Asymptotic spectral analysis and tunnelling for a class of difference operators

We analyze the asymptotic behavior in the limit <tex>\varepsilon \to 0</tex> for a wide class of difference operators <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> with underlying multi-well potential. They act on the square summable functions on the lattice <tex>\varepsilon \mathbb{Z}^d</tex>. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by <tex>H_\varepsilon</tex> and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.