Semigroups for One-Dimensional Schr\"odinger Operators with Multiplicative Gaussian Noise

Let $H:=-\frac{1}{2}\mathrm{\Delta }+V$ be a one-dimensional continuum Schr"odinger operator. Consider $\widehat{H}:=H+\xi$, 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 ${\mathrm{e}}^{-t\widehat{H}}$ 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 $\widehat{H}$ is the stochastic Airy operator.