Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schr\"odinger operators

TL;DR

This paper establishes a scale-free unique continuation principle for Schr"odinger operators, enabling new Wegner estimates for random operators and insights into control costs for heat equations in complex domains.

Contribution

It introduces a scale-free, quantitative unique continuation principle for spectral projectors of Schr"odinger operators, with applications to Wegner estimates and control theory.

Findings

Proves a scale-free unique continuation estimate for spectral projectors.

Derives Wegner estimates for random Schr"odinger operators with nonlinear parameters.

Analyzes control cost dependence in heat equations on multi-scale domains.

Abstract

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector $\chi_{(-\infty,E]}(H_L)$ of a Schr\"odinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schr\"odinger operators with non-linear parameter-dependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-scale domain.