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On the finiteness of the number of eigenvalues of Jacobi operators below the essential spectrum. (English) Zbl 1049.39027

The aim of the paper under review is to determine whether the number of eigenvalues below the essential spectrum of the Jacobi operator on ${\ell }^{2}\left(ℕ\right)$ associated with

$\left(\tau f\right)\left(n\right)=a\left(n\right)f\left(n+1\right)+a\left(n-1\right)f\left(n-1\right)-b\left(n\right)f\left(n\right),$

where $a\left(n\right)\in ℝ\setminus \left\{0\right\},b\left(n\right)\in ℝ$ for $n\in ℕ$, is finite or not. $\tau$ is said to be oscillatory if one (and hence any) real-valued solution of $\tau u=0$ has an infinite number of nodes, that is, points $n\in ℕ$ such that either $u\left(n\right)=0$ or $a\left(n\right)u\left(n\right)u\left(n+1\right)>0$. According to I. M. Glazman [Direct methods of qualitative spectral analysis of singular differential operators. (Translated from the Russian) 1965], the finiteness of the number of eigenvalues is equivalent to the operator being nonoscillatory. The precise relation between the number of eigenvalues and the number of nodes was established by G. Teschl [J. Differ. Equations 129, No. 2, 532–558 (1996; Zbl 0866.39002)]. So, oscillation theory is a natural tool for investigation of questions connected with a number of eigenvalues below the essential spectrum. A positive solution ${u}_{0}\left(n\right)$ of $\tau u=0$ is said to be minimal if $\underset{n\to \infty }{lim}{Q}_{0}\left(n\right)=\infty$, where

${Q}_{0}\left(n\right):=-\sum _{j=0}^{n-1}1/\left(a\left(j\right){u}_{0}\left(j\right){u}_{0}\left(j+1\right)\right)\phantom{\rule{1.em}{0ex}}\left(n\in ℕ\right)·$

The authors prove the following main result.

Theorem. Suppose $b\left(n\right),{b}_{0}\left(n\right)\in ℝ$ and $a\left(n\right)\in ℝ\setminus \left\{0\right\}$ are such that ${a}_{0}<|a\left(n\right)|<{A}_{0}$ for some real constants $0<{a}_{0}<{A}_{0}$. Let ${u}_{0}$ be a nondecreasing minimal positive solution of $a\left(n\right){u}_{0}\left(n+1\right)+a\left(n-1\right){u}_{0}\left(n-1\right)-{b}_{0}\left(n\right){u}_{0}\left(n\right)=0$ and

$B\left(n\right):=-\frac{2a\left(n-1\right)a\left(n+1\right)}{a\left(n-1\right)+a\left(n+1\right)}{u}_{0}^{4}\left(n\right){Q}_{0}^{2}\left(n\right)\left[b\left(n\right)-{b}_{0}\left(n\right)\right]·$

If ${lim inf}_{n\to \infty }B\left(n\right)>-1/4$, then $\tau$ is nonoscillatory; if ${lim sup}_{n\to \infty }B\left(n\right)<-1/4$, then $\tau$ is oscillatory.

As an application of the obtained results, the authors prove a discrete analog of the theorem by A. Kneser [Math. Ann. XLII. 409–435 (1893; JFM 25.0522.01)] for Sturm-Liouville operators.

##### MSC:
 39A70 Difference operators 34B24 Sturm-Liouville theory 34L05 General spectral theory for OD operators 39A12 Discrete version of topics in analysis