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

TL;DR

This paper extends quantitative unique continuation estimates for spectral subspaces of Schr"odinger operators to include singular potentials, using adapted Carleman estimates, with applications in random Schr"odinger operators and control theory.

Contribution

It introduces new Carleman estimates that handle singular potentials in spectral analysis of Schr"odinger operators, broadening previous results.

Findings

Extended unique continuation estimates to singular potentials

Derived applications for Wegner and length scale estimates

Enhanced control theory for heat equations with singular terms

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.