Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators

We analyze a general class of difference operators $H_{\epsilon }=T_{\epsilon }+V_{\epsilon }$ on ${\mathrm{\ell }}^2(\epsilon {\mathbb{Z}}^d)$, where $V_{\epsilon }$ is a multi-well potential and $\epsilon$ is a small parameter. We construct approximate eigenfunctions in neighbourhoods of the different wells and give weighted ${\mathrm{\ell }}^2$-estimates for the difference of these and the exact eigenfunctions of the associated Dirichlet-operators.