Asymptotic Eigenfunctions for a class of Difference Operators

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