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 $-\frac{1}{2}\Delta +V+\xi$, 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 $\xi$.

arXiv comment30 pages; to appear in Annales Henri Poincaré (Theoretical and Mathematical Physics)

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Cited by in corpus (1)

• Semigroups for One-Dimensional Schrödinger Operators with Multiplicative Gaussian Noise