The aim of the paper under review is to determine whether the number of eigenvalues below the essential spectrum of the Jacobi operator on associated with
where for , is finite or not. is said to be oscillatory if one (and hence any) real-valued solution of has an infinite number of nodes, that is, points such that either or . 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 of is said to be minimal if , where
The authors prove the following main result.
Theorem. Suppose and are such that for some real constants . Let be a nondecreasing minimal positive solution of and
If , then is nonoscillatory; if , then 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.