Random Schr\"odinger operator with singular potentials

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.