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

Abstract

We analyze a general class of difference operators ${\mathit{H}}_{\mathit{\epsilon }}={\mathit{T}}_{\mathit{\epsilon }}+{\mathit{V}}_{\mathit{\epsilon }}$ on ${\mathrm{\ell }}^2\mathrm{(}{\mathrm{(}\mathit{\epsilon }\mathbb{Z}\mathrm{)}}^{\mathit{d}}\mathrm{)}$ , where ${\mathit{V}}_{\mathit{\epsilon }}$ is a multi-well potential and ε 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.