Asymptotic Eigenfunctions for a class of difference operators

We analyze a general class of difference operators <tex>H_\varepsilon = T_\varepsilon+ V_\varepsilon</tex> on <tex>\ell^2(\varepsilon\mathbb{Z}^d</tex>), where <tex>V_\varepsilon</tex> is a one-well potential and <tex>\varepsilon</tex> is a small parameter.

We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of <tex>H_\varepsilon</tex>. These are obtained from eigenfunctions or quasimodes for the operator <tex>H_\varepsilon</tex>, acting on <tex>L^2(R^d)</tex>, via restriction to the lattice <tex>\varepsilon Z^d</tex>.