Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas

Abstract

We develop a technique for proving number rigidity (in the sense of Ghosh-Peres) of the spectrum of general random Schrödinger 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 $-\frac{1}{2}\hat{I}”+V+\hat{I}¾$, where $V$ is a deterministic potential and $\hat{I}¾$ 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 $\hat{I}¾$.

30 pages; to appear in Annales Henri Poincaré (Theoretical and Mathematical Physics)