2019/02/13 by Pierre Yves Gaudreau Lamarre, Lamarre, Pierre Yves Gaudreau
Mathematics · #47D08 #47H40 #60J55 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1902.05047
openalex publication_date 2019/02/13 · openalex created_date 2020/12/07 · openalex updated_date 2026/07/28
Let H:=-\tfrac12Δ+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 \mathrm e^-t 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,∞), 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 H is the stochastic Airy operator.