Uniform resolvent estimates for the discrete Schr\"odinger operator in dimension three

TL;DR

This paper establishes uniform resolvent estimates for the three-dimensional discrete Schrödinger operator by analyzing the Fourier decay properties of the Fermi surface measure.

Contribution

It provides the first proof of uniform resolvent estimates in three dimensions for the discrete Schrödinger operator using Fourier decay techniques.

Findings

Proved uniform resolvent estimates in 3D

Analyzed Fourier decay of Fermi surface measure

Established connection between Fourier decay and resolvent bounds

Abstract

In this note, we prove the uniform resolvent estimate of the discrete Schr\"odinger operator with dimension three. To do this, we show a Fourier decay of the surface measure on the Fermi surface.