Asymptotic Eigenfunctions for a class of Difference Operators

Abstract

We analyze a general class of difference operators $H_{\epsilon }=T_{\epsilon }+V_{\epsilon }$ on ${\ell }^2(\epsilon {\mathbb{Z}}^d)$, where $V_{\epsilon }$ is a one-well potential and $\epsilon$ is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of $H_{\epsilon }$. These are obtained from eigenfunctions or quasimodes for the operator $H_{\epsilon }$, acting on $L^2({\mathbb{R}}^d)$, via restriction to the lattice $\epsilon {\mathbb{Z}}^d$.