Tunneling for a Class of Difference Operators

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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 multi-well potential and \({\varepsilon}\) is a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we shall treat the eigenvalue problem for \({H_\varepsilon}\) as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schrödinger operator [see Helffer and Sjöstrand in Commun Partial Differ Equ 9:337–408, 1984], and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.