The continuous Anderson hamiltonian in dimension two

TL;DR

This paper rigorously constructs the two-dimensional Anderson Hamiltonian with singular potentials like Gaussian white noise, proving spectral properties, convergence of approximations, and tail bounds for the ground state.

Contribution

It introduces a method to define and analyze the 2D Anderson Hamiltonian with singular potentials using paracontrolled distributions, establishing spectral discreteness and convergence of regularized operators.

Findings

The spectrum of the operator is discrete with no accumulation points.

The spectrum depends continuously on the enhanced potential.

The regularized operators converge to the singular operator in the resolvent sense.

Abstract

We define the Anderson hamiltonian on the two dimensional torus $\mathbb R^2/\mathbb Z^2$. This operator is formally defined as $\mathscr H:= -\Delta + \xi$ where $\Delta$ is the Laplacian operator and where $\xi$ belongs to a general class of singular potential which includes the Gaussian white noise distribution. We use the notion of paracontrolled distribution as introduced by Gubinelli, Imkeller and Perkowski in [14]. We are able to define the Schr\"odinger operator $\mathscr H$ as an unbounded self-adjoint operator on $L^2(\mathbb T^2)$ and we prove that its real spectrum is discrete with no accumulation points for a general class of singular potential $\xi$. We also establish that the spectrum is a continuous function of a sort of enhancement $\Xi(\xi)$ of the potential $\xi$. As an application, we prove that a correctly renormalized smooth approximations $\mathscr H_\varepsilon:= -\Delta + \xi_\varepsilon+c_\varepsilon$ (where $\xi_\varepsilon$ is a smooth mollification of the Gaussian white noise $\xi$ and $c_\varepsilon$ an explicit diverging renormalization constant) converge in the sense of the resolvent towards the singular operator $\mathscr H$. In the case of a Gaussian white noise $\xi$, we obtain exponential tail bounds for the minimal eigenvalue (sometimes called ground state) of the operator $\mathscr H$ as well as its order of magnitude $\log L$ when the operator is considered on a large box $\mathbb T_L:= \mathbb R^2/(L\mathbb Z)^2$ with $L\to \infty$.