Agmon-Type Estimates for a Class of Difference Operators

Abstract

We analyze a general class of self-adjoint difference operators $H_{\epsilon }=T_{\epsilon }+V_{\epsilon }$ on ${\ell }^2(\epsilon {\mathbb{Z}}^d)$, where $V_{\epsilon }$ is a one-well potential and $\epsilon$ is a small parameter. We construct a Finslerian distance $d$ induced by $H_{\epsilon }$ and show that short integral curves are geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by the Finsler distance to the well. This is analog to semiclassical Agmon estimates for Schrödinger operators.

27 pages

Cited by in corpus (1)

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