Asymptotic eigenfunctions for a class of difference operators

We analyze a general class of difference operators H_ε=T_ε+V_ε on ℓ²((εZ)^d), where V_ε is a one-well potential and ε is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of H_ε. These are obtained from eigenfunctions or quasimodes for the operator H_ε, acting on L²(R^d), via restriction to the lattice (εZ)^d.