to any mode k � 0, since the system is not Galilean invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 (Linearized stability for the Schrödinger-Wave system when k “ 0) Let k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) and let pu, ψ, πq be the solution of (36) associated to an initial data uInit P H1pTdq, ψInit P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq, πInit P L2pTd ˆ Rnq such that ´ Td uInit dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists a constant C ą 0 such that suptě0 }upt, ¨q}H1 ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Again, we use the energetic properties of the linearized equation (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have already remarked that ´ Td upt, xq dx “ 0 for any t ě 0 when ´ Td uInit dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We start by computing d dt "1 2 ˆ Td |∇xu|2 dx ` 1 2 ˆ TdˆRn ´|Btψ|2 c2 ` |∇zψ|2¯ dz dx “ ´iγ 2 ˆ Td Φ∆xpu ´ uq dx ´ γ ˆ TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get d dt ˆ Td Φpu ` uq dx “ ˆ TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx ` i 2 ˆ Td Φ∆xpu ´ uq dx ´ ˆ Td Φk ¨ ∇xpu ` uq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get rid of the last term by assuming k “ 0 and we arrive in this case at d dt "1 2 ˆ Td |∇xu|2 dx ` 1 2 ˆ TdˆRn ´|Btψ|2 c2 ` |∇zψ|2¯ dz dx ` γ ˆ Td Φpu ` uq dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We estimate the coupling term as follows ˇˇˇˇ ˆ Td Φpu ` uq dx ˇˇˇˇ “ ˇˇˇˇ ˆ TdˆRn σ2pzqψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zqσ1 ‹ pu ` uqpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' xq dz dx ˇˇˇˇ ď }σ1 ‹ pu ` uq}L2 ˆ ˆˆ Td ˇˇˇ ˆ Rn σ2pzqψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz ˇˇˇ 2 dx ˙1{2 ď }σ1}L1}u ` u}L2 ˆ ˆˆ Td ˇˇˇ ˆ Rn pσ2pξq pψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ξq dξ p2πqn ˇˇˇ 2 dx ˙1{2 ď 2}σ1}L1}u}L2 ˆ ˆˆ Td ˇˇˇ ˆ Rn pσ2pξq |ξ| |ξ|| pψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ξq| dξ p2πqn ˇˇˇ 2 dx ˙1{2 ď 2}σ1}L1}u}L2 ˆ ˆˆ Rn |pσ2pξq|2 |ξ|2 dξ ˙1{2 ˆ ˆˆ TdˆRn |ξ|2| pψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ξq|2 dξ p2πqn dx ˙1{2 ď 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='κ}σ1}L1}u}L2 ˆ ˆˆ TdˆRn |∇zψpt, x, ξq|2 dz dx ˙1{2 “ 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='κ}σ1}L1}u}L2}∇zψ}L2 ď 1 2γ }∇zψ}2 L2 ` 2κγ}σ1}2 L1}u}2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 22 By using the Poincaré-Wirtinger inequality }u}L2 ď }∇xu}L2, we deduce that 1 2 ˆ Td |∇xupt, xq|2 dx ď E0 1 ´ 4γ2κ}σ1}2 L1 , where E0 depends on the energy of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' While it is natural to start with the linearized operator Lk in (38), it turns out that this formulation is not well-adapted to study the spectral stability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The difficulties relies on the fact that the wave part of the system induces an essential spectrum, reminiscent to the fact that σessp´∆zq “ r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For instance, this is even an obstacle to set up a perturbation argument from the Hartree equation, in the spirit of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We shall introduce later on a more adapted formulation of the linearized equation, which will allow us to overcome these difficulties (and also to go beyond a mere perturbation analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Orbital stability for the Schrödinger-Wave system when k “ 0 In this subsection, we wish to establish the orbital stability of the plane wave solution to (32) obtained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, namely uωpt, xq “ eiωt1pxq, Ψ˚pt, x, zq “ ´γΓpzq @ σ1 D Td, Π˚pt, x, zq “ 0 with k P Zd and ω ą 0 that satisfy the dispersion relation (12) and Γpzq “ p´∆q´1σ2pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The system (32) being invariant under multiplications of u by a phase factor, we define the corresponding orbit through p1pxq, ´γΓpzq @ σ1 D Td, 0q by O1 “ tpeiθ, ´γΓpzq @ σ1 D Td, 0q, θ P Ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As before, orbital stability intuitively means that the solutions of (32) associated to initial data close enough to p1pxq, ´γΓpzq @ σ1 D Td, 0q remain at a close distance to the set O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us introduce, for any k P Zd and ω ą 0 satisfying the dispersion relation (12), the set Sω “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pu, Ψ, Πq P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd, L2pRnqq, Fpuq “ Fp1q “ p2πqd 2 ) , and the functional Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` ωFpuq, (41) intended to serve as a Lyapounov functional, where HSW is the constant of motion defined in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For further purposes, we simply denote Lω “ Lω,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` 1 2i ˆ Td k ¨ ∇u ¯u dx loooooooooomoooooooooon “ dÿ j“1 kjGjpuq ` ´ ω ` k2 2 ¯ Fpuq with HSW defined in (8) and Gjpuq defined in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thanks to the dispersion relation (12), only the second term of this expression depends on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Unfortunately, as pointed out before, the quantities 23 Gjpuq are not constants of the motion so that the dependence on k of the Lyapounov functional (41) cannot be disregarded, in contrast to what we did for the Hartree system in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, as in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2, we need to evaluate the first and second order variations of Lω,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We compute Bpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqHSWpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Πqpv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq “ Re ˆ1 2 ˆ Tdp´∆uqv dx ` γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqΨpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz dy ˙ upxqvpxq dx ˙ ` γ 2 ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz dy ˙ |upxq|2 dy dx ` 1 2 ¨ TdˆRn ´ 1 c2 Π τ ` p´∆zΨq φ dz ¯ dx and B2 pu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqHSWpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Πq ` pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τ 1q ˘ “ Re "1 2 ˆ Tdp´∆vqv1 dx `γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqpφpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zqv1pxq ` φ1pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zqvpxqq dz dy ˙ upxq dx ˙ `γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqΨpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz dy ˙ vpxqv1pxq dx ˙* ` 1 2 ¨ TdˆRn ´ 1 c2 τ τ 1 ` p´∆zφq φ1 dz ¯ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Besides, we have BuFpuqpvq “ Re ˆˆ Td uv dx ˙ , B2 uFpuqpv, v1q “ Re ˆˆ Td vv1 dx ˙ , BuGjpuqpvq “ Im `´ Td Bxjuv dx ˘ , B2 uGpuqpv, v1q “ Im `´ Td Bxjv1v dx ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we are led to Bpu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0qpv, φ, τq “ Re ˆ ´γ2@ σ1 D2 Tdκ ˆ Td v dx ` ´ ω ` k2 2 ¯ ˆ Td v dx ` γ 2 @ σ1 D Td ¨ TdˆRnpσ2 ` ∆zΓq φ dz dx ˙ “ 0 thanks to the dispersion relation (12) and the definition of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, the second order derivative 24 casts as B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv, φ, τq ˘ “ Re ˆ1 2 ˆ Tdp´∆vqv dx ` 1 2 ¨ TdˆRn ´τ 2 c2 ` p´∆zφq φ dz ¯ dx ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ vpxq dx ´ γ2@ σ1 D Td ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqΓpzq dz dy ˙ vpxqvpxq dx ` ´ ω ` k2 2 ¯ ˆ Td vpxqvpxq dx ˙ ` Im ˜ dÿ j“1 kj ˆ Td Bxjvv dx ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The forth and fifth integrals combine as ˆ Td ´ ω ` k2 2 ´ γ2κ @ σ1 D2 Td ¯ vpxqvpxq dx “ 0 which cancels out by virtue of the dispersion relation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence we get B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv, φ, τq ˘ “ Re ˆ1 2 ˆ Tdp´∆vqv dx ` 1 2 ¨ TdˆRn ´τ 2 c2 ` p´∆zφq φ dz ¯ dx ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ vpxq dx ´ i ˆ Td k ¨ ∇v v dx ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Note that the following continuity estimate holds: for any pv, φ, τq P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq, B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv, φ, τq ˘ ď 1 2}∇v}2 L2 ` 1 2c2 }τ}2 L2 ` 1 2}φ}2 L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` 2γκ1{2}σ1}L1}v}L2}φ}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` |k|}∇v}L2}v}L2 ď 1 2 ˆ p1 ` |k|q}v}2 H1 ` 1 c2 }τ}2 L2 ` C}φ}2 L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ˙ ď maxp1{c2, 1 ` |k|, Cq 2 }pv, φ, τq}2 with C “ 1 ` 4γ2κ}σ1}2 L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The functional Lω,k is conserved by the solutions of (32);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' however the difficulty relies on justifying its coercivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are only able to answer positively in the specific case k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the main result of this subsection restricts to this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 (Orbital stability for the Schrödinger-Wave system) Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q is orbitally stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' @ε ą 0, Dδ ą 0, @pvInit, φInit, τ Initq P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq, }vInit ´ 1}H1 ` }φInit ` γΓ @ σ D Td}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` }τ Init}L2 ă δ ñ sup tě0 distppvptq, φptq, τptqq, O1q ă ε (42) 25 where distppv, φ, τq, O1q “ infθPr0,2πr }v ´ eiθ1}H1 ` }φ ` γΓ @ σ D Td}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` }τ}L2 and pt, x, zq ÞÑ pvpt, xq, φpt, x, zq, τpt, x, zqq stands for the solution of (32) with Cauchy data pvInit, φInit, τ Initq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Using the same argument as in the case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, we can reduce the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 to the following coercivity estimate on the Lyapounov functional (and this is where we use that Lω,k is a conserved quantity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exist η ą 0 and c ą 0 such that @pw, ψ, χq P Sω, distppw, ψ, χq, O1q ă η ñ Lω,kppw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq @ σ D Td, 0qq ě cdistppw, ψ, χq, O1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (43) Then the the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q is orbitally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As we have seen before, since Bpu,ψ,ΠqLω,kpp1, ´γΓpzq @ σ D Td, 0qq “ 0, the coercivity estimate (43) can be obtained from an estimate on the bilinear form B2 pu,ψ,ΠqLω,kpp1, ´γ @ σ1 D TdΓ, 0qqppu, φ, τq, pu, φ, τqq for any pu, φ, τq P T1Sω X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here the tangent set to Sω is given by T1Sω “ " u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re ˆˆ Td upxq1pxq dx ˙ “ 0 ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This set is the orthogonal to p1, 0, 0q with respect to the inner product defined in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The tangent set to O1 (which is the orbit generated by the phase multiplications of 1) is T1O1 “ spanRtpi1, 0, 0qu so that pT1O1qK “ " u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re ˆ i ˆ Td upxq1pxq dx ˙ “ 0 ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists ˜c ą 0 B2 pu,ψ,ΠqLω,kpp1, ´γΓpzq @ σ D Td, 0qqppu, φ, τq, pu, φ, τqq ě ˜cp}u}2 H1 ` }φ}2 L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` }τ}2 L2q “ ˜c}pu, φ, τq}2 (44) for any pu, φ, τq P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then there exist η ą 0 and c ą 0 such that (43) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let pw, ψ, χq P Sω such that distppw, ψ, χq, O1q ă η with η ą 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, infθPr0,2πq }w´eiθ1} ă η and, by means of an implicit function theorem argument (see [4, Section 9, Lemma 8]), we obtain that there exists θ P r0, 2πq and v P ␣ u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re ` i ´ Td upxq dx ˘ “ 0 ( such that eiθw “ 1 ` v, inf θPr0,2πq }w ´ eiθ1} ď }v}H1 ď C inf θPr0,2πq }w ´ eiθ1} 26 for some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Denote by φpx, zq “ ψpx, zq`γΓpzq @ σ1 D Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then pv, φ, χq P pT1O1qK and }pv, φ, χq} ď Cη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we use the fact that H1pTdq “ ␣ u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re `´ Td upxq dx ˘ “ 0 ( ‘ spanRt1u to write pv, φ, χq “ pv1, φ, χq ` pv2, 0, 0q with pv1, φ, χq P T1Sω X pT1O1qK and v2 P spanRt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, }v2}H1 ď }v}2 H1 }1}L2 and }v1}H1 ě 1 2}v}H1 provided }v}H1 ď }1}L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, if }v}H1 is small enough, using that B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv1, φ1, τ 1q ˘ ď C}pv, φ, τq}}pv1, φ1, τ 1q}, we obtain B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv1, φ, χq, pv2, 0, 0q ˘ ď C}pv, φ, χq} }v}2 H1 ď C}pv, φ, χq}3, B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv2, 0, 0q, pv2, 0, 0q ˘ ď C}v}4 H1 ď C}pv, φ, χq}4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, χq, pv, φ, χq ˘ “ B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv1, φ, χq, pv1, φ, χq ˘ ` op}pv, φ, χq}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' let pw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq P Sω such that dppw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' O1q ă η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' we have Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kppw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χqq ´ Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kpp1pxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γΓpzq @ σ D Td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0qq “ Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kppeiθw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χqq ´ Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kpp1pxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γΓpzq @ σ D Td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0qq “ B2 pu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqLω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γ @ σ1 D TdΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0q ` pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq ˘ ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q “ B2 pu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqLω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γ @ σ1 D TdΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0q ` pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq ˘ ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q ě ˜c}pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq}2 ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q ě ˜c 2}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq}2 ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q ě ˜c 6dppw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' O1q2 where we use Bpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqLω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γ @ σ1 D TdΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0q “ 0 and pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq P T1Sω X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As before, to prove the orbital stability of the plane solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q it is enough to prove (44) for any pu, φ, τq P T1S1 XpT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let pu, φ, τq P T1S1 XpT1O1qK and write u “ q ` ip with q, p P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ “ Re ˆ1 2 ˆ Tdp´∆uqu dx ` 1 2 ¨ TdˆRn ´τ 2 c2 ` p´∆zφq φ dz ¯ dx ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ upxq dx ´ i ˆ Td k ¨ ∇u u dx ˙ (45) can be reinterpreted as a quadratic form acting on the 4-uplet W “ pq, p, φ, τq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To be specific, it 27 expresses as the following quadratic form on W, QpW, Wq “1 2 ˆ Td |∇p|2 dx ` 1 2c2 ¨ TdˆRn |τ|2 dz dx ` 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dyqpxq dx ˙ ` 2 ˆ Td qk ¨ ∇p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The crossed term ´ Td qk ¨ ∇p dx is an obstacle for proving a coercivity on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For this reason, let us focus on the case k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since pu, φ, τq P T1S1 X pT1O1qK, we have ˆ Td q dx “ 0 and ˆ Td p dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce, when k “ 0 QpW, Wq ě1 4}p}2 H1 ` 1 2c2 }τ}2 L2 ` 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx (46) Next, we expand q, σ1 and φp¨, zq in Fourier series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' qpxq “ ÿ mPZd qmeim¨x, φpx, zq “ ÿ mPZd φmpzqeim¨x and σ1pxq “ ÿ mPZd σ1,meim¨x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that σ1,m “ σ1,m “ σ1,´m since σ1 is real and radially symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, ´ Td q dx “ 0 implies q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, ˆ Td ˆˆ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx “ p2πq2dRe ¨ ˝ ÿ mPZd∖t0u σ1,mqm ˆ Rn σ2pzqφmpzq dz ˛ ‚ which implies 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx “ p2πqd ÿ mPZd∖t0u Re ˆm2 2 q2 m ` 1 2 ˆ Rn |∇zφm|2 dz ` 2p2πqdγσ1,mqm ˆ Rn σ2pzqφmpzq dz ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we remark that for any m P Zd, ˇˇˇˇRe ˆ 2p2πqdγσ1,mqm ˆ Rn σ2pzqφmpzq dz ˙ˇˇˇˇ ď 2p2πqdγσ1,m|qm| ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='κ}∇φm}L2 ď 1 2˜δp4γ2κp2πq2dσ2 1,mqq2 m ` ˜δ 2}∇φm}2 L2 28 for any ˜δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, for any ˜δ P p0, 1q, we get 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx ě p2πqd ÿ mPZd ˆˆm2 2 ´ 1 2˜δ p4γ2κp2πq2dσ2 1,mq ˙ q2 m ` 1 ´ ˜δ 2 }∇φm}2 L2 ˙ (47) As a consequence, we obtain the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists δ P p0, 1q such that 4γ2κp2πq2d σ2 1,m m2 ď δ (48) for all m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists ˜c ą 0 such that B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ ě ˜c}pu, φ, τq}2 (49) for any pu, φ, τq P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If (48) holds, then, for any ˜δ P pδ, 1q, (45)-(46)-(47) lead to B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ ě 1 4}p}2 H1 ` 1 2c2 }τ}2 L2 ` ˜δ ´ δ 2˜δ p2πqd ÿ mPZd∖t0u m2q2 m ` 1 ´ ˜δ 2 p2πqd ÿ mPZd }∇φm}2 L2 “ 1 4}p}2 H1 ` 1 2c2 }τ}2 L2 ` ˜δ ´ δ 2˜δ }∇q}L2 ` 1 ´ ˜δ 2 }φ}L2xH1z ě ˜c}pu, φ, τq}2 where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that ´ Td q dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 gives Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and the orbital stability of the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q in the case k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 The coercivity of B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ on T1S1XpT1O1qK can be recovered from the spectral properties of a convenient unbounded linear operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, as we have seen before, by decomposing u into real and imaginary part, the quadratic form defined by (45) (with k “ 0) can be written as QpW, Wq “ 1 2 ˆ Td |∇p|2 dx ` 1 2c2 ¨ TdˆRn |τ|2 dz dx ` B S ˆq φ ˙ ˇˇˇ ˆq φ ˙F 29 with S : H2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq Ă L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq Ñ L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq the unbounded linear operator given by S ˆ q φ ˙ “ ¨ ˚ ˝ ´1 2∆xq ` γσ1 ‹ ˆ Rn σ2φ dz 1 2φ ` γΓσ1 ‹ q ˛ ‹‚ (where we remind the reader that Γ “ p´∆q´1σ2q) and the inner product Bˆ q φ ˙ ˇˇˇ ˆ q1 φ1 ˙F “ ˆ Td qq1 dx` ˆ TdˆRn ∇zφ¨∇zφ1 dz dx “ ˆ Td qq1 dx` ˆ TdˆRn ˆφpx, ξqˆφ1px, ξq|ξ|2 dξ p2πqn dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq is an Hilbert space with this inner product since n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since ˆ T d ˆ σ1 ‹ ˆ Rn σ2φ dz ˙ pxqq1pxq dx “ ˆ Td ˆˆ TdˆRn σ1px ´ yqσ2pzqφpy, zq dz dy ˙ q1pxq dx “ ˆ TdˆRn φpx, zqσ2pzqpσ1 ‹ q1qpxq dx dz “ ˆ TdˆRn ˆφpx, ξq ˆσ2pξq |ξ|2 pσ1 ‹ q1qpxq dx|ξ|2 dξ p2πqn we can check that S is a self-adjoint operator on L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, σpSq Ă R and one can easily study the spectrum of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' More precisely, using Fourier series, we find that if λ is an eigenvalue of S then there exists at least one m P Zd such that for some pqm, φmq � p0, 0q there holds $ ’ ’ & ’ ’ % ˆm2 2 ´ λ ˙ qm ` γp2πqdσ1,m ˆ Rn σ2pzqφmpzq dz “ 0, ˆ1 2 ´ λ ˙ φmpzq ` γp2πqdΓpzqσ1,mqm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let λ � 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for any m P Zd, qm “ 0 implies φmpzq “ 0 for any z P Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, we may assume qm � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to φmpzq “ ´ γp2πqdσ1,mqm 1{2´λ Γpzq and ˆm2 2 ´ λ ˙ ˆ1 2 ´ λ ˙ ´ γ2p2πq2dσ2 1,mκ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By solving this equation, we obtain λ˘,m “ ´ m2`1 2 ¯ ˘ c´ m2´1 2 ¯2 ` 4γ2p2πq2dσ2 1,mκ 2 so that λ`,m ě 1 4 for any m P Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we remark that λ´,0 “ 1 2 ´ b 1 4 ` 4γ2p2πq2dσ2 1,0κ 2 ă 0 30 since 4γ2κp2πq2dσ2 1,0 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This eigenvalue corresponds to an eigenfunction p˜q, ˜φq with ˜q P spanRt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, ´ Td ˜qpxq dx � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, if (30) holds, λ´,m ě ´ m2`1 2 ¯ ´ c´ m2´1 2 ¯2 ` δm2 2 ě 1 ´ δ 5 for any m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that B S ˆq φ ˙ ˇˇˇ ˆq φ ˙F “ C¨ ˚ ˝ ´1 2∆xq ` γσ1 ‹ ˆ Rn σ2φ dz 1 2φ ` γΓσ1 ‹ q ˛ ‹‚ ˇˇˇ ˆq φ ˙G ě min ˆ1 2, 1 ´ δ 5 ˙ p}q}2 L2 ` }φ}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z q for all pq, φq P tq P L2pTdq, ´ T d q dx “ 0u ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This, together with the Poincaré- Wirtinger inequality, proves the coercivity of B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ on T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5 Discussion about the case k � 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 A new symplectic form of the linearized Schrödinger-Wave system We go back to the linearized problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The viewpoint presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 looks quite natural;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' however, it misses some structural properties of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In order to work in a unified functional framework, we find convenient to change the wave unknown ψ, which is naturally valued in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnq, into p´∆q´1{2φ, where the new unknown φ now lies in L2pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the linearized problem is rephrased as BtX “ LX, where X stands for the 4-uplet pq, p, φ, πq and LX “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xp ´ k ¨ ∇xq 1 2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ ´2c2p´∆q1{2π 1 2p´∆q1{2φ ` γσ2σ1 ‹ q ˛ ‹‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The operator L is seen as an operator on the Hilbert space V “ L2pTdq ˆ L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq, with domain DpLq “ H2pTdq ˆ H2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pRnqq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can start with the following basic information, which has the consequence that the spectral stability amounts to justify that σpLq Ă iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 31 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Let pλ, Xq be an eigenpair of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let Y : px, zq ÞÑ pqp´xq, ´pp´xq, φp´x, zq, ´πp´x, zqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, pλ, Xq, p´λ, Y q and p´λ, Y q are equally eigenpairs of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since L has real coefficients, LX “ λX implies LX “ λX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we check that LY px, zq “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ 1 2∆xp ` k ¨ ∇xq 1 2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz1 ˙ 2c2p´∆q1{2π 1 2p´∆q1{2φ ` γσ2σ1 ‹ q ˛ ‹‹‹‹‹‹‹‚ p´x, zq “ λ ¨ ˚ ˚ ˝ ´qp´x, zq pp´x, zq ´φp´x, zq πp´x, zq ˛ ‹‹‚“ ´λY px, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we make a new symplectic structure appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, let us introduce the blockwise operator J “ ˆJ1 0 0 J2 ˙ , J1 “ ˆ 0 1 ´1 0 ˙ , J2 “ ˆ 0 ´p´∆q1{2 p´∆q1{2 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are thus led to L “ J L with L X “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xq ` k ¨ ∇xp ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ ´1 2∆xp ´ k ¨ ∇xq 1 2φ ` γp´∆q´1{2σ2σ1 ‹ q 2c2π ˛ ‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (50) For further purposes, we also set Ă J “ ˆ ˜ J1 0 0 ˜ J2 ˙ , ˜ J1 “ ˆ0 ´1 1 0 ˙ , ˜ J2 “ ˆ 0 p´∆q´1{2 ´p´∆q´1{2 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (51) The operator J has 0 in its essential spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' nevertheless Ă J plays the role of its inverse since J Ă J “ I “ Ă J J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 The operator L is an unbounded self adjoint operator on V with domain DpL q “ H2pTdq ˆ H2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq, and the operator J is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The space V is endowed with the standard L2 inner product ` X|X1q “ ˆ Tdpqq1 ` pp1q dx ` ¨ TdˆRnpφφ1 ` ππ1q dx dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 32 We get ` L X|X1˘ “ ˆ Td !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´ ´ 1 2∆xq ` k ¨ ∇xp ¯ q1 ` ´ ´ 1 2∆xp ´ k ¨ ∇xq ¯ p1 ) dx `γ ˆ Td σ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ q1 dx ` ¨ TdˆRn ´1 2φφ1 ` 2c2ππ1 ¯ dx dz `γ ¨ TdˆRn ´ p´∆q´1{2σ2σ1 ‹ q ¯ φ1 dx dz “ ˆ Td !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' q ´ ´ 1 2∆xq1 ` k ¨ ∇xp1 ¯ ` p ´ ´ 1 2∆xp1 ´ k ¨ ∇xq1 ¯) dx `γ ¨ TdˆRn φp´∆q´1{2σ2σ1 ‹ q1 dz dx ` ¨ TdˆRn ´1 2φφ1 ` 2c2ππ1 ¯ dx dz `γ ˆ Td qσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ1 dz ˙ dx “ ` X|L X1˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' and ` J X|X1˘ “ ¨ Td ´ pq1 ´ qp1 ¯ dx ` ¨ TdˆRn ´ ´ p´∆q1{2πφ1 ` p´∆q1{2φπ1 ¯ dx dz “ ´ ¨ Td ´ qp1 ´ pq1 ¯ dx ´ ¨ TdˆRn ´ ´ φp´∆q1{2π1 ` πp´∆q1{2φ1 ¯ dx dz “ ´ ` X|J X1˘ As said above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' justifying the spectral stability for the Schrödinger-Wave equation reduces to verify that the spectrum σpLq is purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, the coupling with the wave equation induces delicate subtleties and a direct approach is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Instead, based on the expression L “ J L , we can take advantage of stronger structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the functional framework adopted here allows us to overcome the difficulties related to the essential spectrum induced by the wave equation, which ranges over all the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This approach is strongly inspired by the methods introduced by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Pelinovsky and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Chugunova [9, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The workplan can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It can be shown that the eigenproblem LX “ λX for L is equivalent to a generalized eigenvalue problem AW “ αKW, with α “ ´λ2, see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 below, where the auxiliary operators A and K depend on J , L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, we need to identify negative eigenvalues and complex but non real eigenvalues of the generalized eigenproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we appeal to a counting statement due to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Spectral properties of the operator L The stability analysis relies on the spectral properties of L , collected in the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Let L the linear operator defined by (50) on DpL q Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, the following assertions hold: 33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' σesspL q “ ␣ 1 2, 2c2( , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L has a finite number of negative eigenvalues, with eigenfunctions in DpL q, given by npL q “ 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u `#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, npL q “ 1 when k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The eigenspaces associated to the negative eigenvalues are all finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With X0 “ p0, 1, 0, 0q, we have spanRtX0u Ă KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, given k P Zd∖t0u, let K˚ “ tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 “ 0 and σ1,m “ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, we get dimpKerpL qq “ 1 ` #K˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We remind the reader that σ1 is assumed radially symmetric, see (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Consequently σ1,m “ σ1,´m “ σ1,˘m and both #K˚ and #tm P Zd ∖t0u, m4 ´4pk¨mq2 ď 0 and σ1,m � 0u are necessarily even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since L is self-adjoint, σpL q Ă R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us study the eigenproblem for L : λX “ L X means $ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ % λq “ ´1 2∆xq ` k ¨ ∇xp ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ , λp “ ´1 2∆xp ´ k ¨ ∇xq, λφ “ 1 2φ ` γp´∆q´1{2σ2σ1 ‹ q, λπ “ 2c2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (52) Clearly λ “ 2c2 is an eigenvalue with eigenfunctions of the form p0, 0, 0, πq, π P L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, dimpKerpL ´ 2c2Iqq is not finite and 2c2 P σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We turn to the case λ � 2c2, where the last equation imposes π “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Using Fourier series, we obtain λqm “ m2 2 qm ` ik ¨ mpm ` γp2πqdσ1,m ˆˆ Rnp´∆q´1{2σ2φm dz ˙ , λpm “ m2 2 pm ´ ik ¨ mqm, λφm “ 1 2φm ` γp2πqdp´∆q´1{2σ2σ1,mqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (53) where qm, pm P C are the Fourier coefficients of q, p P L2pTdq while φmpzq “ 1 p2πqd ´ Td φpx, zqe´im¨x dx for all z P Rn and φ P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We split the discussion into several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Case m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, the equations (53) degenerate to λq0 “ γp2πqdσ1,0 ˆˆ Rnp´∆q´1{2σ2φ0 dz ˙ , λp0 “ 0, ´ λ ´ 1 2 ¯ φ0 “ γp2πqdp´∆q´1{2σ2σ1,0q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 34 Combining the first and the third equation yields λ ´ λ ´ 1 2 ¯ q0 “ γ2p2πq2dσ2 1,0κq0, still with κ “ ´ p´∆q´1σ2σ2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It permits us to identify the following eigenvalues: λ “ 0 is an eigenvalue associated to the eigenfunction p0, 1, 0, 0q, since σ1,0 “ 1 p2πqd ´ Td σ1 dx � 0, and p´∆q´1{2σ2 � 0, λ “ 1{2 is an eigenvalue associated to eigenfunctions p0, 0, φ, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As before, since dimpKerpL ´ 1 2Iqq is not finite, 1 2 P σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the roots of λ ´ λ ´ 1 2 ¯ ´ γ2p2πq2dσ2 1,0κ “ λ2 ´ λ 2 ´ γ2p2πq2dσ2 1,0κ “ 0, provide two additional eigenvalues λ˘ “ 1{2 ˘ b 1{4 ` 4γ2p2πq2dσ2 1,0κ 2 , associated to the eigenfunctions p1, 0, γp2πqdσ1,0p´∆q´1{2σ2 λ˘´1{2 , 0q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To sum up, the Fourier mode m “ 0 gives rise to two positive eigenvalues (1/2 and λ`), one negative eigenvalue (λ´) and the eigenvalue 0, the last two being both one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It tells us that dimpKerpL qq ě 1 and npL q ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Case m � 0 with σ1,m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In this case, the m-mode equations (53) for the particle and the wave are uncoupled pλ ´ 1{2qφm “ 0, pMm ´ λq ˆ qm pm ˙ “ 0 where we have introduced the 2 ˆ 2 matrix Mm “ ˆ m2{2 ik ¨ m ´ik ¨ m m2{2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (54) We identify the following eigenvalues: λ “ 1{2 is an eigenvalue associated to the eigenfunction p0, 0, eim¨xφpzq, 0q, for any φ P L2pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Once again, this tells us that 1 2 P σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the eigenvalues λ˘ “ m2˘2k¨m 2 P R of the 2 ˆ 2 matrix Mm, associated to the eigenfunctions peim¨x, ¯ieim¨x, 0, 0q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since trpMmq ą 0, at most only one of these eigenvalues can be negative, which occurs when detpMmq “ m4 4 ´ pk ¨ mq2 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Given k P Zd, we conclude this case by asserting npL q ě 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0, σ1,m “ 0u, 35 and dimpKerpL qq ě 1 ` #tm P Zd ∖ t0u, m2 “ ˘2k ¨ m, σ1,m “ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Case m � 0 with σ1,m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Again, we distinguish several subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' if λ “ 1{2, the third equation on (53) imposes qm “ 0, and we are led to 1 ´ m2 2 pm “ 0, ik ¨ mpm ` γp2πqdσ1,m ˆˆ Rnp´∆q´1{2σ2φm dz ˙ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus, λ “ 1{2 is an eigenvalue associated to the eigenfunctions: p0, 0, eim¨xφpzq, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2, (we recover the same eigenfunctions as for the case m “ 0), p0, eim¨x, 0, 0q if k ¨ m “ 0, m2 “ 1, and ´ 0, ´γp2πqdκσ1,m ik ¨ m eim¨x, p´∆q´1{2σ2pzqeim¨x, 0 ¯ if k ¨ m � 0, m2 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' if λ “ m2 2 � 1 2, (53) becomes 0 “ ik ¨ mpm ` γp2πqdσ1,m ˆˆ Rnp´∆q´1{2σ2φm dz ˙ , 0 “ ´ik ¨ mqm, m2 ´ 1 2 φm “ γp2πqdp´∆q´1{2σ2σ1,mqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' There is no non-trivial solution when k ¨ m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Otherwise, we see that λ “ m2{2 is an eigenvalue associated to the eigenfunctions: p0, eim¨x, 0, 0q if λ � t1 2, m2 2 u, we set µ “ λ ´ m2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We see that a non trivial solution of (53) exists if its component qm does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We combine the equations in (53) to obtain Ppµqqm “ 0 where P is the third order polynomial Ppµq “ µ3 ` bµ2 ` cµ ` d, b “ m2 ´ 1 2 ě 0, c “ ´ppk ¨ mq2 ` γ2κp2πq2dσ2 1,mq ă 0, d “ ´pk ¨ mq2 m2 ´ 1 2 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Observe that d “ ´pk ¨ mq2b and pk ¨ mq2 ă |c| ă pk ¨ mq2 ` 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We thus need to examine the roots of this polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we compute the discriminant D “ 18bcd ´ 4b3d ` b2c2 ´ 4c3 ´ 27d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A tedious, but elementary, computation allows us to reorganize terms as follows D “ 4pk ¨ mq2` pk ¨ mq2 ´ b2˘2 ` b2σ2 1,mγp20pk ¨ mq2 ` γσ2 1,mq `4pk ¨ mq2σ2 1,mγp2pk ¨ mq2 ` γσ2 1,mq ` 4σ2 1,mγ ` pk ¨ mq4 ` 2pk ¨ mq2σ2 1,mγ ` σ4 1,mγ2˘ , 36 where we have set γ “ γ2κp2πq2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since σ1,m � 0, we thus have D ą 0 and P has 3 distinct real roots, µ1 ă µ2 ă µ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In order to bring further information about the location of the roots, we observe that limµÑ˘8 Ppµq “ ˘8 while Pp0q “ d ď 0 and P 1p0q “ c ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, studying the zeroes of P 1pµq “ 3µ2 ` 2bµ ` c, we see that µmax “ ´b´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' b2´3c 3 ă 0 is a local maximum and µmin “ ´b` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' b2´3c 3 ą 0 is a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, P 2pµq “ 6µ ` 2b, showing that P is convex on the domain p´pm2 ´ 1q{6, `8q, concave on p´8, ´pm2 ´ 1q{6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A typical shape of the polynomial P is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' From this discussion, we infer µ1 ă µmax ă µ2 ď 0 ă µmin ă µ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 6 5 4 3 2 1 0 1 2 3 40 30 20 10 0 10 20 30 40 50 P( ) Figure 1: Typical graph for µ ÞÑ Ppµq, with its roots µ1 ă µ2 ă µ3 and local extrema µmax, µmin Coming back to the issue of counting the negative eigenvalues of L , we are thus wondering whether or not λj “ µj ` m2{2 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We already know that µ3 ą µmin ą 0, hence µ3 ą ´m2{2 and we have at most 2 negative eigenvalues for each Fourier mode m � 0 such that σ1,m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To decide how many negative eigenvalues should be counted, we look at the sign of Pp´m2{2q (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 1): i) if Pp´m2{2q ą 0 then µ1 ă ´m2{2 ă µ2, ii) if Pp´m2{2q “ 0 then either ´m2{2 ă µmax, in which case µ1 “ ´m2{2 ă µ2, or ´m2{2 ą µmax, in which case µ2 “ ´m2{2 ą µ1, iii) if Pp´m2{2q ă 0 then either ´m2{2 ă µmax, in which case ´m2{2 ă µ1 ă µ2, or ´m2{2 ą µmax, in which case µ1 ă µ2 ă ´m2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, we remark that Pp´m2{2q “ ´m6 8 ` m4pm2 ´ 1q 8 ` m2 2 ppk ¨ mq2 ` γσ2 1,mq ´ m2 ´ 1 2 pk ¨ mq2 “ ´m4 8 ´ 1 ´ 4γσ2 1,m m2 ¯ ` pk ¨ mq2 2 “ ´1 8pm4 ´ 4pk ¨ mq2 ´ 4m2γσ2 1,mq, (55) 37 where, by virtue of (9), m � 0 and σ1m � 0, 1 ą 4 γσ2 1,m m2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be combined together with P 1p´m2{2q “3m4 4 ´ m2pm2 ´ 1q 2 ´ pk ¨ mq2 ´ γσ2 1,m “ m4 4 ` m2 2 ´ pk ¨ mq2 ´ γσ2 1,m “1 4 ` m4 ´ 4pk ¨ mq2 ´ 4m2γσ2 1,m ˘ ` m2γσ2 1,m ` m2 2 ´ γσ2 1,m “ ´ 2Pp´m2{2q ` m2 2 ` pm2 ´ 1qγσ2 1,m ą ´2Pp´m2{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, P 2p´m2{2q “ ´2m2 ´ 1 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, Pp´m2{2q ă 0 implies P 1p´m2{2q ą 0, while P 2p´m2{2q ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This shows that ´m2{2 ă µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, in case iii), the only remaining possibility is the situation where Pp´m2{2q ă 0 with ´m2{2 ă µ1 ă µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a conclusion, if Pp´m2{2q ă 0, all eigenvalues λj are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we claim that case ii) cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, Pp´m2{2q “ 0 if and only if pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq “ 4m2γσ2 1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the term on the left hand side of this equality has to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is possible if and only if both factors, which belong to Z, are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In this case, according to the sign of k ¨ m, one of them is ě m2 so that m2 ď 4m2γσ2 1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This contradicts the smallness condition (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that Pp´m2{2q � 0 implies λj � 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' m-modes with m � 0 and σ1,m � 0 cannot generate elements of KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a conclusion, negative eigenvalues only come from case i) and for each m-mode such that Pp´m2{2q ą 0 we have exactly one negative eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Going back to (55), in this case, we have pm4 ´ 4pk ¨ mq2q “ pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq ă m24γσ2 1,m ă m2 owing to (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This excludes the possibility that m4´4pk¨mq2 ą 0, since we noticed above that whenever this term is positive, it is ě m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, case i) holds if and only if m4´4pk¨mq2 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This ends the counting of the negative eigenvalues of L in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that the associated eigenspaces are spanned by ´ eim¨x, ´ ik ¨ m λ ´ m2{2eim¨x, eim¨x σ1,mγp2πqdp´∆zq´1{2σ2 λ ´ 1{2 , 0 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The discussion has permitted us to find the elements of KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To be specific, the equation 38 L X “ 0 yields π “ 0 and the following relations for the Fourier coefficients m2 2 pm ´ ik ¨ mqm “ 0, φm 2 ` p2πqdγp´∆q´1{2σ2σ1,mqm “ 0, m2 2 qm ` ik ¨ mpm ` p2πqdγσ1,m ˆ p´∆q´1{2σ2φm dz “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have seen that the mode m “ 0 gives rise the eigenspace spanned by p0, 1, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m � 0, ele- ments of KerpL q can be obtained only in the case σ1,m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, the condition m2 “ ˘2k ¨m has to be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In such a case, peim¨x, ¯ieim¨x, 0, 0q P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, it remains to prove that σesspL q “ ␣1 2, 2c2( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have already noticed that ␣ 1 2, 2c2( Ă σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose, by contradiction, that there exists λ P σesspL q ∖ ␣ 1 2, 2c2( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, by Weyl’s criterion [42, Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='14], there exists a sequence pXνqνPN with Xν “ pqν, pν, φν, πνq P DpL q such that, as ν goes to 8, }pL ´ λIqXν} Ñ 0, }Xν} “ 1 and Xν á 0 weakly in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (56) Since λ � 1 2 and λ � 2c2, from (52) and (56) we have }πν}L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pRnqq Ñ 0 and φν “ ´ ˆ1 2 ´ λ ˙´1 γp´∆q´1{2σ2σ1 ‹ qν ` εν with εν P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq such that limνÑ8 }εν}L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pRnqq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to ››››´1 2∆xqν ´ λqν ` k ¨ ∇xpν ´ γ2κ 1{2 ´ λΣ ‹ qν ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2εν dz ˙›››› L2pTdq ÝÝÝÑ νÑ8 0, ››››´1 2∆xpν ´ λpν ´ k ¨ ∇xqν ›››› L2pTdq ÝÝÝÑ νÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Using the fact that the sequence ppqν, pν, ενqqνPN is bounded in L2pTdq ˆ L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq, we deduce that pqν, pνqνPN is bounded in H2pTdq ˆ H2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, reasoning on Fourier series, this amounts to estimate ÿ mPZd |m|4p|qν,m|2 ` |pν,m|2q ď 2 ÿ mPZd ` |m2qν,m ` 2ik ¨ mpν,m|2 ` |m2pν,m ´ 2ik ¨ mqν,m|2q `8 ÿ mPZd p|k ¨ mpν,m|2 ` |k ¨ mqν,m|2q ď 2 ›› ´ ∆xqν ` 2k ¨ ∇xpν ›› L2pTdq ` 2 ›› ´ ∆xpν ´ 2k ¨ ∇xqν ›› L2pTdq `4 δ|k|4 ÿ mPZd ` |qν,m|2 ` |pν,m|2˘ ` 4δ ÿ mPZd |m|4p|qν,m|2 ` |pν,m|2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Choosing 0 ă δ ă 1{4 and using the already known estimates, we conclude that }∆xqν}2 L2 ` }∆xpν}2 L2 “ ř mPZd |m|4` |qν,m|2 ` |pν,m|2˘ is bounded, uniformly with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, because of the compact Sobolev embedding of H2pTdq into L2pTdq, we have that pqν, pνqνPN has a (strongly) convergent subsequence in L2pTdq ˆ L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, the sequence pXνqνPN has a conver- 39 gent subsequence in V , which contradicts (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A consequence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 is that 0 is an isolated eigenvalue of L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the restriction of L to the subspace pKerpL qqK is, by definition, injective, it makes sense to define on it its inverse L ´1, with domain RanpL q Ă pKerpL qqK Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In fact, 0 being an isolated eigenvalue, RanpL q is closed and coincides with pKerpL qqK, [42, Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be shown by means of spectral measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Given X P pKerpL qqK, the support of the associated spectral measure dµX does not meet the interval p´ǫ, `ǫq for ǫ ą 0 small enough, independent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we get }L X}2 “ ˆ `8 ´8 λ2 dµXpλq “ ˆ |λ|ěǫ λ2 dµXpλq ě ǫ2}X}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the Fredholm alternative applies: for any Y P pKerpL qqK, there exists a unique X P pKerpL qqK such that L X “ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We will denote X “ L ´1Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For further purposes, let us set X0 “ p0, 1, 0, 0q P KerpL q and Y0 “ ´J X0 “ p1, 0, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that Y0 P pKerpL qqK, so that it makes sense to consider the equation L U0 “ Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We find πm “ 0, φm “ ´2γp2πqdp´∆q´1{2σ2σ1,mqm, m2pm “ 2ik ¨ mqm, and m2qm ` 2ik ¨ mpm ` 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz “ δ0,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It yields, for m � 0, pm4 4 ´ pk ¨ mq2 ´ γ|σ1,m|2m2¯ qm “ 0 and q0 “ ´ 1 2γ2p2πq2d|σ1,0|2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, we can set U0 “ L ´1Y0 “ ´ 1 2γ2p2πq2d|σ1,0|2κ ` 1, 0, ´2γp2πqdp´∆q´1{2σ2σ1,0, 0 ˘ , solution of L U0 “ Y0 such that U0 P pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We note that pU0, Y0q “ ´ 1 2γ2p2πqd|σ1,0|2κ ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (57) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Reformulation of the eigenvalue problem, and counting theo- rem The aim of the section is to introduce several reformulations of the eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This will allow us to make use of general counting arguments, set up by [9, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 Let us set M “ ´J L J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The coupled system M Y “ ´λX, L X “ λY, (58) admits a solution with λ � 0, X P DpL q ∖ t0u, Y P DpJ L J q ∖ t0u iff there exists two vectors X˘ P DpLq ∖ t0u that satisfy LX˘ “ ˘λX˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 40 Let P stand for the orthogonal projection from V to pKerpL qqK Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 Let us set A “ PM P and K “ PL ´1P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us define the following Hilbert space H “ DpM q X pKerpL qqK Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The coupled system (58) has a pair of non trivial solutions p˘λ, X, ˘Y q, with λ � 0 iff the gener- alized eigenproblem AW “ αKW, W P H , (59) admits the eigenvalue α “ ´λ2 � 0, with at least two linearly independent eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Recall that the plane wave solution obtained Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 is spectrally stable, if the spectrum of L is contained in iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In view of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, this happens if and only if all the eigenvalues of the generalized eigenproblem (59) are real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In other words, the presence of spectrally unstable directions corresponds to the existence of negative eigenvalues or complex but non real eigenvalues of the generalized eigenproblem (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Our goal is then to count the eigenvalues α of the generalized eigenvalue problem (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular we define the following quantities: N ´ n , the number of negative eigenvalues N 0 n, the number of eigenvalues zero N p n, the number of positive eigenvalues of (59), counted with their algebraic multiplicity, the eigenvectors of which are associated to non- positive values of the the quadratic form W ÞÑ pKW|Wq “ pL ´1PW|PWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, let NC` be the number of eigenvalues α P C with Impαq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As pointed out above, the eigenvalues counted by N ´ n and NC` correspond to cases of instabil- ities for the linearized problem (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that to prove the spectral stability, it is enough to show that the generalized eigenproblem (59) does not have negative eigenvalues and NC` “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, as a consequence of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, if α P C ∖ R is an eigenvalue of (59), then ¯α is an eigenvalue too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, if NC` “ 0, then the generalized eigenproblem (59) does not have solutions in C ∖ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, for using the counting argument introduce by Chugunova and Pelinovsky in [9], we need the following information on the essential spectrum of A, see [43, Lemma 2-(H1’) and Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 Let M “ ´J L J be defined on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then σesspM q “ r0, `8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let A “ PM P and K “ PL ´1P be defined on H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then σesspAq “ r0, `8q and we can find δ˚, d˚ ą 0 such that for any real number 0 ă δ ă δ˚, A`δK admits a bounded inverse and we have σesspA`δKq Ă rd˚δ, `8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We check that J L J X “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ∆xq 2 ´ k ¨ ∇xp ∆xp 2 ` k ¨ ∇xq ` γσ1 ‹ ˆ p´∆zq´1{2σ2p´∆zq1{2π dz 2c2∆zφ ∆zπ 2 ` γσ2σ1 ‹ p ˛ ‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 41 As a matter of fact, for any φ P H2pRnq, the vector Xe “ p0, 0, φ, 0q lies in pKerpL qqK and satisfies J L J Xe “ ¨ ˚ ˚ ˝ 0 0 2c2∆zφ 0 ˛ ‹‹‚P pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Consequently M Xe “ AXe “ ´J L J Xe “ p0, 0, ´2c2∆zφ, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It indicates that a Weyl sequence for A ´ λI, λ ą 0, can be obtained by adapting a Weyl sequence for p´∆z ´ µIq, µ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us consider a sequence of smooth functions ζν P C8 c pRnq such that supppζνq Ă Bp0, ν ` 1q, ζνpzq “ 1 for x P Bp0, νq and }∇zζν}L8pRnq ď C0 ă 8, }D2 zζν}L8pRnq ď C0 ă 8, uniformly with respect to ν P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We set φνpzq “ ζνpzqeiξ¨z{p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2cq for some ξ P Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get p´|ξ|2 ´ 2c2∆zqφνpzq “ eiξ¨z{p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2cq´ 2i ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2cξ ¨ ∇zζν ` 2c2∆zζν ¯ pzq, which is thus bounded in L8pRnq and supported in Bp0, ν ` 1q ∖ Bp0, νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that }p´|ξ|2 ´ 2c2∆zqφν}2 L2pRnq ≲ νn´1, while }φν}2 L2pRnq ≳ νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we obtain }φν}2 L2pRnq }p´|ξ|2´2c2∆zqφν}2 L2pRnq ≳ ν Ñ 8 as ν Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, φν equally provides a Weyl sequence for M ´ |ξ|2I and A ´ |ξ|2I, showing the inclusions r0, 8q Ă σesspM q and r0, 8q Ă σesspAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, let λ � r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We suppose that we can find a Weyl sequence pXνqνPN for M , such that M Xν ´ λXν “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´λqν ´ ∆xqν 2 ` k ¨ ∇xpν ´λpν ´ ∆xpν 2 ´ k ¨ ∇xqν ´ γσ1 ‹ ˆ p´∆zq´1{2σ2p´∆zq1{2πν dz ´λφν ´ 2c2∆zφν ´λπν ´ ∆zπν 2 ´ γσ2σ1 ‹ pν ˛ ‹‹‹‹‹‹‚ “ ¨ ˚ ˚ ˝ q1 ν p1 ν φ1 ν π1 ν ˛ ‹‹‚ÝÝÝÑ νÑ8 0, with, moreover, }Xν} “ 1 and Xν á 0 weakly in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, we can set x φνpx, ξq “ x φ1νpx, ξq 2c2|ξ|2 ´ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (60) It defines a sequence which tends to 0 strongly L2pTd ˆ Rnq since, writing λ “ a ` ib P C ∖ r0, 8q, we get |2c2|ξ|2 ´λ|2 “ |2c2|ξ|2 ´a|2 `b2 which is ě b2 ą 0 when λ � R, and, in case b “ 0, ě a2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, we can write x πνpx, ξq “ 2x π1νpx, ξq |ξ|2 ´ 2λ loooomoooon “hνpx,ξqPL2pTdˆRnq `γ 2x σ2pξq |ξ|2 ´ 2λ loooomoooon PL2pRnq σ1 ‹ pν, (61) 42 where hν tends to 0 strongly L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are led to the system ¨ ˚ ˝ ´ ´ λ ` ∆x 2 ¯ qν ` k ¨ ∇xpν ´k ¨ ∇xqν ´ ´ λ ` ∆x 2 ¯ pν ´ 2γ2 ˆ |x σ2|2 p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν ˛ ‹‚ “ ¨ ˝ q1 ν p1 ν ´ γσ1 ‹ ˆ x σ2pξq |ξ| hνpx, ξq dξ p2πqn ˛ ‚ÝÝÝÑ νÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (62) Reasoning as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-1), we conclude that Xν converges strongly to 0 in V , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, λ P C ∖ r0, 8q cannot belong to σesspM q and the identification σesspM q “ r0, 8q holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-3) identifies KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us introduce the mapping Ă P : ˆq p ˙ P L2pTdqˆL2pTdq ÞÝÑ ¨ ˚ ˚ ˝ ÿ mPK˚, k¨mą0 pqm ´ ipmqeim¨x ` ÿ mPK˚, k¨mă0 pqm ` ipmqeim¨x p0 ` i ÿ mPK˚, k¨mą0 pqm ´ ipmqeim¨x ´ i ÿ mPK˚, k¨mă0 pqm ` ipmqeim¨x ˛ ‹‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, X “ ¨ ˚ ˚ ˝ q p φ π ˛ ‹‹‚ÞÝÑ ¨ ˚ ˚ ˝ Ă P ˆ q p ˙ 0 0 ˛ ‹‹‚ is the projection of V on KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we realize that P does not modify the last two components of a vector X “ pq, p, φ, πq P V , and for X P pKerpL qqK, we have p0 “ 0, and qm “ ˘ipm for any m P K˚, depending on the sign of k ¨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Now, let λ P C ∖ r0, 8q and suppose that we can exhibit a Weyl sequence pXνqνPN for A ´ λI: Xν P H Ă pKerpL qqK, PXν “ Xν, }Xν} “ 1, Xν á 0 in V and limνÑ8 }pA ´ λIqXν} “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can apply the same reasoning as before for the last two components of pA ´ λIqXν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it leads to (60) and (61), where, using λ � r0, 8q, φν and hν converge strongly to 0 in L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We arrive at the following analog to (62) pI ´ Ă Pq ¨ ˚ ˝ ´ ´ λ ` ∆x 2 ¯ qν ` k ¨ ∇xpν ´k ¨ ∇xqν ´ ´ λ ` ∆x 2 ¯ pν ´ 2γ2 ˆ |x σ2|2 p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν ˛ ‹‚ “ ˆq1 ν p1 ν ˙ ´ pI ´ Ă Pq ¨ ˝ 0 γσ1 ‹ ˆ x σ2pξq |ξ| hνpx, ξq dξ p2πqn ˛ ‚ÝÝÝÑ νÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (63) In order to derive from (63) an estimate in a positive Sobolev space as we did in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-1), we should consider the Fourier coefficients arising from ´ 1 2∆xqν ` k ¨ ∇xpν and ´ 1 2∆xpν ´k¨∇xqν, namely Qm “ m2 2 qν,m`ik¨mpν,m and Pm “ m2 2 pν,m´ik¨mqν,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Only the coef- ficients belonging to K˚ are affected by the action of Ă P, which leads to Qm ´ pQm ¯ iPmq “ ˘iPm and Pm ¯ ipQm ¯ iPmq “ ¯iQm, according to the sign of k ¨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, we bear in mind that qm “ ˘ipm when m P K˚ with ˘k ¨ m ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for coefficients in K˚, the contributions of the differential operators reduces to ˘im2pm “ ˘m2qm and ¯im2qm “ ˘m2pm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note 43 also that for these coefficients there is no contributions coming from the convolution with σ1 in (63) since σ1,m “ 0 for m P K˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, reasoning as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-1) for coefficients m P Zd ∖ K˚, we can obtain a uniform bound on ř mPZd |m|4p|qν,m|2 ` |pν,m|2q, which provides a uniform H2 bound on qν and pν, leading eventually to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that σesspAq “ r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let δ ą 0 and consider the shifted operator A ` δK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='10, we will see that KerpA ` δKq “ t0u for any δ ą 0: 0 is not an eigenvalue for A ` δK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' let us justify it does not belong to the essential spectrum neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we need to detail the expression of the operator K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Given X P H , we wish to find X1 P H satisfying L X1 “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xq1 ` k ¨ ∇xp1 ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ1 dz ˙ ´1 2∆xp1 ´ k ¨ ∇xq1 1 2φ1 ` γp´∆q´1{2σ2σ1 ‹ q1 2c2π1 ˛ ‹‹‹‹‹‹‚ “ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We infer π1 “ π 2c2 and the relation φ1 “ 2φ ´ 2γp´∆zq´1{2σ2σ1 ‹ q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, the Fourier coefficients of q1, p1 are required to satisfy ˆ m2{2 ´ 2γ2κp2πq2d|σ1,m|2 ik ¨ m ´ik ¨ m m2{2 ˙ ˆ q1 m p1 m ˙ “ ¨ ˝qm ´ 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz pm ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When m � 0, m � K˚, the matrix of this system has its determinant equal to det “ m4 4 ` 1 ´ 4γ2κp2πq2d |σ1,m|2 m2 ˘ ´ pk ¨ mq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Owing to (9), since pk ¨ mq2 takes values in N, it does not vanish and we obtain q1 m, p1 m by solving the system q1 m “ 1 det ˆm2 2 ´ qm ´ 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz ¯ ´ ik ¨ mpm ˙ , p1 m “ 1 det ˆ `ik ¨ m ´ qm ´ 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz ¯ ` ´m2 2 ´ 2γ2κp2πq2d|σ1,m|2¯ pm ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m P K˚ we find a solution in pKerpL qqK by setting p1 m “ pm m2 , q1 m “ ˘ip1 m, according to the sign of k ¨m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' if m “ 0, we set p1 0 “ 0 and q1 0 “ 1 2γ2κp2πq2d|σ1,0|2 ` q0 ´2γp2πqdσ1,0 ´ p´∆q´1{2σ2φ0 dz ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This defines X1 “ KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, the last two components of pA ` δK ´ λIqX read p2δ ´ λqφ ´ 2c2∆zφ ´ 2δγp´∆q´1{2σ2σ1 ‹ q1, ´ δ 2c2 ´ λ ¯ π ´ 1 2∆zπ ´ γσ2σ1 ‹ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, when λ does not belong to rδd˚, 8q, with d˚ “ minp2, 1 2c2q, we can repeat the analysis performed above to establish that λ � σesspA ` δKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular the essential spectrum of A has been shifted away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are now able to apply the results of Chugunova and Pelinovsky [9] (see also [43]), to obtain 44 the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 [9, Theorem 1] Let L be defined by (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With the operators M , A, K defined as in Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, the following identity holds N ´ n ` N 0 n ` N ` n ` NC` “ npL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us now detail the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, adapted from [43, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 1 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The goal is to establish connections between the following three problems: (Ev) the eigenvalue problem LX “ λX, with L “ J L , (Co) the coupled problem L X “ λY , M Y “ ´λX, with M “ ´J L J , (GEv) the generalized eigenvalue problem AW “ αKW, with A “ PM P, K “ PL ´1P, the projection P on pKerpL qqK, and W P H “ DpM q X pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The proof of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 follows from the following sequence of arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (i) By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, we already know that if there exists a solution pλ, X`q of (Ev), with λ � 0 and X` � 0, then, there exists X´ � 0, such that p´λ, X´q satisfies (Ev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Being eigenvectors associated to distinct eigenvalues, X` and X´ are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that only this part of the proof uses the specific structure of the operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (ii) From these eigenpairs for L, we set X “ X` ` X´ 2 , Y “ Ă J ˆX` ´ X´ 2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since X` and X´ are linearly independent, we have X � 0, Y � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, X “ X``X´ 2 and J Y “ X`´X´ 2 are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get L X “ Ă J LX “ Ă J ˆλ 2pX` ´ X´q ˙ “ λY, M Y “ ´J L ˆX` ´ X´ 2 ˙ “ ´L ˆX` ´ X´ 2 ˙ “ ´λ 2pX` ` X´q “ ´λX, so that pλ, X, Y q satisfies (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (iii) If pλ, X, Y q is a solution (Co), then p´λ, X, ´Y q satisfies (Co) too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (iv) Let pλ, X, Y q be a solution (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Set X1 “ J Y, Y 1 “ Ă J X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We observe that M Y 1 “ ´J L J Ă J X “ ´J L X “ ´J pλY q “ ´λX1, L X1 “ L J Y “ Ă J J L J Y “ ´ Ă J M Y “ λ Ă J X “ λY 1, which means that pλ, J Y, Ă J Xq is a solution of (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, if X and J Y are linearly independent, Y and Ă J X are linearly independent too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 45 (v) Let pλ, X, Y q be a solution (Co), with X � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get LpX ˘ J Y q “ J L X ˘ J L J Y “ J L X ¯ M Y “ J pλY q ˘ λX “ ˘λpX ˘ J Y q, so that p˘λ, X ˘ J Y q satisfy (Ev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In the situation where X and J Y are linearly inde- pendent, we have X ˘ J Y � 0 and p˘λ, X ˘ J Y q are eigenpairs for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Otherwise, one of the vectors X ˘ J Y might vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Nevertheless, since only one of these two vectors can be 0, we still obtain an eigenvector X˘ � 0 of L, associated to either ˘λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to i), we conclude that ¯λ is an eigenvalue too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Items i) to v) justify the equivalence stated in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (vi) Let pλ, X, Y q be a solution (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' From L X “ λY , we infer Y P RanpL q Ă pKerpL qqK so that PY “ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The relation thus recasts as X “ λPL ´1PY ` ˜Y, ˜Y P KerpL q, P ˜Y “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (Here, PL ´1PY stands for the unique solution of L Z “ Y which lies in pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=') We obtain PM Y “ Pp ´ λXq “ ´λPpλPL ´1PY ` ˜Y q “ ´λ2PL ´1PY “ ´λ2KY “ PM PY “ AY, so that p´λ2, Y q satisfies (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Going back to iv), we know that p´λ2, Ă J Xq is equally a solution to (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If X and J Y are linearly independent, we obtain this way two linearly independent vectors, Y and Ă J X, solutions of (GEv) with α “ ´λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (vii) Let pα, Wq satisfy (GEv), with α � 0, W � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We set X “ ´M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have Ă J X “ ´ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α Ă J M W “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α Ă J J L J W “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αL J W which lies in RanpL q Ă pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus, using P Ă J X “ Ă J X, we compute K Ă J X “ PL ´1P Ă J X “ PL ´1 Ă J X “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPL ´1L J W “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we observe that A Ă J X “ PM P Ă J X “ ´PJ L J Ă J X “ ´PJ L X “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L M W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, we can use PW “ W (since W P H Ă pKerpL qqK) and the fact that, for any vector Z, L Z “ L pI ´ PqZ ` L PZ “ 0 ` L PZ, which yields A Ă J X “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L PM PW “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L AW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αPJ L KW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L PL ´1PW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L L ´1W “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that A Ă J X “ αK Ă J X: pα, Ă J Xq satisfies (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (viii) Let pα, Wq satisfy (GEv), with α � 0, W � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have PpM PW ´ αL ´1PWq “ 0 and thus M PW ´ αL ´1PW “ ˜Y P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 46 Let us set Y “ PW P pKerpL qqK, X “ ´M PW ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α “ ´1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αp˜Y ` αL ´1PWq, so that L X “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αPW “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αY, M Y “ M PW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X, Y q satisfies (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By v), p˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X ˘ J Y q satisfy (Ev), and at least one of the vectors X ˘J Y does not vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' using i), we thus obtain eigenpairs p˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X˘q of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With ii), we construct solutions of (Co) under the form ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X``X´ 2 , Ă J ` X`´X´ 2 ˘˘ , which, owing to iv) and vi), provide the linearly independent solutions ` α, Ă J `X`˘X´ 2 ˘˘ of (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The dimension of the linear space of solutions of (GEv) is at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' At least one of these vectors X˘ is given by the formula ˜X˘ “ ´ M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ˘ J W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By the way, we indeed note that AW “ αKW, with W P H , can be cast as L J L J W “ ´αW (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 below) so that L ´ ´ M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ˘ J W ¯ “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αJ pL J L J Wq ˘ J L J W “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αJ W ¯ M W “ ˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ´ ´ M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ˘ J W ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With these manipulations we have checked that p˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, ˜X˘q satisfy (Ev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If both vectors ˜X˘ are non zero, we get X˘ “ ˜X˘ and we recover W “ Ă J ` X`´X´ 2 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If ˜X˘ “ 0, then, we get ˜X¯ “ ¯J W � 0, and we directly obtain X¯ “ ˜X¯, W “ ¯ Ă J X¯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In any cases, W lies in the space spanned by X` and X´ and the dimension of the space of solutions of (GEv) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This ends the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 Spectral instability We are going to compute the terms arising in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Eventually, it will allow us to identify the possible unstable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In what follows, we find convenient to work with the operator M ´αL ´1 instead of PpM ´ αL ´1qP “ A ´ αK, owing to to the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 Let α � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In the space H “ DpM q X pKerpL qqK, the two subspaces KerpA ´ αKq and KerpM ´ αL ´1q coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let X P H satisfy M X “ αL ´1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, we have X “ PX and, thus, pA ´ αKqX “ PpM ´ αL ´1qPX “ PpM X ´ αL ´1Xq “ 0, showing the inclusion KerpM ´ αL ´1q X H Ă KerpA ´ αKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Conversely, the equation pA ´ αKqX “ 0, with X “ PX P pKerpL qqK means that pM ´ αL ´1qX “ Y P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Applying L then yields L M X “ αX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since both terms of this relation lie in pKerpL qqK, it is legitimate to apply L ´1, showing that M X “ αL ´1X: we have shown KerpA ´ αKq X H Ă KerpM ´ αL ´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 47 Therefore, we shall consider the solutions of the generalized eigenvalue problem M X “ αL ´1X, with X P H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We rewrite the equation by introducing an auxiliary unknown: M X “ α ˜X, L ˜X “ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='9 Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' N 0 n “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are interested in the solutions of ´1 2∆xq ` k ¨ ∇xp “ 0, ´1 2∆xp ´ k ¨ ∇xq ´ γσ1 ‹ ˆ σ2π dz “ 0, ´2c2∆zφ “ 0, ´1 2∆zπ ´ γσ2σ1 ‹ p “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We infer φpx, zq “ 0 and pπpx, ξq “ 2γ x σ2pξq |ξ|2 σ1 ‹ ppxq, and, next, ´1 2∆xq ` k ¨ ∇xp “ 0, ´1 2∆xp ´ k ¨ ∇xq ´ 2γ2κΣ ‹ p “ 0 with Σ “ σ1 ‹ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In terms of Fourier coefficients, it becomes m2 2 qm ` ik ¨ mpm “ 0, m2 2 pm ´ ik ¨ mqm ´ 2p2πq2dγ2κ|σ1,m|2pm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, we get p0 “ 0 and we find the eigenfunction p1, 0, 0, 0q “ Y0 “ ´J X0 with X0 “ p0, 1, 0, 0q P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m � 0 with σ1,m � 0, we get m4 ´ 4pk ¨ mq2 “ 2p2πq2dγ2κ|σ1,m|2 loooooooooomoooooooooon Pp0,1q m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' which cannot hold (see the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m � 0 with σ1,m “ 0, we get Mm ˆqm pm ˙ “ 0 with Mm defined in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As far as m4 ´4pk ¨mq2 � 0, Mm is invertible and the only solution is pm “ 0 “ qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m4 ´4pk ¨mq2 “ 0, we find the eigenfunctions peik¨m, ˘ieik¨m, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These functions belong to KerpL q, and thus do not lie in the working space H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that KerpM q “ spanRtY0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, this vector Y0 does not belong to RanpM q so that the algebraic multiplicity of the eigenvalue 0 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, bearing in mind (57), which can be recast as pKY0|Y0q ă 0, we arrive at N 0 n “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='10 Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The generalized eigenproblem (59) does not admit negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, N ´ n “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 48 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let α ă 0, X “ pq, p, φ, πq and ˜X “ p˜q, ˜p, ˜φ, ˜πq satisfy ´1 2∆xq ` k ¨ ∇xp “ α˜q, ´1 2∆xp ´ k ¨ ∇xq ´ γσ1 ‹ ˆ σ2π dz “ α˜p, ´2c2∆zφ “ α˜φ, ´1 2∆zπ ´ γσ2σ1 ‹ p “ α˜π, (64) where q “ ´1 2∆x˜q ` k ¨ ∇x˜p ` γσ1 ‹ ˆ p´∆zq´1{2σ2 ˜φ dz, p “ ´1 2∆x˜p ´ k ¨ ∇x˜q, φ “ 1 2 ˜φ ` γp´∆zq´1{2σ2σ1 ‹ ˜q, π “ 2c2˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (65) This leads to solve an elliptic equation for π ´|α| c2 ´ ∆z ¯ π “ 2γσ2σ1 ‹ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In other words, we get, by means of Fourier transform pπpx, ξq “ 2γσ1 ‹ ppxq ˆ x σ2pξq |ξ|2 ` |α|{c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' On the same token, we obtain ´|α| c2 ´ ∆z ¯ ˜φ “ ´2γp´∆zq1{2σ2σ1 ‹ ˜q, which yields p˜φpx, ξq “ ´2γσ1 ‹ ˜qpxq ˆ |ξ|x σ2pξq |ξ|2 ` |α|{c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For λ ą 0, we introduce the symbol 0 ď κλ “ ˆ |x σ2pξq|2 |ξ|2 ` λ ď κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It turns out that ´1 2∆xq ` k ¨ ∇xp “ α˜q, ´1 2∆xp ´ k ¨ ∇xq ´ 2γ2κ|α|{c2Σ ‹ p “ α˜p, with q “ ´1 2∆x˜q ` k ¨ ∇x˜p ´ 2γ2κ|α|{c2Σ ‹ ˜q, p “ ´1 2∆x˜p ´ k ¨ ∇x˜q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 49 For the Fourier coefficients, it casts as m2 2 qm ` ik ¨ mpm ��� α˜qm, m2 2 pm ´ ik ¨ mqm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2pm “ α˜pm, with qm “ m2 2 ˜qm ` ik ¨ m˜pm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2˜qm, pm “ m2 2 ˜pm ´ ik ¨ m˜qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are going to see that these equations do not have non trivial solutions with α ă 0: If m “ 0, we get p0 “ 0, ˜q0 “ 0, and, consequently, ˜p0 “ 0, q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for α ă 0, we cannot find an eigenvector with a non trivial 0-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m � 0 and σ1,m “ 0, we see that pqm, pmq and p˜qm, ˜pmq are related by Mm ˆqm pm ˙ “ α ˆ˜qm ˜pm ˙ , ˆqm pm ˙ “ Mm ˆ˜qm ˜pm ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (66) It means that α is an eigenvalue of M2 m “ ˜ m4 4 ` pk ¨ mq2 im2k ¨ m ´im2k ¨ m m4 4 ` pk ¨ mq2 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The roots of the characteristic polynomial of M2 m are pm2 2 ˘ k ¨ mq2 ě 0, which contradicts the assumption α ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the case where m � 0 and σ1,m � 0, we introduce the shorthand notation am “ 2γ2p2πq2d|σ1,m|2κ|α|{c2, bearing in mind that 0 ă am ă m2 2 by virtue of the smallness condi- tion (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are led to the systems ˆ Mm ´ ˆ 0 0 0 am ˙˙ ˆ qm pm ˙ “ α ˆ ˜qm ˜pm ˙ , ˆ qm pm ˙ “ ˆ Mm ´ ˆ am 0 0 0 ˙˙ ˆ ˜qm ˜pm ˙ , which imply that α is an eigenvalue of the matrix ˆ Mm ´ ˆ 0 0 0 am ˙˙ ˆ Mm ´ ˆ am 0 0 0 ˙˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However the eigenvalues of this matrix read ` b m2 2 pm2 2 ´ amq ˘ pk ¨ mq2˘2 ě 0, contradicting that α is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='11 Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' N ` n “ #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 50 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We should consider the system of equations (64)-(65), now with α ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For Fourier coefficients, it casts as m2 2 qm ` ik ¨ mpm “ α˜qm, m2 2 pm ´ ik ¨ mqm ´ γp2πqdσ1,m ˆ σ2πm dz “ α˜pm, ´2c2∆zφm “ α˜φm, ´1 2∆zπm ´ γp2πqdσ1,mσ2pm “ α˜πm, where qm “ m2 2 ˜qm ` ik ¨ m˜pm ` γp2πqdσ1,m ˆ p´∆zq´1{2σ2 ˜φm dz, pm “ m2 2 ˜pm ´ ik ¨ m˜qm, φm “ 1 2 ˜φm ` γp2πqdp´∆zq´1{2σ2σ1,m˜qm, πm “ 2c2˜πm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, we obtain p0 “ 0, ˜q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence π0 satisfies p´α{c2 ´ ∆zqπ0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here, `α{c2 lies in the essential spectrum of ´∆z and the only solution in L2 of this equation is π0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, this implies ˜p0 “ 0, p´α{c2 ´ ∆zqφ0 “ 0, and thus φ0 “ 0, q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for α ą 0, we cannot find an eigenvector with a non trivial 0-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When m � 0 and σ1,m “ 0, we are led to p´α{c2 ´ ∆qφm “ 0, p´α{c2 ´ ∆qπm “ 0 that imply φm “ 0, πm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, we get (66) for qm, pm, ˜qm, ˜pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This holds iff α is an eigenvalue of M2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m4 � 4pk ¨ mq2, we find two eigenvalues αm,˘ “ pm2 2 ˘ k ¨ mq2 ą 0, with associated eigenvectors Xm,˘ “ peim¨x, ¯ieim¨x, 0, 0q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To decide whether these modes should be counted, we need to evaluate the sign of pL ´1Xm,˘|Xm,˘q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We start by solving L X1 m,˘ “ Xm,˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It yields φ1 m,˘ 2 “ 0, 2c2π1 m,˘ “ 0 and Mm ˆq1 m,˘ p1 m,˘ ˙ “ ˆ 1 ¯i ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We obtain q1 m,˘ “ 2 m2 ˘ 2k ¨ m, π1 m,˘ “ ¯2i m2 ˘ 2k ¨ m, so that pL ´1Xm,˘|Xm,˘q “ 2 m2 ˘ 2k ¨ m ˆˆ Td eim¨xe´im¨x dx ` ˆ Tdp¯iqeim¨x˘ie´im¨x dx ˙ “ 4p2πqd m2 ˘ 2k ¨ m, the sign of which is determined by the sign of m2 ˘2k¨m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We count only the situation where these quantities are negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' reproducing a discussion made in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3, we conclude that N ` n ě #tm P Zd ∖ t0u, σ1,m “ 0 and m4 ´ 4pk ¨ mq2 ă 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 51 When m4 ´ 4pk ¨ mq2 “ 0, the eigenvalues of M2 n are 0 and m4, and we just have to consider the positive eigenvalue α “ m4, associated to the eigenvector Xm “ peim¨x, ˘ieim¨x, 0, 0q (depending whether m2 2 “ ¯k ¨ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The equation L Ym “ Xm has infinitely many solu- tions of the form p2{m2eim¨x, 0, 0, 0q ` zp˘ieim¨x, eim¨x, 0, 0q, with z P C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We deduce that pL ´1Xm|Xmq “ 2p2πqd m2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus these modes do not affect the counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When m � 0 and σ1,m � 0, we are led to the relations p´α{c2 ´ ∆zqπm “ 2σ2γp2πqdσ1,mpm, p´α{c2 ´∆zq˜φm “ ´2p´∆zq1{2σ2γp2πqdσ1,m˜qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The only solutions with square integrability on Rn are πm “ 0, ˜φm “ 0, pm “ 0, ˜qm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be seen by means of Fourier transform: p´α{c2 ´ ∆zqφ “ σ amounts to pφpξq “ pσpξq |ξ|2´α{c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' due to (H4) this function has a singularity which cannot be square-integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, this equally implies φm “ 0 and ˜πm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, we arrive at m2 2 qm “ 0 and ´ik ¨ mqm “ α˜pm, together with qm “ ik ¨ m˜pm and m2 2 ˜pm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that α ą 0 cannot be an eigenvalue associated to a m-mode such that m � 0 and σ1,m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can now make use of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7, together with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to 0 ` 1 ` #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u ` NC` “ N ´ n ` N 0 n ` N ` n ` NC` “ npL q “ 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u `#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u so that NC` “ #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the eigenvalue problem (59) does not have negative (real) eigenvalues, this is the only source of instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a matter of fact, when k “ 0, we obtain NC` “ 0, which yields the following statement, (hopefully!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=') consistent with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='12 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q is spectrally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In contrast to what happens for the Hartree equation, for which the eigenvalues are purely imaginary, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2, we can find unstable modes, despite the smallness condition (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us consider the following two examples in dimension d “ 1, with k P Z ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='13 Suppose σ1,0 � 0, and σ1,1 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, the set tm P Z ∖ t0u, m4 ´ 4k2m2 ď 0 and σ1,m � 0u contains t´1, `1u (since 4k2 ě 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let k P Z ∖ t0u and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peiωteikx, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='14 Let m˚ P Z ∖ t0u be the first Fourier mode such that σ1,m˚ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let k P Z and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, for all k P Z such that 4k2 ă m2 ˚, the plane wave solution peiωteikx, ´γΓpzq @ σ D Td, 0q is spectrally stable, while for all k P Z such that 4k2 ě m2 ˚, the plane wave solution peiωteikx, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 52 In general, if k P Zd ∖ t0u, the set tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u contains ´k and k provided σ1,k � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='15 Let k P Zd ∖ t0u and ω ą 0 such that the dispersion relation (12) is satis- fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds and σ1,m � 0 for all m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peipωt`k¨xq, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='16 (Orbital instability) Given Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='15, it is natural to ask whether in this case the plane wave solution peipωt`k¨xq, ´γΓpzq @ σ1 D Td, 0q is orbitally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that, if σ1,m � 0 for all m P Zd ∖ t0u, we deduce from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 that npLq ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, the arguments used in [22] to prove the orbital instability (see also [38, 41]) do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It seems then necessary to work directly with the propagator generated by the linearized operator as in [23, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, one has to establish Strichartz type estimates for the propagator of L (a task we leave for future work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A Scaling of the model and physical interpretation It is worthwhile to discuss the meaning of the parameters that govern the equations and the asymptotic issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Going back to physical units, the system reads ˆ iℏBtU ` ℏ2 2m∆xU ˙ pt, xq “ ˆˆ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ upt, xq, (67a) pB2 ttΨ ´ κ2∆zΨqpt, x, zq “ ´σ2pzq ˆˆ Td σ1px ´ yq|Upt, yq|2 dy ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (67b) The quantum particle is described by the wave function pt, xq ÞÑ Upt, xq: given Ω Ă Td, the integral ´ Ω |Upt, xq|2 dx gives the probability of presence of the quantum particle at time t in the domain Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' this is a dimensionless quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In (67a), ℏ stands for the Planck constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' its homogeneity is MassˆLength2 Time (and its value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='055 ˆ 10´34 Js) and m is the inertial mass of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us introduce mass, length and time units of observations: M, L and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It helps the intuition to think of the z directions as homogeneous to a length, but in fact this is not necessarily the case: we denote by Ψ and Z the (unspecified) units for Ψ and the zj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, κ is homogeneous to the ratio Z T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The coupling between the vibrational field and the particle is driven by the product of the form functions σ1σ2, which has the same homogeneity as ℏ TΨLdZn from (67a) and as Ψ LdT2 from (67b), both are thus measured with the same units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' From now on, we denote by ς this coupling unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, we are led to the following dimensionless quantities U1pt1, x1q “ Upt1T, x1Lq c Ld m M, Ψ1pt1, x1, z1q “ 1 ΨΨpt1T, x1L, z1Zq, σ1 1px1qσ2pz1q “ 1 ς σ1px1Lqσ2pz1Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Bearing in mind that u is a probability density, we note that ˆ Td |U1pt1, x1q|2 dx1 “ m M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 53 Dropping the primes, (67a)-(67b) becomes, in dimensionless form, ˆ iBtU ` ℏT mL2 1 2∆xU ˙ pt, xq “ ςΨLdZnT ℏ ˆˆ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ Upt, xq, (68a) ´ B2 ttΨ ´ κ2T2 Z2 ∆zΨ ¯ pt, x, zq “ ´ςT2 Ψ M mσ2pzq ˆˆ Td σ1px ´ yq|Upt, yq|2 dy ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (68b) Energy conservation plays a central role in the analysis of the system: the total energy is defined by using the reference units and we obtain E0 “ ´ ℏT mL2 ¯2 1 2 ˆ Td |∇xU|2 dx ` Ψ2LdZn ML2 1 2 ¨ TdˆRn ´ |BtΨ|2 ` κ2T2 Z2 |∇zΨ|2¯ dz dx `ς ΨLdZnT2 mL2 ¨ TdˆRn |U|2σ2σ1 ‹ Ψ dz dx, with E0 dimensionless (hence the total energy of the original system is E0 ML2 T2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, we see that the dynamics is encoded by four independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In what follows, we get rid of a parameter by assuming ℏT mL2 “ 1, and we work with the following three independent parameters α “ ςΨLdZnT2 mL2 mL2 ℏT , β “ ςZ2 κ2Ψ M m, c “ κT Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It leads to ˆ iBtU ` 1 2∆xU ˙ pt, xq “ α ˆˆ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ Upt, xq, (69a) ´ 1 c2 B2 ttΨ ´ ∆zΨ ¯ pt, x, zq “ ´βσ2pzq ˆˆ Td σ1px ´ yq|Upt, yq|2 dy ˙ (69b) together with E0 “ 1 2 ˆ Td |∇xU|2 dx ` 1 2 α β ¨ TdˆRn ´ 1 c2 |BtΨ|2 ` |∇zΨ|2¯ dz dx `α ¨ TdˆRn |U|2σ2σ1 ‹ Ψ dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This relation allows us to interpret the scaling parameters as weights in the energy balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Now, for notational convenience, we decide to work with a m M b α β Ψ instead of Ψ and b M mU instead of U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it leads to (3a)-(3c) and (8) with γ “ b M m ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we shall implicitly work with solutions with amplitude of magnitude unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The regime where c Ñ 8, with α, β fixed leads, at least formally, to the Hartree system (1a)-(1b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' arguments are sketched in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The smallness condition (9) makes a threshold appear on the coefficients in order to guaranty the stability: since it involves the product M mαβ, it can be interpreted as a condition on the strength of the coupling between the particle and the environment, and on the amplitude of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We shall see in the proof that a sharper condition can be derived, expressed by means of the Fourier coefficients of the form function σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 54 B From Schödinger-Wave to Hartree In this Section we wish to justify that solutions – hereafter denoted Uc – of (3a)-(3c) converge to the solution of (1a)-(1b) as c Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We adapt the ideas in [10] where this question is investigated for Vlasov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Throughout this section we consider a sequence of initial data UInit c , ΨInit c , ΠInit c such that sup cą0 ˆ Td |UInit c |2 dx “ M0 ă 8, (70a) sup cą0 ˆ Td |∇xUInit c |2 dx “ M1 ă 8, (70b) sup cą0 " 1 2c2 ¨ TdˆRn |ΠInit c |2 dz dx ` 1 2 ¨ TdˆRn |∇zΨInit c |2 dz dx “ M2 ă 8, (70c) sup cą0 ¨ |UInit c |2σ1 ‹ σ2|ΨInit c | dz dx “ M3 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (70d) There are several direct consequences of these assumptions: The total energy is initially bounded uniformly with respect to c ą 0, In fact, we shall see that the last assumption can be deduced from the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the L2 norm of Uc is conserved by the equation, we already know that Uc is bounded in L8p0, 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we reformulate the expression of the potential, separating the contribution due to the initial data of the wave equation and the self-consistent part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By using the linearity of the wave equation, we can split Φc “ ΦInit,c ` ΦCou,c where ΦInit,c is defined from the free-wave equation on Rn and initial data ΨInit c , ΠInit c : 1 c2 B2 ttΥc ´ ∆zΨ “ 0, pΥc, BtΥcq ˇˇ t“0 “ pΨInit c , ΠInit c q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (71) Namely, we set ΦInit,cpt, xq “ ˆ Rn σ2pzqσ1 ‹ Υcpt, x, zq dz “ ˆ Rn ´ cospc|ξ|tqσ1 ‹ pΨInit c px, |ξq ` sinpc|ξ|t c|ξ| σ1 ‹ pΨInit c px, |ξq ¯pσ2pξq dξ p2πqn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly rΨc “ Ψc ´ Υc satisfies 1 c2 B2 tt rΨc ´ ∆z rΨc´ “ ´γσ2σ1 ‹ |Uc|2, prΨc, Bt rΨcq ˇˇ t“0 “ p0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (72) 55 and we get ΦCou,cpt, xq “ γ ˆ Rn σ2pzqσ1 ‹ rΨcpt, x, zq dz “ γ2c2 ˆ t 0 ˆ Rn sinpc|ξ|sq c|ξ| Σ ‹ |Uc|2pt ´ s, xq|pσ2pξq|2 dξ p2πqn ds “ γ2 ˆ ct 0 ˆˆ Rn sinpτ|ξ|q |ξ| |pσ2pξq|2 dξ p2πqn ˙ looooooooooooooooooomooooooooooooooooooon “ppτq Σ ‹ |Uc|2pt ´ τ{c, xq dτ, where it is known that the kernel p is integrable on r0, 8q [10, Lemma 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 There exists a constant Mw ą 0 such that sup c,t,x |ΦInit,cpt, xq| ď Mw, sup c,t,x |ΦCou,cpt, xq| ď Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Combining the Sobolev embedding theorem (mind the condition n ě 3) and the standard energy conservation for the free linear wave equation, we obtain }Υc}L8p0,8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2n{pn´2qpRnqqq ď C}∇zΥc}L8p0,8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdˆRnqq ď C a 2M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Applying Hölder’s inequality, we are thus led to: |ΦInit,cpt, xq| ď C}σ2}L2n{pn`2qpRnq}σ1}L2pRdq a 2M2, (73) which proves the first part of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Incidentally, it also shows that (70d) is a consequence of (70a) and (70c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get |ΦCou,cpt, xq| ď γ}Σ}L8pTdq}Uc}L8pr0,8q,L2pTdqq ˆ 8 0 |ppτq| dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 There exists a constant MS ą 0 such that sup c,t }∇Ucpt, ¨q}L2pTdq ď MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is a consequence of the energy conservation (the total energy being bounded by virtue of (70b)-(70d)) where the coupling term ˆ TdpΦInit,c ` ΦCou,cq|Uc|2 dx can be dominated by 2MwM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to BtUc “ ´ 1 2i∆xUc ` γ i pΦInit,c ` ΦCou,cqUc (74) 56 we see that BtUc is bounded in L2p0, 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H´1pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Combining the obtained estimates with Aubin- Simon’s lemma [44, Corollary 4], we deduce that Uc is relatively compact in in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' LppTdqq, 1 ď p ă 2d d ´ 2, for any 0 ă T ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, possibly at the price of extracting a subsequence, we can suppose that Uc converges strongly to U in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It remains to pass to the limit in (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The difficulty consists in letting c go to 8 in the potential term and to justify the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 For any ζ P C8 c pp0, 8q ˆ Tdq, we have lim cÑ8 ˆ 8 0 ˆ TdpΦInit,c ` ΦCou,cqUcζ dx dt “ γκ ˆ 8 0 ˆ Td Σ ‹ |Uc|2 Ucζ dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We expect that ΦCou,c converges to γκΣ ‹ |U|2: ˇˇΦCou,cpt, xq ´ γκΣ ‹ |U|2pt, xq ˇˇ “ γ ˇˇˇˇ ˆ ct 0 Σ ‹ |Uc|2pt ´ τ{c, xqppτq dτ ´ κΣ ‹ |U|2pt, xq ˇˇˇˇ ď γ ˆ ct 0 ˇˇˇΣ ‹ |Uc|2pt ´ τ{c, xq ´ Σ ‹ |U|2pt, xq ˇˇˇ |ppτq| dτ ` γ ˆ 8 ct |ppτq| dτ ˆ }Σ ‹ |U|2}L8pp0,8qˆTdq ď γ ˆ ct 0 Σ ‹ ˇˇ|Uc|2 ´ |U|2ˇˇpt ´ τ{c, xq |ppτq| dτ `γ ˆ ct 0 Σ ‹ ˇˇ|U|2pt ´ τ{c, xq ´ |U|2pt, xq ˇˇ |ppτq| dτ `γ ˆ 8 ct |ppτq| dτ }Σ}L8pTdq}U}L8pp0,8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us denote by Icpt, xq, IIcpt, xq, IIIcptq, the three terms of the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since p P L1pr0, 8qq, for any t ą 0, IIIcptq tends to 0 as c Ñ 8, and it is dominated by }p}L1pr0,8q}Σ}L8pTdqM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we have |Icpt, xq| ď }p}L1pr0,8q}Σ}L8pTdq sup sě0 ˆ Td ˇˇ|Uc|2 ´ |U|2ˇˇps, yq dy ď }p}L1pr0,8q}Σ}L8pTdq sup sě0 ˆˆ Td |Uc ´ U|2ps, yq dy ` 2Re ˆ TdpUc ´ UqUps, yq dy ˙ which also goes to 0 as c Ñ 8 and is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Eventually, we get |IIcpt, xq| ď }Σ}L8pTdq ˆ ct 0 ˆˆ Td ˇˇ|U|2pt ´ τ{c, yq ´ |U|2pt, yq ˇˇ dy ˙ |ppτq| dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since U P C0pr0, 8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq, with }Upt, ¨q}L2pTdq ď M0, we can apply the Lebesgue theorem to show that IIcpt, xq tends to 0 for any pt, xq fixed, and it is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This allows us to pass to the limit in ˆ 8 0 ˆ Td ΦCou,cUcζ dx dt ´ κ ˆ 8 0 ˆ Td Σ ‹ |U|2Uζ dx dt “ ˆ 8 0 ˆ Td ΦCou,cpUc ´ Uqζ dx dt ` ˆ 8 0 ˆ Td ´ ΦCou,c ´ γκΣ ‹ |U|2¯ Uζ dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 57 It remains to justify that lim cÑ8 ˆ 8 0 ˆ Td Φinit,cUcζ dx dt “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The space variable x is just a parameter for the free wave equation (71), which is equally satisfied by σ1 ‹ Υc, with initial data σ1 ‹ pΨInit c , ΠInit c q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We appeal to the Strichartz estimate for the wave equation, see [26, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3] or [45, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2, for the case n “ 3],which yields c1{p ˜ˆ 8 0 ˆˆ Rn |σ1 ‹ Υcpt, x, yq|q dy ˙p{q dt ¸1{p ď C ˆ 1 c2 ˆ Rn |σ1 ‹ ΠInit c px, zq|2 dz ` ˆ Rn |σ1 ‹ ∇yΨInit c px, zq|2 dz ˙1{2 , for any admissible pair: 2 ď p ď q ď 8, 1 p ` n q “ n 2 ´ 1, 2 p ` n ´ 1 q ď n ´ 1 2 , pp, q, nq � p2, 8, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The L2 norm with respect to the space variable of the right hand side is dominated by b }σ1}L1pTdq M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that ˆ Td ˜ˆ 8 0 ˆˆ Rn |σ1 ‹ Υcpt, x, zq|q dz ˙p{q dt ¸2{p dx ď C2}σ1}L1pRdq M2 1 c2{p ÝÝÝÑ cÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Repeated use of the Hölder inequality (with 1{p ` 1{p1 “ 1) leads to ˇˇˇˇ ˆ 8 0 ˆ Td UcζΦInit,c dx dt ˇˇˇˇ ď ˜ˆ Td ˆˆ 8 0 |Ucζpt, xq|p1 dt ˙2{p1 dx ¸1{2 ˜ˆ Td ˆˆ 8 0 |ΦInit,cpt, xq|p dt ˙2{p dx ¸1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' On the one hand, assuming that ζ is supported in r0, Rs ˆ Td and p ą 2, we have ˆ Td ˆˆ 8 0 |Ucζ|p1 dt ˙2{p1 dx ď ˆ Td ˆˆ R 0 |Uc|2 dt ˙ ˆˆ R 0 |ζ|2p1{p2´p1q dt ˙p2´p1q{p1 dx ď R1`p2´p1q{p1}ζ}L8pp0,8qˆTdq}Uc}L8pp0,8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdqq which is thus bounded uniformly with respect to c ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' On the other hand, we get ˆ Td ˆˆ 8 0 |ΦInit,cpt, xq|p dt ˙2{p dx “ ˆ Td ˆˆ 8 0 ˇˇˇ ˆ Rn σ2pzqσ1 ‹ Υcpt, x, zq dz ˇˇˇ p dt ˙2{p dx ď }σ2}Lq1pRnq ˆ Td ˆˆ 8 0 ˇˇˇ ˆ Rn |σ1 ‹ Υcpt, x, zq|q dz ˇˇˇ p{q dt ˙2{p dx which is of the order Opc´2{pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 58 C Well-posedness of the Schrödinger-Wave system The well-posedness of the Schrödinger-Wave system is justified by means of a fixed point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The method described here works as well for the problem set on Rd, and it is simpler than the approach in [21] since it avoids the use of “dual” Strichartz estimates for the Schrödinger and the wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We define a mapping that associates to a function pt, xq P r0, Ts ˆ Td ÞÑ V pt, xq P C: first, the solution Ψ of the linear wave equation 1 c2 B2 ttΨ ´ ∆zΨ “ ´σ2σ1 ‹ |V |2, pΨ, BtΨq ˇˇ t“0 “ pΨ0, Ψ1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' next, the potential Φ “ σ1 ‹ ´ Rn σ2Ψ dz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' and finally the solution of the linear Schrödinger equation iBtU ` 1 2∆xU “ γΦU, U ˇˇ t“0 “ UInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These successive steps define a mapping S : V ÞÝÑ U and we wish to show that this mapping admits a fixed point in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq, which, in turn, provides a solution to the non linear system (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In this discussion, the initial data UInit, Ψ0, Ψ1 are fixed once for all in the space of finite energy: UInit P H1pTdq, Ψ0 P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq, Ψ1 P L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We observe that d dt ˆ Td |U|2 dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the mapping S applies the ball Bp0, }UInit}L2pTdqq of C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq in itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' we thus consider U “ SpV q for V P C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq such that }V pt, ¨q}L2pTdq ď }UInit}L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, we can split Ψ “ Υ ` rΨ with Υ solution of the free wave equation 1 c2 B2 ttΥ ´ ∆zΥ “ 0, pΥ, BtΥq ˇˇ t“0 “ pΨ0, Ψ1q, and 1 c2 B2 tt rΨ ´ ∆z rΨ “ 0, pΥ, Bt rΨq ˇˇ t“0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We write Φ “ ΦI ` rΦ for the associated splitting of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the standard energy conservation for the wave equation tells us that 1 2c2 ¨ TdˆRn |BtΥ|2 dz dx ` 1 2 ¨ TdˆRn |∇zΥ|2 dz dx “ 1 2c2 ¨ TdˆRn |Ψ1|2 dz dx ` 1 2 ¨ TdˆRn |∇zΨ0|2 dz dx “ M2 59 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that |ΦIpt, xq| ď C}σ2}L2n{pn`2pRnq}σ1}L2pTdq a 2M2 by using Sobolev’s embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we obtain rΦpt, xq “ ˆ Rn σ2pzqσ1 ‹ rΨpt, x, zq dz “ γ ˆ ct 0 ˆˆ Rn sinpτ|ξ|q |ξ| |pσ2pξq|2 dξ p2πqn ˙ looooooooooooooooooomooooooooooooooooooon “ppτq Σ ‹ |V |2pt ´ τ{c, xq dτ, which thus satisfies sup xPTd |rΦpt, xq| ď γ}Σ}L8pTdq ˆ ct 0 |ppτq| ˆˆ Td |V |2pt ´ τ{c, yq dy ˙ dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular |rΦpt, xq| ď γ}Σ}L8pTdq}p}L1pp0,8qq}V }C0pr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdqq ď γ}Σ}L8pTdq}p}L1pp0,8qq}UInit}L2pTdq lies in L8pp0, Tq ˆ Tdq, and thus Φ P L8pp0, Tq ˆ Rdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This observation guarantees that U “ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='SpV q is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus, let us pick V1, V2 in this ball of C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq and consider Uj “ SpVjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have iBtpU2 ´ U1q ` 1 2∆xpU2 ´ U1q “ γΦ2pU2 ´ U1q ` γpΦ2 ´ Φ1qU1, pU2 ´ U1q ˇˇ t“0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that d dt ˆ Td |U2 ´ U1|2 dx “ 2γIm ˆˆ TdpΦ2 ´ Φ1qU 1pU2 ´ U1q dx ˙ ď 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }Φ2 ´ Φ1}L8pTdq “ 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }rΦ2 ´ rΦ1}L8pTdq ď 2γ2}Σ}L8pTdq}UInit}L2pTdq }U2 ´ U1}L2pTdq ˆ ct 0 |ppτq| ˆˆ Td ˇˇ|V2|2 ´ |V1|2ˇˇpt ´ τ{c, yq dy ˙ dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We use the elementary estimate ˆ Td ˇˇ|V2|2´|V1|2ˇˇ dy “ ˆ Td ˇˇ|V2´V1|2`2RepV2´V1qV1 ˇˇ dy ď }V2´V1}2 L2pTdq`2}V2´V1}L2pTdq }V1}L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Combining this with Cauchy-Schwarz and Young inequalities, we arrive at d dt ˆ Td |U2 ´ U1|2 dx ď 2γ2}Σ}L8pTdq}UInit}L2pTdq ˆ 2}UInit}L2pTdq ˆ ct 0 |ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ `}U2 ´ U1}L2pTdq2}UInit}L2pTdq ˆ ct 0 |ppτq|}V2 ´ V1}pt ´ τ{cqL2pTdq dτ ˙ ď 2γ2}Σ}L8pTdq}UInit}2 L2pTdq ´ }U2 ´ U1}2 L2pTdq `p2 ` }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8qq ˆ ct 0 |ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 60 Set L “ 2γ2}Σ}L8pTdq}UInit}2 L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We deduce that }U2 ´ U1}ptq2 L2pTdq ď p2 ` }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8qqL ˆ t 0 eLpt´sq ˆ cs 0 |ppτq|}V2 ´ V1}2ps ´ τ{cqL2pTdq dτ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We use this estimate for 0 ď t ď T ă 8 and we obtain }U2 ´ U1}ptq2 L2pTdq ď p4 ` }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8qqLTeLT }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8q sup 0ďsďT }V2 ´ V1}2psqL2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence for T small enough, S is a contraction in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq, and consequently it admits a unique fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the fixed point still has its L2 norm equal to }UInit}L2pTdq, the solution can be extended on the whole interval r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The argument can be adapted to handle the Hartree system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' References [1] B.' metadata={'source': 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