Spectral rigidity of random Schr\"odinger operators via Feynman-Kac formulas

TL;DR

This paper introduces a new technique using Feynman-Kac formulas to prove spectral number rigidity for a broad class of one-dimensional random Schr"odinger operators with Gaussian noise, under mild assumptions.

Contribution

It develops a novel method leveraging Feynman-Kac formulas to establish spectral number rigidity for general RSOs with minimal regularity assumptions.

Findings

Proves number rigidity for a class of 1D RSOs with Gaussian noise.

Uses Feynman-Kac formulas to estimate variance of spectral statistics.

Applicable under mild regularity and boundary conditions.

Abstract

We develop a technique for proving number rigidity (in the sense of Ghosh-Peres) of the spectrum of general random Schr\"odinger operators (RSOs). Our method makes use of Feynman-Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman-Kac formulas for RSOs with multiplicative white noise by Gorin, Shkolnikov and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $-\frac12\Delta+V+\xi$, where $V$ is a deterministic potential and $\xi$ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential $V$, and the singularity of the noise $\xi$.