Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise

Abstract

Let $H:=-\frac{1}{2}\mathrm{\Delta }+V$ be a one-dimensional continuum Schrödinger operator. Consider $\hat{H}:=H+\mathrm{\xi }$, where ξ is a translation invariant Gaussian noise. Under some assumptions on ξ, 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 $(0,\mathrm{\infty })$, 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.