Tunneling for a Class of Difference Operators: Complete Asymptotics

Abstract

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 derive full asymptotic expansions of the prefactor of the exponentially small eigenvalue splitting due to interactions between two “wells” (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both the case where there is a single minimal geodesic (with respect to the natural Finsler metric induced by the leading symbol $h_0(x,\xi )$ of $H_\varepsilon$) connecting the two minima and the case where the minimal geodesics form an $\ell +1$ dimensional manifold, $\ell \ge 1$. These results on the tunneling problem are as sharp as the classical results for the Schrödinger operator in Helffer and Sjöstrand (Commun PDE 9:337–408, 1984). Technically, our approach is pseudo-differential and we adapt techniques from Helffer and Sjöstrand [Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique), Mémoires de la S.M.F., 2 series, tome 34, pp 1–113, 1988)] and Helffer and Parisse (Ann Inst Henri Poincaré 60(2):147–187, 1994) to our discrete setting.