Title: Asymptotic Eigenfunctions for a class of Difference Operators,
Abstract: 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
$\varepsilon\mathbb{Z}^d$.