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In this paper, we consider self-adjoint difference equations of the form-\Delta(a_{n-1}\Delta y_{n-1})+b_{n}y_{n}=\lambday_{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 associatedHamburger moment problem is determined), we prove that the existence of anexponentially bounded solution of \eqref{eq:abstract} implies a bound on thedistance from $\lambda$ to the spectrum of the associated self-adjointoperator, and that if a solution of \eqref{eq:abstract} is bounded by a powerof n for n sufficiently large, then $\lambda\in\sigma(B)$. Here, $B$ is acertain self-adjoint operator generated by \eqref{eq:abstract}. These resultsare the difference equation version of differential operator results of Shnol'.We use this to then prove that the spectrum of the associated orthogonalpolynomials contains the closure of the set of $\lambda$ for which we can finda polynomially bounded solution. We further conjecture that under thehypotheses 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 aresult concerning the invariance of the essential spectrum under weakperturbation of the coefficients.

Spectral Analysis of a Class of Self-Adjoint Difference Equations