On the spectral analysis of self adjoint operators generated by second order difference equations

Abstract

SynopsisIn this paper, I shall consider operators generated by difference equations of the formwhere Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.