Spectral Analysis of a Class of Self-Adjoint Difference Equations

TL;DR

This paper analyzes the spectral properties of a class of self-adjoint difference equations, establishing bounds on the spectrum based on the growth of solutions and extending differential operator spectral results to difference equations.

Contribution

It proves a spectral characterization for self-adjoint difference equations using bounded solutions, extending Shnol's theorem to the discrete setting, and discusses spectrum invariance under perturbations.

Findings

Existence of exponentially bounded solutions bounds the spectrum.

Polynomially bounded solutions imply spectral inclusion.

Spectrum contains the closure of λ-values with polynomially bounded solutions.

Abstract

In this paper, we consider self-adjoint difference equations of the form -\Delta(a_{n-1}\Delta y_{n-1})+b_{n}y_{n}=\lambda y_{n},n=0,1,...\label{eq:abstract} where $a_{n-1}>0$ for all $n\ge0$ and $b_{n}$ are real and $\lambda$ is complex. Under the assumption that $a_{n-1}$ satisfies certain growth conditions and is limit point (that is, the associated Hamburger moment problem is determined), we prove that the existence of an exponentially bounded solution of \eqref{eq:abstract} implies a bound on the distance from $\lambda$ to the spectrum of the associated self-adjoint operator, and that if a solution of \eqref{eq:abstract} is bounded by a power of n for n sufficiently large, then $\lambda\in\sigma(B)$. Here, $B$ is a certain self-adjoint operator generated by \eqref{eq:abstract}. These results are the difference equation version of differential operator results of Shnol'. We use this to then prove that the spectrum of the associated orthogonal polynomials contains the closure of the set of $\lambda$ for which we can find a polynomially bounded solution. We further conjecture that under the hypotheses in the paper, the spectrum is precisely the closure of this set of $\lambda$ for which there is a polynomially bounded solution. We also present a result concerning the invariance of the essential spectrum under weak perturbation of the coefficients.