Semigroups for One-Dimensional Schrödinger Operators with Multiplicative Gaussian Noise

Abstract

Let $H:=-\frac{1}{2}\Delta +V$ be a one-dimensional continuum Schrödinger operator. Consider $\hat{H}:=H+\xi$, where $\xi$ is a translation invariant Gaussian noise. Under some assumptions on $\xi$, we prove that if $V$ is locally integrable, bounded below, and grows faster than $\log$ at infinity, then the semigroup ${\mathrm{e}}^{-t\hat{H}}$ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line $\mathbb{R}$, the half line $(0,\infty )$, or a bounded interval $(0,b)$, with a variety of boundary conditions. Our method of proof consists of a comprehensive generalization of techniques recently developed in the random matrix theory literature to tackle this problem in the special case where $\hat{H}$ is the stochastic Airy operator.

47 pages, 2 figures. Final version, accepted for publication in the Electronic Journal of Probability

References in corpus (4)

• From Random Matrices to Stochastic Operators
• Rigidity of Determinantal Point Processes with the Airy, the Bessel and the Gamma Kernel
• Operator limits of random matrices
• Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas

Cited by in corpus (3)

• Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model
• Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas
• Killed Rough Super-Brownian Motion