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

Published online by Cambridge University Press:  14 November 2011

Dale T. Smith
Dale T. Smith
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203, U.S.A
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Synopsis

In this paper, I shall consider operators generated by difference equations of the form

where 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 1-2 , 1991 , pp. 139 - 151
Copyright
Copyright © Royal Society of Edinburgh 1991

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