Quantitative unique continuation for spectral subspaces of Schr\"odinger operators with singular potentials

Abstract

Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schr\"odinger operators are extended to allow singular potentials such as certain $L^p$-functions. The proof is based on accordingly adapted Carleman estimates. Applications include Wegner and initial length scale estimates for random Schr\"odinger operators and control theory for the controlled heat equation with singular heat generation term.