Localization for Schr\"odinger operators with Poisson random potential

TL;DR

This paper establishes exponential and dynamical localization for Schr"odinger operators with Poisson random potentials at the spectrum's bottom, demonstrating finite eigenvalue multiplicity and localization in specific energy intervals.

Contribution

It provides new proofs of localization phenomena for Poisson random potentials, including finite multiplicity of eigenvalues and energy-dependent localization results.

Findings

Proves exponential and dynamical localization at the spectrum's bottom

Shows eigenvalues have finite multiplicity in localized regions

Establishes localization in energy intervals with high Poisson process density

Abstract

We prove exponential and dynamical localization for the Schr\"odinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.