Scale-free uncertainty principles and Wegner estimates for random breather potentials

TL;DR

This paper develops scale-free uncertainty principles for Schrödinger operators and applies them to derive Wegner estimates for random breather potentials, addressing challenges from their non-linear dependence on randomness.

Contribution

It introduces new scale-free unique continuation principles and extends Wegner estimates to high energies for non-linear random breather potentials.

Findings

Established scale-free unique continuation principles.

Derived Wegner estimates valid at arbitrarily high energies.

Addressed non-linear dependence challenges in random potentials.

Abstract

We present new scale-free quantitative unique continuation principles for Schr\"odinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a prescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas-Molina & Veseli\'c, and Klein. We apply the scale-free unique continuation principle to obtain a Wegner estimate for a random Schr\"odinger operator of breather type. It holds for arbitrarily high energies. Schr\"odinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challenges arising from this non-linear dependence.