The landscape law for tight binding Hamiltonians

TL;DR

This paper extends landscape theory to tight-binding Schrödinger operators on rac{Z}{d}, providing bounds for the integrated density of states using the localization landscape, advancing understanding of spectral properties in lattice systems.

Contribution

It introduces bounds for the integrated density of states in tight-binding models using the localization landscape, expanding the applicability of landscape theory.

Findings

Established upper and lower bounds for the integrated density of states.

Extended landscape theory to lattice Schrödinger operators.

Provided analytical tools for spectral analysis in tight-binding systems.

Abstract

The present paper extends the landscape theory pioneered in [FM, ADFJM2, DFM] to the tight-binding Schr\"odinger operator on $\Z^d$. In particular, we establish upper and lower bounds for the integrated density of states in terms of the counting function based upon the localization landscape.