Spectral Analysis of a Class of Self-Adjoint Difference Equations

Dale T. Smith

Published 2012-04-15, updated 2012-08-24Version 4

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.