Quantitative Anderson localization of Schr\"odinger eigenstates under disorder potentials

TL;DR

This paper investigates how disorder in high-amplitude potentials affects the spectral properties of Schr"odinger operators, demonstrating the existence of spectral gaps and localized eigenstates with decay rates linked to potential geometry.

Contribution

It provides a rigorous analysis of spectral gaps and localization phenomena in Schr"odinger operators with disordered potentials, using convergence theory and domain decomposition methods.

Findings

Existence of spectral gaps among lowermost eigenvalues

Emergence of exponentially localized eigenstates

Decay rates quantified by geometric parameters

Abstract

This paper concerns spectral properties of linear Schr\"odinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.