# Agmon-Type Estimates for a Class of Difference Operators

## TL;DR

This paper extends Agmon estimates to a class of difference operators, demonstrating exponential decay of eigenfunctions governed by a Finslerian metric, analogous to semiclassical Schrödinger operators.

## Contribution

It introduces a Finslerian distance for difference operators and proves exponential decay of eigenfunctions based on this metric, generalizing classical Agmon estimates.

## Key findings

- Constructed a Finslerian distance for difference operators
- Proved eigenfunction decay rates are controlled by this distance
- Established the geometric structure of geodesics for the operators

## Abstract

We analyze a general class of self-adjoint difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct a Finslerian distance $d$ induced by $H_\varepsilon$ 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 Schr\"odinger operators.