Random Schr\"odinger operator with singular potentials

TL;DR

This paper reviews the localization theory of random Schr"odinger operators with singular potentials, focusing on methods for both H"older-continuous and Bernoulli distributions in lattice and continuum models.

Contribution

It provides a comprehensive survey of techniques like Wegner estimates and unique continuation principles for singular potentials, expanding understanding beyond regular distributions.

Findings

Wegner estimates enable multiscale analysis for H"older-continuous laws.

Unique continuation principles address Bernoulli laws where spectral averaging fails.

The discussion covers both lattice and continuum models.

Abstract

We survey the localization theory of random Schr\"odinger operators with singular single-site distributions, focusing on two regimes: (i) H\"older-continuous laws, where quantitative Wegner estimates enable the classical multiscale analysis (MSA); and (ii) purely atomic (Bernoulli) laws, where the failure of spectral averaging is overcome via quantitative unique continuation principles (UCP). Our discussion covers both lattice and continuum settings and highlights the analytic and combinatorial mechanisms that replace regularity of the single-site measure.