  Let $ H:=-\tfrac12\Delta+V$ be a one-dimensional continuum Schr\"odinger
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.
