Tunneling for a Class of Difference Operators

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 Schr?dinger operator [see Helffer and Sj?strand 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.