Discrete Schrödinger operators and Finsler metric (Spectral and Scattering Theory and Related Topics)

Abstract

In this article, we review the results in [9] on the Agmon estimate for discrete Schrödinger operators. We first discuss the semiclassical analysis for discrete Schrödinger operators with emphasis on the microlocal analysis on the torus. We discretize a semiclassical continuous Schrödinger operator with mesh size proportional to the semiclassical parameter. Under this setting, we show the Agmon estimate for eigenfunctions. The natural Agmon metric for the discrete Schrödinger operator is a Finsler metric rather than a Riemannian metric. It turned out that Klein-Rosenberger (2008) already discussed the semiclassical Agmon estimate in terms of the same Finsler metric by a different argument in the special case of a potential minimum. We also show the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schrödinger operators in the non-semiclassical standard setting.