1207/1207.3632.md

RELATIVE OSCILLATION THEORY FOR JACOBI MATRICES EXTENDED

KERSTIN AMMANN

ABSTRACT. We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-)open and closed spectral intervals, simplify the proof, and establish the comparison theorem.

1. INTRODUCTION

Jacobi operators appear at numerous occasions in mathematics as well as in physical models. For example, they are intimately related to the theory of orthogonal polynomials or constitute as a simple one-band tight binding model in quantum mechanics. They can be viewed as the discrete analog of Sturm–Liouville operators and their investigation has many similarities with Sturm–Liouville theory. Moreover, spectral and inverse spectral theory for Jacobi operators plays a fundamental role in the investigation of the Toda lattice and its modified counterpart, the Kac–van Moerbeke lattice. For further information we refer, e.g., to [16]. Let $a, b \in \ell(\mathbb{Z}) = {\varphi \mid \varphi : \mathbb{Z} \rightarrow \mathbb{R}}$ , where $a(n) < 0$ holds for all $n \in \mathbb{Z}$ . Then, the Jacobi matrix

is self-adjoint and $\sigma(J)$ is real and simple. The corresponding Jacobi difference equation

$$\tau u=zu,(1.2)\backslash tau\; u\; =\; zu,\; \backslash quad\; (1.2)$$

where

---

2000 Mathematics Subject Classification. Primary 39A21, 47B36; Secondary 34C10, 34L05.

Key words and phrases. Jacobi matrices, oscillation theory, Wronskian.

Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.$z \in \mathbb{R}$ , and $\partial u(n) = u(n+1) - u(n)$ is the usual forward difference operator, is the discrete analog of the Sturm-Liouville differential equation.

We call $u(z)$ a solution of (1.2) if $(\tau - z)u(z) = 0$ and $u(z) \not\equiv 0$ . Whenever the spectral parameter is evident from the context we abbreviate $u = u(z)$ . To any two initial values $u(n_0), u(n_0+1), n_0 \in \mathbb{Z}$ , there exists a unique 'solution' $u$ of (1.2) which vanishes if and only if $(u(n_0), u(n_0+1)) = (0, 0)$ . We exclude this case and thus, a solution of (1.2) cannot have two consecutive zeros. We call $n$ a node of $u$ if

$$u(n)=0\text{or}a(n)u(n)u(n+1)>0(1.3)u(n)\; =\; 0\; \backslash quad\; \backslash text\{or\}\; \backslash quad\; a(n)u(n)u(n+1)\; >\; 0\; \backslash quad\; (1.3)$$

and say that a node $n$ of $u$ lies between $m$ and $l$ if either $m < n < l$ or if $n = m$ and $u(m) \neq 0$ . The number of nodes of $u$ between $m$ and $l$ is denoted as $#_{(m,l)}(u)$ . From classical oscillation theory originating in the seminal work of Sturm from 1836 [15] we know that the $n$ -th eigenfunction of a Sturm-Liouville operator has $n-1$ nodes. This also holds for eigensequences of Jacobi operators, cf. e.g. [6], [12], or [18]. Our aim now is to show that the number of nodes of the Wronskian determinant of two (suitable) solutions $u_j(\lambda_j)$ of $(\tau_j - \lambda_j)u_j = 0$ , $j = 0, 1$ , equals the difference of the number of eigenvalues of $J_0$ and $J_1$ in $(\lambda_0, \lambda_1)$ .

In [2] (cf. also [1]) Teschl and myself considered the special case $a_0 = a_1$ which is now generalized to arbitrary perturbations. We still assume $a_0, a_1 < 0$ . This is no restriction since altering the sign of one or more elements of $a$ does not affect the spectrum of the corresponding matrices, their similarity can easily be shown. Nevertheless, the signs of the solutions of the underlying difference equations depend on the signs of $a$ and therefore we assume $a < 0$ to simplify (1.5).

The Wronskian is given by $W(u_0, u_1) \in \ell(\mathbb{Z})$ , where

$$W_n(u_0,u_1)=u_0(n)a_1(n)u_1(n+1)-u_1(n)a_0(n)u_0(n+1)\mathrm{.}(1.4)W\_n(u\_0,\; u\_1)\; =\; u\_0(n)a\_1(n)u\_1(n+1)\; -\; u\_1(n)a\_0(n)u\_0(n+1).\; \backslash quad\; (1.4)$$

We set

and say the Wronskian has a (weighted) node at $n$ if $#_n(u_0, u_1) \neq 0$ . We denote the number of weighted nodes of the Wronskian between $m$ and $n$ , $m < n$ , by

$${\mathrm{\#}}_{[m,n]}(u_0,u_1)=\sum _{j=m}^{n-1}{\mathrm{\#}}_j(u_0,u_1)(1.6)\backslash \#\_\{[m,n]\}(u\_0,\; u\_1)\; =\; \backslash sum\_\{j=m\}^\{n-1\}\; \backslash \#\_j(u\_0,\; u\_1)\; \backslash quad\; (1.6)$$

and set

and

Here we slightly changed the notation compared to [2]: $#{(m,n)}$ from [2] is now denoted as $#{[m,n]}$ .

We find the following

Theorem 1.1 (Relative Oscillation Theorem). Let $a_0, a_1 < 0$ , then,

and

where $E_\Omega(J)$ is the number of eigenvalues of $J$ in $\Omega \subseteq \mathbb{R}$ and $u_{-/+}$ are solutions fulfilling the left/right Dirichlet boundary condition of $J$ , i.e. $u_-(0) = u_+(N) = 0$ .

Equation (1.9) generalizes Theorem 1.2 from [2] to different $a$ 's. Analogous results for Sturm–Liouville operators were first given by Krüger and Teschl [9, 10]. For the case of Dirac operators see Stadler and Teschl in [14]. For extensions to symplectic eigenvalue problems see Elyseeva [3–5].

In the sequel (Sections 2–4) we prove Theorem 1.1 using the discrete Prüfer transformation. Compared to [2, 9, 10, 14] we also present a simplified proof which eliminates the need to interpolate between operators. This is of particular importance in the present case, since $a_0 < a_1$ does not imply the corresponding relation for the operators, which would make the interpolation step more difficult. In addition, (1.10) is new. The proofs for regular Sturm–Liouville operators [9, Theorem 2.3] and regular Dirac operators [14, Theorem 3.3] can be shortened in the same manner and both theorems can be extended to (half-)open and closed spectral intervals analogously to (1.10) (for the first case cf. also [17]).

An extension of Sturm's classical comparison theorem for nodes of solutions to nodes of Wronskians has first been established for Sturm–Liouville operators by Krüger and Teschl in [9, 10]. In Section 5 we show that an analogous comparison theorem holds for Wronskians of solutions of Jacobi difference equations in the case $a_0 = a_1$ , therefore confer also [1]. Moreover, we give Sturm-type comparison theorems for arbitrary perturbations of Jacobi matrices, where, unlike the case of Sturm–Liouville operators, we do not obtain a direct dependence on the coefficients of the operators because $a_0 \leq a_1$ doesn't imply $J_0 \leq J_1$ .

An extension of Theorem 1.1 to Jacobi operators on the half line and on the line is in preparation. This will fill the gap that classical oscillation theory is only applicable below the essential spectrum, while relative oscillation theory works perfectly insidegaps of the essential spectrum. We hope that this will stimulate further research, e.g. to find new relative oscillation criteria as in the Sturm–Liouville–case, [7, 8]. We’d be remiss not to mention that several other extensions of relative oscillation theory are thinkable, e.g. to CMV matrices. Only recently, Šimon Hilscher pointed out in [13] that an extension to the case of Jacobi difference equations with a nonlinear dependence on the spectral parameter would be of particular interest.

2. THE WRONSKIAN

Definition 2.1. We define the (modified) Wronskian (also referred to as Wronski determinant or Casorati determinant) by

where $\mathbb{D}$ denotes the space of difference equations, such that

We abbreviate $\Delta b := b_0 - b_1$ , $\Delta a := a_0 - a_1$ , and $W_n(\varphi, \psi) = W_n^{\tau_0, \tau_1}(\varphi, \psi)$ whenever the corresponding difference equations are evident from the context. Clearly, if $a_0 = a_1$ holds, then $W$ equals the Wronskian from [2]. We have

for all $c \in \mathbb{R}$ and $\varphi, \tilde{\varphi}, \psi, \tilde{\psi} \in \ell(\mathbb{Z})$ .

Lemma 2.2. Green’s Formula. We find

Proof. Just a short calculation. $\square$

Corollary 2.3. Let $(\tau_j - z)u_j = 0$ , then

$$W_m(u_0,u_1)-W_{n-1}(u_0,u_1)(2.5)W\_m(u\_0,\; u\_1)\; -\; W\_\{n-1\}(u\_0,\; u\_1)\; \backslash quad\; (2.5)$$

If $u$ and $\tilde{u}$ solve $\tau u = zu$ , then $W(u, \tilde{u})$ is constant (and vanishes iff $u$ and $\tilde{u}$ are linearly dependent).3. PRÜFER TRANSFORMATION

From now on let $u$ be a solution of (1.2) and let $u_{-/+}$ moreover fulfill the left/right Dirichlet boundary condition of $J$ .

Lemma 3.1. [16]. The Jacobi matrix $J$ has $N - 1$ real and simple eigenvalues. Moreover,

$$z\in \sigma (J)⟺u_{-}(z,N)=0⟺u_{+}(z,0)=0.(3.1)z\; \backslash in\; \backslash sigma(J)\; \backslash iff\; u\_-(z,\; N)\; =\; 0\; \backslash iff\; u\_+(z,\; 0)\; =\; 0.\; \backslash quad\; (3.1)$$

Proof. Since $J$ is Hermitian all eigenvalues are real: let $z \in \sigma(J)$ , $Jv = zv$ and $|v| = 1$ . Then $z = \langle v, zv \rangle = \langle v, Jv \rangle = \langle Jv, v \rangle = \overline{z}$ . It can easily be seen that every eigenvector $u$ corresponding to $z$ fulfills $\tau u = zu$ and $u(0) = 0$ . Hence, by $W_0(u_-(z), u) = 0$ , $u_-(z)$ and $u$ are linearly dependent. $\square$

Theorem 3.2. [18], [16, Theorem 4.7]. For all $\lambda \in \mathbb{R}$

$$E_{(-\mathrm{\infty },\lambda )}(J)={\mathrm{\#}}_{(0,N)}(u_{-}(\lambda ))={\mathrm{\#}}_{(0,N)}(u_{+}(\lambda ))(3.2)E\_\{(-\backslash infty,\; \backslash lambda)\}(J)\; =\; \backslash \#\_\{(0,\; N)\}(u\_-(\backslash lambda))\; =\; \backslash \#\_\{(0,\; N)\}(u\_+(\backslash lambda))\; \backslash quad\; (3.2)$$

holds.

Lemma 3.3. If $u(n) = 0$ , then $u(n - 1)u(n + 1) < 0$ .

Proof. Since (1.2) is a three-term-recursion and $u \not\equiv 0$ , all zeros of $u$ are simple and

holds. $\square$

By $(u(n), u(n + 1)) \neq (0, 0)$ for all $n \in \mathbb{Z}$ we can introduce Prüfer variables: let $\rho_u, \theta_u \in \ell(\mathbb{Z})$ denote sequences so that

and $\rho_u(n) > 0$ holds for all $n \in \mathbb{Z}$ . Choose $\theta_u(n_0) \in (-\pi, \pi]$ at the initial position $n_0$ and assume

$$\lceil {\theta }_u(n)\mathrm{/}\pi \rceil \le \lceil {\theta }_u(n+1)\mathrm{/}\pi \rceil \le \lceil {\theta }_u(n)\mathrm{/}\pi \rceil +1(3.5)\backslash lceil\; \backslash theta\_u(n)/\backslash pi\; \backslash rceil\; \backslash leq\; \backslash lceil\; \backslash theta\_u(n\; +\; 1)/\backslash pi\; \backslash rceil\; \backslash leq\; \backslash lceil\; \backslash theta\_u(n)/\backslash pi\; \backslash rceil\; +\; 1\; \backslash quad\; (3.5)$$

for all $n \in \mathbb{Z}$ , then both sequences are well-defined and unique. Here, $x \mapsto \lceil x \rceil = \min{n \in \mathbb{Z} \mid n \geq x}$ denotes the ceiling function, a left-continuous analog to the well-known floor function $x \mapsto \lfloor x \rfloor = \max{n \in \mathbb{Z} \mid n \leq x}$ which itself is a right-continuous step function.

We follow [9] and use the slightly refined (compared to [2, 16, 18]) definition of Prüfer variables by taking the secondary diagonals $a$ into account. By $-a > 0$ this will not influence the herein recalled claims on the nodes of solutions, but it simplifies our calculations as soon as we look at the nodes of the Wronskian.

Lemma 3.4. Fix some $n \in \mathbb{Z}$ , then $\exists k \in \mathbb{Z}$ s.t. $\theta_u(n) = k\pi + \gamma$ and $\theta_u(n + 1) = k\pi + \Gamma$ , where

holds. Moreover,

$${\theta }_u(n)=k\pi +\frac{\pi }{2}⟺{\theta }_u(n+1)=(k+1)\pi \mathrm{.}(3.7)\backslash theta\_u(n)\; =\; k\backslash pi\; +\; \backslash frac\{\backslash pi\}\{2\}\; \backslash iff\; \backslash theta\_u(n\; +\; 1)\; =\; (k\; +\; 1)\backslash pi.\; \backslash quad\; (3.7)$$Proof. Abbreviate $\theta = \theta_u$ . Choose $k \in \mathbb{Z}$ such that $\theta(n) = k\pi + \gamma$ , $\gamma \in (0, \pi]$ holds. By (3.5) we have $\Gamma \in (0, 2\pi]$ . If $u(n)u(n+1) \neq 0$ , then $\sin \gamma \cos \gamma > 0$ iff $n$ is not a node of $u$ and $\sin \gamma \cos \gamma < 0$ iff $n$ is a node of $u$ , hence (3.6) clearly holds for $\gamma$ . By $\sin \Gamma \cos \gamma > 0$ we have $\sin \Gamma > 0$ iff $n$ is not a node of $u$ and $\sin \Gamma < 0$ iff $n$ is a node of $u$ , thus, (3.6) also holds for $\Gamma$ .

Now, suppose we have $u(n+1) = 0$ , then $n$ is not a node of $u$ and either $\Gamma = \pi$ or $\Gamma = 2\pi$ holds. By Lemma 3.3 we have $u(n)u(n+2) < 0$ , hence $\sin \theta(n) \cos \theta(n+1) = (-1)^k \sin \gamma (-1)^k \cos \Gamma < 0$ . Thus, by $\cos \Gamma < 0$ , we have $\Gamma = \pi$ . From $-a(n)u(n+1) = \rho(n) \cos \theta(n) = 0$ we conclude that $(-1)^k \cos \gamma = 0$ , thus $\gamma = \frac{\pi}{2}$ and hence (3.6) and (3.7) hold. If $u(n) = 0$ , then $n$ is a node of $u$ , $\gamma = \pi$ , and (3.6) holds by $\sin \theta(n+1) \cos \theta(n) > 0$ , i.e. $(-1)^k \sin \Gamma (-1)^k \cos \gamma > 0$ . $\square$

Corollary 3.5. For all $n \in \mathbb{Z}$ we have

Now we are able to count nodes of solutions of the Jacobi difference equation using Prüfer variables and the number of nodes in an interval $(m, n)$ is given by

Theorem 3.6. [18, Lemma 2.5]. We have

$${\mathrm{\#}}_{(m,n)}(u)=\lceil {\theta }_u(n)\mathrm{/}\pi \rceil -\lfloor {\theta }_u(m)\mathrm{/}\pi \rfloor -1.(3.9)\backslash \#\_\{(m,n)\}(u)\; =\; \backslash lceil\; \backslash theta\_u(n)/\backslash pi\; \backslash rceil\; -\; \backslash lfloor\; \backslash theta\_u(m)/\backslash pi\; \backslash rfloor\; -\; 1.\; \backslash quad\; (3.9)$$

Proof. We use mathematical induction: let $n = m+1$ , then if $u(m) = 0$ , $u(n) \neq 0$ we have $#_{(m,n)}(u) = 0$ and by Corollary 3.5

$$\lceil {\theta }_u(n)\mathrm{/}\pi \rceil =\lceil {\theta }_u(m+1)\mathrm{/}\pi \rceil =\underset{\in \mathbb{Z}}{\underbrace{\lceil {\theta }_u(m)\mathrm{/}\pi \rceil }}+1=\lfloor {\theta }_u(m)\mathrm{/}\pi \rfloor +1(3.10)\backslash lceil\; \backslash theta\_u(n)/\backslash pi\; \backslash rceil\; =\; \backslash lceil\; \backslash theta\_u(m+1)/\backslash pi\; \backslash rceil\; =\; \backslash underbrace\{\backslash lceil\; \backslash theta\_u(m)/\backslash pi\; \backslash rceil\}\_\{\backslash in\; \backslash mathbb\{Z\}\}\; +\; 1\; =\; \backslash lfloor\; \backslash theta\_u(m)/\backslash pi\; \backslash rfloor\; +\; 1\; \backslash quad\; (3.10)$$

holds. If $u(m) \neq 0$ holds, then by Corollary 3.5 we have

The inductive step follows again from Corollary 3.5. $\square$

Let $s_{-/+}(z)$ denote the solution of $\tau s = zs$ fulfilling

$$s_{-}(z,0)=0,s_{-}(z,1)=1,\text{resp. }{s}_{+}(z,N)=0,s_{+}(z,N+1)=1(3.12)s\_-(z,\; 0)\; =\; 0,\; s\_-(z,\; 1)\; =\; 1,\; \backslash quad\; \backslash text\{resp.\; \}\; s\_+(z,\; N)\; =\; 0,\; s\_+(z,\; N+1)\; =\; 1\; \backslash quad\; (3.12)$$

and let $n_0$ denote the base point, i.e. $n_0 = 0$ , resp. $n_0 = N$ . Then, by $s_{\pm}(n_0) = 0$ we have $\sin \theta_{\pm}(n_0) = 0$ and by $s_{\pm}(n_0+1) = 1$ we have $-a(n_0)s_{\pm}(n_0+1) = \rho_s(n_0) \cos \theta_{\pm}(n_0) > 0$ , hence $\theta_{\pm}(n_0) = 0$ holds by $\theta_{\pm}(n_0) \in (-\pi, \pi]$ .

Corollary 3.7. We have

$${\mathrm{\#}}_{(0,N)}(s_{-})=\lceil {\theta }_{s_{-}}(N)\mathrm{/}\pi \rceil -1\text{and}{\mathrm{\#}}_{(0,N)}(s_{+})=-\lfloor {\theta }_{s_{+}}(0)\mathrm{/}\pi \rfloor -1.(3.13)\backslash \#\_\{(0,N)\}(s\_-)\; =\; \backslash lceil\; \backslash theta\_\{s\_-\}(N)/\backslash pi\; \backslash rceil\; -\; 1\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash \#\_\{(0,N)\}(s\_+)\; =\; -\backslash lfloor\; \backslash theta\_\{s\_+\}(0)/\backslash pi\; \backslash rfloor\; -\; 1.\; \backslash quad\; (3.13)$$Lemma 3.8. We find

where $\Delta_{u,v} = \theta_v - \theta_u \in \ell(\mathbb{Z})$ .

Proof. Abbreviate $s_{j,\pm} = s_{j,\pm}(\lambda_j)$ . By Theorem 3.2, Corollary 3.7, and $-[x] = \lfloor -x \rfloor$ for all $x \in \mathbb{R}$ we have

By Lemma 3.1 and (3.4) we have

and hence

$$\begin{array}{cccc}& E_{(-\mathrm{\infty },{\lambda }_1)}(J_1)-E_{(-\mathrm{\infty },{\lambda }_0)}(J_0)=\lceil {\mathrm{\Delta }}_{s_{0,\pm },s_{1,\mp }}(N)\mathrm{/}\pi \rceil -\lceil {\mathrm{\Delta }}_{s_{0,\pm },s_{1,\mp }}(0)\mathrm{/}\pi \rceil -1& & \text{(3.15)}\end{array}E\_\{(-\backslash infty,\; \backslash lambda\_1)\}(J\_1)\; -\; E\_\{(-\backslash infty,\; \backslash lambda\_0)\}(J\_0)\; =\; \backslash lceil\; \backslash Delta\_\{s\_\{0,\backslash pm\},\; s\_\{1,\backslash mp\}\}(N)/\backslash pi\; \backslash rceil\; -\; \backslash lceil\; \backslash Delta\_\{s\_\{0,\backslash pm\},\; s\_\{1,\backslash mp\}\}(0)/\backslash pi\; \backslash rceil\; -\; 1\; \backslash tag\{3.15\}$$

holds by

The rest now follows analogously. $\square$

4. NODES OF THE WRONSKIAN

It remains to investigate the sign-changes of $W(u_0, u_1)$ . We will express them in terms of Prüfer angles of the involved solutions to finally gain their connection to the difference of the spectra of the corresponding matrices by Lemma 3.8.

Therefore let $u_j$ be solutions of $\tau_j - z$ , $j = 0, 1$ , where $\rho_j, \theta_j \in \ell(\mathbb{Z})$ are their Prüfer variables from (3.4). They correspond to the same spectral parameter $z$ , which is no restriction, since we can always replace $b_1$ by $b_1 - (z_1 - z_0)$ . We abbreviate

$$\begin{array}{cccc}& \mathrm{\Delta }:={\mathrm{\Delta }}_{u_0,u_1}={\theta }_1-{\theta }_0\in \mathrm{\ell }(\mathbb{Z})& & \text{(4.1)}\end{array}\backslash Delta\; :=\; \backslash Delta\_\{u\_0,\; u\_1\}\; =\; \backslash theta\_1\; -\; \backslash theta\_0\; \backslash in\; \backslash ell(\backslash mathbb\{Z\})\; \backslash tag\{4.1\}$$

and adopt Lemma 4.1 and Lemma 4.2 from [2]:

Lemma 4.1. [2]. Fix some $n \in \mathbb{Z}$ , then $\exists k_j \in \mathbb{Z}, j = 0, 1$ , s.t.

where(1): either $u_0$ and $u_1$ have a node at $n$ or both do not have a node at $n$ , then

$${\gamma }_1-{\gamma }_0\in (-\frac{\pi }{2},\frac{\pi }{2})\text{and}{\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_0\in (-\pi ,\pi )\mathrm{.}(4.3)\backslash gamma\_1\; -\; \backslash gamma\_0\; \backslash in\; \backslash left(-\backslash frac\{\backslash pi\}\{2\},\; \backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash Gamma\_1\; -\; \backslash Gamma\_0\; \backslash in\; (-\backslash pi,\; \backslash pi).\; \backslash quad\; (4.3)$$

(2): $u_1$ has no node at $n$ , but $u_0$ has a node at $n$ , then

$${\gamma }_1-{\gamma }_0\in (-\pi ,0)\text{and}{\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_0\in (-2\pi ,0)\mathrm{.}(4.4)\backslash gamma\_1\; -\; \backslash gamma\_0\; \backslash in\; (-\backslash pi,\; 0)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash Gamma\_1\; -\; \backslash Gamma\_0\; \backslash in\; (-2\backslash pi,\; 0).\; \backslash quad\; (4.4)$$

(3): $u_1$ has a node at $n$ , but $u_0$ has no node at $n$ , then

$${\gamma }_1-{\gamma }_0\in (0,\pi )\text{and}{\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_0\in (0,2\pi )\mathrm{.}(4.5)\backslash gamma\_1\; -\; \backslash gamma\_0\; \backslash in\; (0,\; \backslash pi)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash Gamma\_1\; -\; \backslash Gamma\_0\; \backslash in\; (0,\; 2\backslash pi).\; \backslash quad\; (4.5)$$

Proof. Use Lemma 3.4. $\square$

Lemma 4.2. [2]. We have

$$\lceil \mathrm{\Delta }(n)\mathrm{/}\pi \rceil -1\le \lceil \mathrm{\Delta }(n+1)\mathrm{/}\pi \rceil \le \lceil \mathrm{\Delta }(n)\mathrm{/}\pi \rceil +1.(4.6)\backslash lceil\; \backslash Delta(n)/\backslash pi\; \backslash rceil\; -\; 1\; \backslash leq\; \backslash lceil\; \backslash Delta(n+1)/\backslash pi\; \backslash rceil\; \backslash leq\; \backslash lceil\; \backslash Delta(n)/\backslash pi\; \backslash rceil\; +\; 1.\; \backslash quad\; (4.6)$$

Proof. Let $k := k_1 - k_0$ , $n \in \mathbb{Z}$ . By Lemma 4.1 we have either

$$\begin{array}{r}{\displaystyle \mathrm{\Delta }(n)\in (k\pi -\frac{\pi }{2},k\pi +\frac{\pi }{2})\text{and}\mathrm{\Delta }(n+1)\in (k\pi -\pi ,k\pi +\pi ),}\\ {\displaystyle \mathrm{\Delta }(n)\in (k\pi -\pi ,k\pi )\text{and}\mathrm{\Delta }(n+1)\in (k\pi -2\pi ,k\pi ),\text{ or}}\\ {\displaystyle \mathrm{\Delta }(n)\in (k\pi ,k\pi +\pi )\text{and}\mathrm{\Delta }(n+1)\in (k\pi ,k\pi +2\pi )\mathrm{.}}\end{array}(4.7)\backslash begin\{aligned\}\; \backslash Delta(n)\; \backslash in\; (k\backslash pi\; -\; \backslash frac\{\backslash pi\}\{2\},\; k\backslash pi\; +\; \backslash frac\{\backslash pi\}\{2\})\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash Delta(n+1)\; \backslash in\; (k\backslash pi\; -\; \backslash pi,\; k\backslash pi\; +\; \backslash pi),\; \backslash \backslash \; \backslash Delta(n)\; \backslash in\; (k\backslash pi\; -\; \backslash pi,\; k\backslash pi)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash Delta(n+1)\; \backslash in\; (k\backslash pi\; -\; 2\backslash pi,\; k\backslash pi),\; \backslash text\{\; or\}\; \backslash \backslash \; \backslash Delta(n)\; \backslash in\; (k\backslash pi,\; k\backslash pi\; +\; \backslash pi)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash Delta(n+1)\; \backslash in\; (k\backslash pi,\; k\backslash pi\; +\; 2\backslash pi).\; \backslash end\{aligned\}\; \backslash quad\; (4.7)$$

In each case the lemma holds. $\square$

Lemma 4.3. We have

$$W_n(u_0,u_1)={\rho }_0(n){\rho }_1(n)\sin \mathrm{\Delta }(n),(4.8)W\_n(u\_0,\; u\_1)\; =\; \backslash rho\_0(n)\backslash rho\_1(n)\; \backslash sin\; \backslash Delta(n),\; \backslash quad\; (4.8)$$

$$W_n(u_0,u_1)u_0(n+1)u_1(n+1)=p\sin ({\gamma }_1-{\gamma }_0)\cos {\gamma }_0\cos {\gamma }_1,(4.9)W\_n(u\_0,\; u\_1)u\_0(n+1)u\_1(n+1)\; =\; p\; \backslash sin(\backslash gamma\_1\; -\; \backslash gamma\_0)\; \backslash cos\; \backslash gamma\_0\; \backslash cos\; \backslash gamma\_1,\; \backslash quad\; (4.9)$$

$$W_{n+1}(u_0,u_1)u_0(n+1)u_1(n+1)=\tilde{p}\sin ({\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_0)\cos {\gamma }_0\cos {\gamma }_1,(4.10)W\_\{n+1\}(u\_0,\; u\_1)u\_0(n+1)u\_1(n+1)\; =\; \backslash tilde\{p\}\; \backslash sin(\backslash Gamma\_1\; -\; \backslash Gamma\_0)\; \backslash cos\; \backslash gamma\_0\; \backslash cos\; \backslash gamma\_1,\; \backslash quad\; (4.10)$$

where $p, \tilde{p} > 0$ .

Proof. Consider

and set $p = \frac{\rho_0(n)^2 \rho_1(n)^2}{a_0(n)a_1(n)}$ and $\tilde{p} = \frac{\rho_0(n)\rho_1(n)\rho_0(n+1)\rho_1(n+1)}{a_0(n)a_1(n)}$ . $\square$

Lemma 4.4. We have

Proof. The first claim holds trivially. For the second claim just observe that by Lemma 3.3

$$W_n(u_0,u_1)W_{n+1}(u_0,u_1)=-u_0(n)u_0(n+2)a_0(n+1)a_1(n)u_1(n+1{)}^2>0(4.13)W\_n(u\_0,\; u\_1)W\_\{n+1\}(u\_0,\; u\_1)\; =\; -u\_0(n)u\_0(n+2)a\_0(n+1)a\_1(n)u\_1(n+1)^2\; >\; 0\; \backslash quad\; (4.13)$$

holds if $u_0(n+1) = 0, u_1(n+1) \neq 0$ and

$$W_n(u_0,u_1)W_{n+1}(u_0,u_1)=-u_1(n)u_1(n+2)a_0(n)a_1(n+1)u_0(n+1{)}^2>0(4.14)W\_n(u\_0,\; u\_1)W\_\{n+1\}(u\_0,\; u\_1)\; =\; -u\_1(n)u\_1(n+2)a\_0(n)a\_1(n+1)u\_0(n+1)^2\; >\; 0\; \backslash quad\; (4.14)$$

holds if $u_0(n+1) \neq 0, u_1(n+1) = 0$ . $\square$Corollary 4.5. If $W_n(u_0, u_1)W_{n+1}(u_0, u_1) < 0$ or $W_n(u_0, u_1) = 0, W_{n+1}(u_0, u_1) \neq 0$ or $W_n(u_0, u_1) \neq 0, W_{n+1}(u_0, u_1) = 0$ , then

$$u_0(n+1)u_1(n+1)\mathrm{\ne }0(4.15)u\_0(n+1)u\_1(n+1)\; \backslash neq\; 0\; \backslash quad\; (4.15)$$

and moreover $\Delta a(n) \neq 0$ or $\Delta b(n+1) \neq 0$ holds.

To shorten notation we denote

Lemma 4.6. Let $n \in \mathbb{Z}$ , then

$$(0)⟺\text{otherwise.}(4.19)(0)\; \backslash iff\; \backslash text\{otherwise.\}\; \backslash quad\; (4.19)$$

Proof. If (+1), then we either have case (1) of Lemma 4.1 and $\gamma_1 - \gamma_0 \in (-\frac{\pi}{2}, 0]$ , $\Gamma_1 - \Gamma_0 \in (0, \pi)$ or we have case (3) of Lemma 4.1 and $\gamma_1 - \gamma_0 \in (0, \pi)$ , $\Gamma_1 - \Gamma_0 \in (\pi, 2\pi)$ . Clearly, by (4.8), in either case we have

Hence, by Corollary 4.5 we have $u_0(n+1)u_1(n+1) \neq 0$ and thus $\cos \gamma_0 \cos \gamma_1 \neq 0$ . In case (1) of Lemma 4.1 we have $\sin(\Gamma_1 - \Gamma_0) > 0$ and $\cos \gamma_0 \cos \gamma_1 > 0$ by Lemma 3.4. Hence, by (4.10) $W_{n+1}(u_0, u_1)u_0(n+1)u_1(n+1) > 0$ holds. In case (3) of Lemma 4.1 we have $\sin(\Gamma_1 - \Gamma_0) < 0$ and $\cos \gamma_0 \cos \gamma_1 < 0$ by Lemma 3.4. Hence, by (4.10)

$$W_{n+1}(u_0,u_1)u_0(n+1)u_1(n+1)>0(4.21)W\_\{n+1\}(u\_0,\; u\_1)u\_0(n+1)u\_1(n+1)\; >\; 0\; \backslash quad\; (4.21)$$

holds.

If (-1), then we either have case (1) of Lemma 4.1 and $\gamma_1 - \gamma_0 \in (0, \frac{\pi}{2})$ , $\Gamma_1 - \Gamma_0 \in (-\pi, 0]$ or we have case (2) of Lemma 4.1 and $\gamma_1 - \gamma_0 \in (-\pi, 0)$ , $\Gamma_1 - \Gamma_0 \in (-2\pi, -\pi]$ . Clearly, by (4.8), in either case we have

Hence, by Corollary 4.5 we have $u_0(n+1)u_1(n+1) \neq 0$ and thus $\cos \gamma_0 \cos \gamma_1 \neq 0$ . In case (1) of Lemma 4.1 we have $\sin(\gamma_1 - \gamma_0) > 0$ and $\cos \gamma_0 \cos \gamma_1 > 0$ by Lemma 3.4. Hence, by (4.9) $W_n(u_0, u_1)u_0(n+1)u_1(n+1) > 0$ holds. In case (2) of Lemma 4.1 we have $\sin(\gamma_1 - \gamma_0) < 0$ and $\cos \gamma_0 \cos \gamma_1 < 0$ by Lemma 3.4. Hence, by (4.9)

$$W_n(u_0,u_1)u_0(n+1)u_1(n+1)>0(4.23)W\_n(u\_0,\; u\_1)u\_0(n+1)u\_1(n+1)\; >\; 0\; \backslash quad\; (4.23)$$

holds.

On the other hand, if $W_n(u_0, u_1)W_{n+1}(u_0, u_1) < 0$ by (4.8) we have either (+1) or (-1). If, use (4.9),

$$W_n(u_0,u_1)u_0(n+1)u_1(n+1)=p\sin ({\gamma }_1-{\gamma }_0)\cos {\gamma }_0\cos {\gamma }_1>0,(4.24)W\_n(u\_0,\; u\_1)u\_0(n+1)u\_1(n+1)\; =\; p\; \backslash sin(\backslash gamma\_1\; -\; \backslash gamma\_0)\; \backslash cos\; \backslash gamma\_0\; \backslash cos\; \backslash gamma\_1\; >\; 0,\; \backslash quad\; (4.24)$$then we have either case (1) or case (2) of Lemma 4.1 and in each case we have (0) or $(-1)$ . Hence,

If, use (4.9),

then we have either case (1) or case (3) of Lemma 4.1 and in each case we have (0) or $(+1)$ . Hence,

If $W_n(u_0, u_1) = 0$ , $W_{n+1}(u_0, u_1) \neq 0$ , then we have case (1) of Lemma 4.1 and by Corollary 4.5 we have $\cos \gamma_0 \cos \gamma_1 > 0$ . Hence, if $W_{n+1}(u_0, u_1)u_0(n+1)u_1(n+1) > 0$ , then (4.10) implies $\sin(\Gamma_1 - \Gamma_0) > 0$ , thus, $(+1)$ holds by case (1) of Lemma 4.1. If $W_n(u_0, u_1) \neq 0$ , $W_{n+1}(u_0, u_1) = 0$ , then by Corollary 4.5 we have $\cos \gamma_0 \cos \gamma_1 \neq 0$ . If additionally $W_n(u_0, u_1)u_0(n+1)u_1(n+1) > 0$ holds, then by (4.9) $\cos \gamma_0 \cos \gamma_1$ and $\sin(\gamma_1 - \gamma_0)$ are of the same sign. Hence, we have case (1) of Lemma 4.1 and $(-1)$ or case (2) of Lemma 4.1 and $(-1)$ .

Thus, (4.17) and (4.18) hold and clearly by Lemma 4.2 we have (0) otherwise. $\square$

Remark 4.7. Consider (1.5), then

by Corollary 4.5. Moreover, if $W_n(u_0, u_1) = 0$ and $W_{n+1}(u_0, u_1) \neq 0$ holds, then $u_0(n) = 0 \iff u_1(n) = 0$ .

That (1.5) is a generalization of the counting method established in [2, (1.8)], where $\Delta a = 0$ holds, follows from (2.6). From Lemma 4.6 we conclude

$${\mathrm{\#}}_n(u_0,u_1)=\lceil \mathrm{\Delta }(n+1)\mathrm{/}\pi \rceil -\lceil \mathrm{\Delta }(n)\mathrm{/}\pi \rceil ,(4.27)\backslash \#\_n(u\_0,\; u\_1)\; =\; \backslash lceil\; \backslash Delta(n+1)/\backslash pi\; \backslash rceil\; -\; \backslash lceil\; \backslash Delta(n)/\backslash pi\; \backslash rceil,\; \backslash quad\; (4.27)$$

$${\mathrm{\#}}_{[m,n]}(u_0,u_1)=\lceil \mathrm{\Delta }(n)\mathrm{/}\pi \rceil -\lceil \mathrm{\Delta }(m)\mathrm{/}\pi \rceil \mathrm{.}(4.28)\backslash \#\_\{[m,n]\}(u\_0,\; u\_1)\; =\; \backslash lceil\; \backslash Delta(n)/\backslash pi\; \backslash rceil\; -\; \backslash lceil\; \backslash Delta(m)/\backslash pi\; \backslash rceil.\; \backslash quad\; (4.28)$$

Lemma 4.8. We have

$${\mathrm{\#}}_{(m,n]}(u_0,u_1)=\lceil \mathrm{\Delta }(n)\mathrm{/}\pi \rceil -\lfloor \mathrm{\Delta }(m)\mathrm{/}\pi \rfloor -1,(4.29)\backslash \#\_\{(m,n]\}(u\_0,\; u\_1)\; =\; \backslash lceil\; \backslash Delta(n)/\backslash pi\; \backslash rceil\; -\; \backslash lfloor\; \backslash Delta(m)/\backslash pi\; \backslash rfloor\; -\; 1,\; \backslash quad\; (4.29)$$

$${\mathrm{\#}}_{[m,n)}(u_0,u_1)=\lfloor \mathrm{\Delta }(n)\mathrm{/}\pi \rfloor -\lceil \mathrm{\Delta }(m)\mathrm{/}\pi \rceil +1,\text{ and}(4.30)\backslash \#\_\{[m,n)\}(u\_0,\; u\_1)\; =\; \backslash lfloor\; \backslash Delta(n)/\backslash pi\; \backslash rfloor\; -\; \backslash lceil\; \backslash Delta(m)/\backslash pi\; \backslash rceil\; +\; 1,\; \backslash text\{\; and\}\; \backslash quad\; (4.30)$$

$${\mathrm{\#}}_{(m,n)}(u_0,u_1)=\lfloor \mathrm{\Delta }(n)\mathrm{/}\pi \rfloor -\lfloor \mathrm{\Delta }(m)\mathrm{/}\pi \rfloor \mathrm{.}(4.31)\backslash \#\_\{(m,n)\}(u\_0,\; u\_1)\; =\; \backslash lfloor\; \backslash Delta(n)/\backslash pi\; \backslash rfloor\; -\; \backslash lfloor\; \backslash Delta(m)/\backslash pi\; \backslash rfloor.\; \backslash quad\; (4.31)$$

Proof. By (4.8) we have $W_j(u_0, u_1) = 0 \iff \Delta(j)/\pi \in \mathbb{Z}$ and hence by (4.28)

holds. The second and the third claim follow analogously. $\square$

Lemma 4.9. We have

$${\mathrm{\#}}_{[m,n]}(u_0,u_1)=-{\mathrm{\#}}_{(m,n)}(u_1,u_0),{\mathrm{\#}}_{(m,n]}(u_0,u_1)=-{\mathrm{\#}}_{[m,n)}(u_1,u_0)\mathrm{.}(4.33)\backslash \#\_\{[m,n]\}(u\_0,\; u\_1)\; =\; -\backslash \#\_\{(m,n)\}(u\_1,\; u\_0),\; \backslash quad\; \backslash \#\_\{(m,n]\}(u\_0,\; u\_1)\; =\; -\backslash \#\_\{[m,n)\}(u\_1,\; u\_0).\; \backslash quad\; (4.33)$$

If $W_m(u_0, u_1) \neq 0$ and $W_n(u_0, u_1) \neq 0$ , then $#{[m,n]}(u_0, u_1) = -#{[m,n]}(u_1, u_0)$ .

Proof. Use $\lceil x \rceil = -\lfloor -x \rfloor$ and Lemma 4.8. $\square$Proof of Theorem 1.1. By Lemma 3.8 and Lemma 4.8 we have

$$\begin{array}{r}{\displaystyle E_{(-\mathrm{\infty },{\lambda }_1)}(J_1)-E_{(-\mathrm{\infty },{\lambda }_0)}(J_0)}\\ {\displaystyle =\lceil {\mathrm{\Delta }}_{s_0,\pm ({\lambda }_0),s_{1,\mp }({\lambda }_1)}(N)\mathrm{/}\pi \rceil -\lfloor {\mathrm{\Delta }}_{s_0,\pm ({\lambda }_0),s_{1,\mp }({\lambda }_1)}(0)\mathrm{/}\pi \rfloor -1}\\ {\displaystyle ={\mathrm{\#}}_{(0,N]}(s_{0,\pm }({\lambda }_0),s_{1,\mp }({\lambda }_1))={\mathrm{\#}}_{(0,N]}(u_{0,\pm }({\lambda }_0),u_{1,\mp }({\lambda }_1))\mathrm{.}}\end{array}(4.34)\backslash begin\{aligned\}\; E\_\{(-\backslash infty,\; \backslash lambda\_1)\}(J\_1)\; -\; E\_\{(-\backslash infty,\; \backslash lambda\_0)\}(J\_0)\; \backslash \backslash \; =\; \backslash lceil\; \backslash Delta\_\{s\_0,\; \backslash pm(\backslash lambda\_0),\; s\_\{1,\; \backslash mp\}(\backslash lambda\_1)\}(N)/\backslash pi\; \backslash rceil\; -\; \backslash lfloor\; \backslash Delta\_\{s\_0,\; \backslash pm(\backslash lambda\_0),\; s\_\{1,\; \backslash mp\}(\backslash lambda\_1)\}(0)/\backslash pi\; \backslash rfloor\; -\; 1\; \backslash \backslash \; =\; \backslash \#\_\{(0,\; N]\}(s\_\{0,\; \backslash pm\}(\backslash lambda\_0),\; s\_\{1,\; \backslash mp\}(\backslash lambda\_1))\; =\; \backslash \#\_\{(0,\; N]\}(u\_\{0,\; \backslash pm\}(\backslash lambda\_0),\; u\_\{1,\; \backslash mp\}(\backslash lambda\_1)).\; \backslash end\{aligned\}\; \backslash quad\; (4.34)$$

Equations (1.10) can be shown analogously. $\square$

Remark 4.10. By Theorem 1.1 we have

$${\mathrm{\#}}_{[0,N]}(u_{0,\pm }(\lambda ),u_{1,\mp }({\lambda }_1))=-{\mathrm{\#}}_{[0,N]}(u_{1,\pm }(\lambda ),u_{0,\mp }(\lambda )),(4.35)\backslash \#\_\{[0,\; N]\}(u\_\{0,\; \backslash pm\}(\backslash lambda),\; u\_\{1,\; \backslash mp\}(\backslash lambda\_1))\; =\; -\backslash \#\_\{[0,\; N]\}(u\_\{1,\; \backslash pm\}(\backslash lambda),\; u\_\{0,\; \backslash mp\}(\backslash lambda)),\; \backslash quad\; (4.35)$$

$$\begin{array}{r}{\displaystyle {\mathrm{\#}}_{[0,N]}(u_{0,+}(\lambda ),u_{3,-}(\lambda ))}\\ {\displaystyle ={\mathrm{\#}}_{[0,N]}(u_{0,+}(\lambda ),u_{1,-}(\lambda ))+{\mathrm{\#}}_{[0,N]}(u_{1,-}(\lambda ),u_{2,+}(\lambda ))+{\mathrm{\#}}_{(0,N]}(u_{2,+}(\lambda ),u_{3,-}(\lambda )),}\end{array}\backslash begin\{aligned\}\; \backslash \#\_\{[0,\; N]\}(u\_\{0,\; +\}(\backslash lambda),\; u\_\{3,\; -\}(\backslash lambda))\; \backslash \backslash \; =\; \backslash \#\_\{[0,\; N]\}(u\_\{0,\; +\}(\backslash lambda),\; u\_\{1,\; -\}(\backslash lambda))\; +\; \backslash \#\_\{[0,\; N]\}(u\_\{1,\; -\}(\backslash lambda),\; u\_\{2,\; +\}(\backslash lambda))\; +\; \backslash \#\_\{(0,\; N]\}(u\_\{2,\; +\}(\backslash lambda),\; u\_\{3,\; -\}(\backslash lambda)),\; \backslash end\{aligned\}$$

and

$$\begin{array}{r}{\displaystyle {\mathrm{\#}}_{[0,N]}(u_{0,-}(\lambda ),u_{3,+}(\lambda ))}\\ {\displaystyle ={\mathrm{\#}}_{(0,N]}(u_{0,-}(\lambda ),u_{1,+}(\lambda ))+{\mathrm{\#}}_{[0,N]}(u_{1,+}(\lambda ),u_{2,-}(\lambda ))+{\mathrm{\#}}_{[0,N]}(u_{2,-}(\lambda ),u_{3,+}(\lambda ))\mathrm{.}}\end{array}\backslash begin\{aligned\}\; \backslash \#\_\{[0,\; N]\}(u\_\{0,\; -\}(\backslash lambda),\; u\_\{3,\; +\}(\backslash lambda))\; \backslash \backslash \; =\; \backslash \#\_\{(0,\; N]\}(u\_\{0,\; -\}(\backslash lambda),\; u\_\{1,\; +\}(\backslash lambda))\; +\; \backslash \#\_\{[0,\; N]\}(u\_\{1,\; +\}(\backslash lambda),\; u\_\{2,\; -\}(\backslash lambda))\; +\; \backslash \#\_\{[0,\; N]\}(u\_\{2,\; -\}(\backslash lambda),\; u\_\{3,\; +\}(\backslash lambda)).\; \backslash end\{aligned\}$$

5. TRIANGLE INEQUALITY AND COMPARISON THEOREM

In this section we establish the Triangle Inequality and the Comparison Theorem for Wronskians which generalize Theorem 5.12 and Theorem 5.13 from [1] to different $a$ 's. Moreover, Theorem 5.3 generalizes and sharpens Theorem 5.11 from [1].

Theorem 5.1 (Comparison Theorem for Wronskians I). Let $J_1 \geq J_2$ , then,

$${\mathrm{\#}}_{[0,N]}(u_{0,\pm }(\lambda ),u_{2,\mp }(\lambda ))\ge {\mathrm{\#}}_{[0,N]}(u_{0,\pm }(\lambda ),u_{1,\mp }(\lambda )),(5.1)\backslash \#\_\{[0,\; N]\}(u\_\{0,\; \backslash pm\}(\backslash lambda),\; u\_\{2,\; \backslash mp\}(\backslash lambda))\; \backslash geq\; \backslash \#\_\{[0,\; N]\}(u\_\{0,\; \backslash pm\}(\backslash lambda),\; u\_\{1,\; \backslash mp\}(\backslash lambda)),\; \backslash quad\; (5.1)$$

where $#{[0, N]}$ can be replaced by $#{(0, N]}$ , $#{[0, N]}$ , or $#{(0, N]}$ .

Proof. Let $\sigma(J_1) = {\lambda_1, \dots, \lambda_{N-1}}$ and $\sigma(J_2) = {\tilde{\lambda}1, \dots, \tilde{\lambda}{N-1}}$ , then $\lambda_i \geq \tilde{\lambda}i$ for all $i$ by $J_1 \geq J_2$ , cf. [11, Theorem 8.7.1], and hence we have $E{(-\infty, \lambda)}(J_2) \geq E_{(-\infty, \lambda)}(J_1)$ . Thus, by Theorem 1.1

The other claims follow analogously from $E_{(-\infty, \lambda]}(J_2) \geq E_{(-\infty, \lambda]}(J_1)$ and Theorem 1.1. $\square$

Corollary 5.2. Let $a_0 = a_1 = a_2$ and $b_0(j) \geq b_1(j) \geq b_2(j)$ for all $j = 1, \dots, N-1$ . If 0 and $N-1$ are positive nodes of $W(u_{0, \pm}(\lambda), u_{1, \mp}(\lambda))$ , then $W(u_{0, \pm}(\lambda), u_{2, \mp}(\lambda))$ has at least two positive nodes at $0, \dots, N-1$ .

Theorem 5.3. Let $m < n$ , then

$$\mathrm{\mid }{\mathrm{\#}}_{[m,n]}(u_0,u_1)-({\mathrm{\#}}_{(m,n)}(u_1)-{\mathrm{\#}}_{(m,n)}(u_0))\mathrm{\mid }\le 1,(5.3)|\backslash \#\_\{[m,\; n]\}(u\_0,\; u\_1)\; -\; (\backslash \#\_\{(m,\; n)\}(u\_1)\; -\; \backslash \#\_\{(m,\; n)\}(u\_0))|\; \backslash leq\; 1,\; \backslash quad\; (5.3)$$

where $#{[m, n]}$ can be replaced by $#{(m, n]}$ or $#_{[m, n]}$ .

Proof. For all $x, y \in \mathbb{R}$ we have

$$0\le \lceil x-y\rceil -(\lceil x\rceil -\lceil y\rceil )\le 1\text{and}-1\le \lfloor x-y\rfloor -(\lfloor x\rfloor -\lfloor y\rfloor )\le 0.(5.4)0\; \backslash leq\; \backslash lceil\; x\; -\; y\; \backslash rceil\; -\; (\backslash lceil\; x\; \backslash rceil\; -\; \backslash lceil\; y\; \backslash rceil)\; \backslash leq\; 1\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; -1\; \backslash leq\; \backslash lfloor\; x\; -\; y\; \backslash rfloor\; -\; (\backslash lfloor\; x\; \backslash rfloor\; -\; \backslash lfloor\; y\; \backslash rfloor)\; \backslash leq\; 0.\; \backslash quad\; (5.4)$$

Hence, by (4.28), Theorem 3.6, and $-\lceil x \rceil = \lfloor -x \rfloor$ we have

Moreover, by Lemma 4.8 and Theorem 3.6 we have

and

□

Theorem 5.4 (Triangle Inequality for Wronskians). We have

$$\mathrm{\mid }{\mathrm{\#}}_{[m,n]}(u_0,u_2)-({\mathrm{\#}}_{[m,n]}(u_0,u_1)+{\mathrm{\#}}_{[m,n]}(u_1,u_2))\mathrm{\mid }\le 1,(5.6)|\backslash \#\_\{[m,n]\}(u\_0,\; u\_2)\; -\; (\backslash \#\_\{[m,n]\}(u\_0,\; u\_1)\; +\; \backslash \#\_\{[m,n]\}(u\_1,\; u\_2))|\; \backslash leq\; 1,\; \backslash quad\; (5.6)$$

where $#{[m,n]}$ can be replaced by $#{(m,n]}$ .

Proof. Abbreviate $\Delta_{i,j} := \Delta_{u_i, u_j}$ , then $\Delta_{0,1} + \Delta_{1,2} = \Delta_{0,2}$ . By (4.28) we have $#{[m,n]}(u_0, u_2) = \lceil \Delta{0,2}(n)/\pi \rceil - \lceil \Delta_{0,2}(m)/\pi \rceil$ , hence

holds by $\lceil x + y \rceil \leq \lceil x \rceil + \lceil y \rceil \leq \lceil x + y \rceil + 1$ for all $x, y \in \mathbb{R}$ . Further, by Lemma 4.8 and $\lfloor x + y \rfloor - 1 \leq \lfloor x \rfloor + \lfloor y \rfloor \leq \lfloor x + y \rfloor$ we have

and $#{(m,n]}(u_0, u_2) \leq #{(m,n]}(u_0, u_1) + #_{(m,n]}(u_1, u_2) + 1$ . □

Theorem 5.5 (Comparison Theorem for Wronskians II). If either

A: $W_j(u_0, u_1)u_0(j+1)u_1(j+1) \leq 0$ and $W_j(u_1, u_2)u_1(j+1)u_2(j+1) \leq 0$ for all $j = 0, \dots, N-2$ or

B: $a_0 = a_1 = a_2$ and $b_0(j) \geq b_1(j) \geq b_2(j)$ for all $j = 1, \dots, N-1$

holds and 0 and $N-2$ are (positive) nodes of $W(u_0, u_1)$ , then $W(u_0, u_2)$ has at least one positive node at $0, \dots, N-2$ .

Proof. In either case we have $#_j(u_0, u_1) \geq 0$ and $#_j(u_1, u_2) \geq 0$ for all $j = 0, \dots, N-2$ and hence from Theorem 5.4 we conclude

$${\mathrm{\#}}_{[0,N-1]}(u_0,u_2)\ge \underset{\ge 2}{\underbrace{{\mathrm{\#}}_{[0,N-1]}(u_0,u_1)}}+\underset{\ge 0}{\underbrace{{\mathrm{\#}}_{[0,N-1]}(u_1,u_2)}}-1.(5.9)\backslash \#\_\{[0,N-1]\}(u\_0,\; u\_2)\; \backslash geq\; \backslash underbrace\{\backslash \#\_\{[0,N-1]\}(u\_0,\; u\_1)\}\_\{\backslash geq\; 2\}\; +\; \backslash underbrace\{\backslash \#\_\{[0,N-1]\}(u\_1,\; u\_2)\}\_\{\backslash geq\; 0\}\; -\; 1.\; \backslash quad\; (5.9)$$

□

Acknowledgments. I wish to thank Gerald Teschl for several useful discussions.REFERENCES

• [1] K. Ammann, Relative oscillation theory for Jacobi operators, diploma thesis, http://othes.univie.ac.at/2534/, 2008.
• [2] K. Ammann and G. Teschl, Relative oscillation theory for Jacobi matrices, Proceedings of the 14th International Conference on Difference Equations and Applications, M. Bohner (ed) et al., 105–115, Uğur–Bahçeşehir Univ. Publ. Co., Istanbul, 2009.
• [3] Yu. V. Eliseeva, Spectra of Discrete Symplectic Eigenvalue Problems with Separated Boundary Conditions, (Russian) Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, pp. 84–88, 2011; translation in Russian Math. (Iz VUZ) 55, No. 11, 71–75, 2011.
• [4] J. Elyseeva, A note on relative oscillation theory for symplectic difference systems with general boundary conditions, Applied Mathematics Letters (2012), to appear.
• [5] J. Elyseeva, On relative oscillation theory for symplectic eigenvalue problems, Appl. Math. Lett. 23, 1231–1237, 2010.
• [6] T. Fort, Finite Differences and Difference Equations in the Real Domain, Oxford University Press, London, 1948.
• [7] H. Krüger, On Perturbations of Quasi-Periodic Schrödinger operators, J. Diff. Eq. 249:6, 1305–1321, 2010.
• [8] H. Krüger and G. Teschl, Effective Prüfer angles and relative oscillation criteria, J. Diff. Eq. 245, 3823–3848, 2008.
• [9] H. Krüger and G. Teschl, Relative oscillation theory for Sturm–Liouville operators extended, J. Funct. Anal. 254–6, 1702–1720, 2008.
• [10] H. Krüger and G. Teschl, Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function, Commun. Math. Phys. 287:2, 613–640, 2009.
• [11] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Sc. and Applied Math., Academic Press, San Diego, 1985.
• [12] B. Simon, Sturm oscillation and comparison theorems, Sturm–Liouville Theory: Past and Present, 29–43, Birkhäuser Verlag, Basel, 2005.
• [13] R. Šimon Hilscher, Spectral and oscillation theory for general second order Sturm–Liouville difference equations, in: ‘Oscillation of Difference, Differential, and Dynamic Equations’, M. Bohner, Z. Dosla, and S. Pinelas, editors, Advances in Difference Equations 2012, 2012:82, to appear.
• [14] R. Stadler and G. Teschl, Relative oscillation theory for Dirac operators, J. Math. Anal. Appl. 371, 638–648, 2010.
• [15] J.C.F. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1, 106–186, 1836.
• [16] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000.
• [17] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Grad. Stud. Math. 140, Amer. Math. Soc., Rhode Island, 2012.
• [18] G. Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Diff. Eqs. 129, 532–558, 1996.

FACULTY OF MATHEMATICS, UNIVERSITY OF VIENNA, NORDBERGSTRASSE 15, 1090 WIEN, AUSTRIA

E-mail address: Kerstin.Ammann@univie.ac.at

URL: http://www.mat.univie.ac.at/~kerstin/