Semiclassical analysis and the Agmon-Finsler metric for discrete Schr\"odinger operators

TL;DR

This paper investigates the Agmon estimates for discrete Schr"odinger operators, revealing that the decay of eigenfunctions is governed by a Finsler metric, extending prior results to a broader setting.

Contribution

It establishes Agmon estimates and optimal decay rates for discrete Schr"odinger operators using microlocal analysis and introduces the Agmon-Finsler metric framework.

Findings

Agmon estimates hold for discrete Schr"odinger operators.

Eigenfunction decay is characterized by a Finsler metric.

Optimal anisotropic exponential decay is proven.

Abstract

The Agmon estimate for multi-dimensional discrete Schr\"{o}dinger operators is studied with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schr\"{o}dinger operators are discretized with the mesh width proportional to the semiclassical parameter. Under this setting, the Agmon estimate for eigenfunctions is described by an Agmon metric, which is a Finsler metric rather than a Riemannian metric. Klein-Rosenberger (2008) proved this by a different argument in the case of a potential minimum. We also prove the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schr\"{o}dinger operators in the non-semiclassical standard setting.