Analysis of the Anderson operator

Abstract

We consider the continuous Anderson operator H = ∆ + ξ on a two dimensional closed Riemannian manifold S. We provide a short self-contained functional analysis construction of the operator as an unbounded operator on L 2 (S) and give almost sure spectral gap estimates under mild geometric assumptions on the Riemannian manifold. We prove a sharp Gaussian small time asymptotic for the heat kernel of H that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called Anderson Gaussian free field, and prove that the law of its random partition function characterizes the law of the spectrum of H. We also give a simple and short construction of the polymer measure on path space and relate the Wick square of the Anderson Gaussian free field to the occupation measure of a Poisson process of loops of polymer paths. We further prove large deviation results for the polymer measure and its bridges. Contents c 1+ln(µ(M)/2) 1/α as a lower bound for the spectral radius. We also note that our method of proof gives gives the same conclusion as in Theorem 48 when working on a manifold equipped with a positive smooth density and its associated Laplace operator, or when working on a graph or a Dirichlet space.

Variational methods for some singular stochastic elliptic PDEs

“…We will not need in the present work any of the technical details associated with the use of paracontrolled calculus or regularity structures. We only mention from the works [11,3] the following facts that we will freely use below. Recall α − 2 < −1 stands for the almost sure regularity of white noise.…”

“…(See e.g. Section 2 of [11] or Section 3 of [3].) -There exists a random constant c such that the quadratic form associated with the operator −H + c is positive definite.…”

“…Mouzard further simplified the approach of [8] and constructed the operator over an arbitrary closed two dimensional Riemannian manifold. A deep study of the Anderson operator and some associated objects was done recently by Bailleul, Dang & Mouzard in [3]. In any case one is able to define H as a closed symmetric unbounded operator with random domain and compact resolvent.…”

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

“…One can compare the initial condition in [18,43] and that in this paper and in [12,13,41,42], where the initial data is chosen to have a specific structure e −Y v 0 with v 0 belonging to a weighted H 2 space. For more discussions on the Anderson Hamiltonian, see [1,3,10,29,32].…”