Let H := − 1 2 ∆ + V be a one-dimensional continuum Schrödinger operator. Consider H := H +ξ, 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 e −tĤ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line R, the half line (0, ∞), 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 whereĤ is the stochastic Airy operator.
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