Abstract
Let H:=−12Δ+V be a one-dimensional continuum Schrödinger operator. Consider Hˆ:=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−tHˆ 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 where Hˆ is the stochastic Airy operator.
Highlights
Let I ⊂ R be an open interval and V : I → R be a function.denote a Schrödinger operator with potential V acting on functions f : I → R with prescribed boundary conditions when I has a boundary
Provided the potentials under consideration are sufficiently well behaved, there is a remarkable connection between Schrödinger semigroups and the theory of stochastic processes that can be expressed in the form of the Feynman-Kac formula (e.g., [41, Theorem A.2.7]): Assuming I = R for simplicity, for every f ∈ L2(R), t > 0, and x ∈ R, one has t e−tH f (x) = Ex exp − V B(s) ds f B(t) where B is a Brownian motion and Ex signifies that we are taking the expected value with respect to B conditioned on the starting point B(0) = x
Our purpose in this paper is to lay out the foundations of a general semigroup theory for random Schrödinger operators of the form (1.1)
Summary
Let I ⊂ R be an open interval (possibly unbounded) and V : I → R be a function. denote a Schrödinger operator with potential V acting on functions f : I → R with prescribed boundary conditions when I has a boundary. Semigroups for 1D Schrödinger operators with Gaussian noise distribution, i.e., a centered Gaussian process on an appropriate function space with covariance. Provided the potentials under consideration are sufficiently well behaved, there is a remarkable connection between Schrödinger semigroups and the theory of stochastic processes that can be expressed in the form of the Feynman-Kac formula (e.g., [41, Theorem A.2.7]): Assuming I = R for simplicity, for every f ∈ L2(R), t > 0, and x ∈ R, one has t e−tH f (x) = Ex exp − V B(s) ds f B(t). Our purpose in this paper is to lay out the foundations of a general semigroup theory (or Feynman-Kac formulas) for random Schrödinger operators of the form (1.1). Since we consider very irregular noises (i.e., in general ξ is not a proper function that can be evaluated at points in R), this undertaking is not a direct application or a trivial extension of the classical theory; see Section 1.1 for more details.
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