Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise

Abstract

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.

Rigidity of the Stochastic Airy Operator

“…The proof for the B t,u (x, y) part is similar, see also [17,Lemma 5.6] for a similar statement. for some constants c, C > 0 independent of t. Thus by independence we have E e 4At,u(x,y) Therefore, we can find constants C, c > 0 such that for any t ∈ (0, 1] and u > 0 fixed we have (0,∞) 2 E e 4At,u(x,y) 1/4 e −(x−y) 2 /c dxdy ≤ C (0,∞) 2 e −yu/2−(x−y) 2 /c dxdy < ∞.…”

“…The spectrum of the stochastic Airy operator was first constructed using the form (2.1) in [3,30]. The following statement is a special case of [17, Propositions 3.2 and 3.4] (see also [26]): 3,17,26,30]). Let β > 0 and w ∈ (−∞, ∞].…”

“…We use L 0 t (Z) (t > 0) to denote the boundary local time of a process Z (or its conditioned versions) on the time interval [0, t], that is, 17,19,23]). For every t > 0,…”

“…Given that Brownian motion sample paths are not differentiable, a rigorous definition of H β is nontrivial. Nevertheless, it can be shown that once a suitable boundary condition is fixed, H β is a bonafide semibounded self-adjoint operator with compact resolvent when defined through its quadratic form [3,17,26,30]. We refer to Section 2.1 below for more details.…”

The stochastic Airy operator at large temperature

“…. of multiplicity one and the corresponding eigenfunctions (ψ k ) k≥1 , normalized in L 2 (0, ∞), are Hölder functions of regularity index 3/2 − , see [RRV11,Gau19]. Up to rescaling the eigenvalues / eigenfunctions appropriately (see Remark 1.1 below), it is equivalent to consider the operator…”