A comprehensive proof of localization for continuous Anderson models with singular random potentials

TL;DR

This paper proves a comprehensive form of localization for continuous Anderson models with singular potentials, including spectral, dynamical, and correlation decay properties, without regularity assumptions on the potential distribution.

Contribution

It establishes a strong localization result for continuous Anderson Hamiltonians with minimal assumptions on the potential distribution, extending previous results to singular cases.

Findings

Proves Anderson localization with pure point spectrum and exponential decay of eigenfunctions.

Demonstrates dynamical localization with no wave packet spreading.

Shows decay of eigenfunction correlations and Fermi projections.

Abstract

We study continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any regularity condition on the single site probability distribution. We prove the existence of a strong form of localization at the bottom of the spectrum, which includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) with finite multiplicity of eigenvalues, dynamical localization (no spreading of wave packets under the time evolution), decay of eigenfunctions correlations, and decay of the Fermi projections. We also prove log-H\" older continuity of the integrated density of states at the bottom of the spectrum.