The continuous Anderson hamiltonian in dimension two

Abstract

We define the Anderson hamiltonian on the two dimensional torus ${\mathbb{R}}^2/{\mathbb{Z}}^2$. This operator is formally defined as $\mathcal{H}:=-\Delta +\xi$ 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 Schrödinger operator $\mathcal{H}$ 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 ${\mathcal{H}}_{\epsilon }:=-\Delta +{\xi }_{\epsilon }+c_{\epsilon }$ (where ${\xi }_{\epsilon }$ is a smooth mollification of the Gaussian white noise $\xi$ and $c_{\epsilon }$ an explicit diverging renormalization constant) converge in the sense of the resolvent towards the singular operator $\mathcal{H}$. 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 $\mathcal{H}$ 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 $L\to \infty$.

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• Beta ensembles, stochastic Airy spectrum, and a diffusion
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