The continuous Anderson hamiltonian in $d\le 3$

TL;DR

This paper constructs the continuous Anderson Hamiltonian in dimensions up to three using regularity structures, establishing its self-adjointness, spectrum properties, and tail estimates of eigenvalues, including the necessity of renormalization in higher dimensions.

Contribution

It provides a rigorous construction of the Anderson Hamiltonian in dimensions up to three, including renormalization techniques and spectral analysis, extending previous work to higher dimensions.

Findings

Constructed a self-adjoint Anderson Hamiltonian in $d\,\leq\,3$

Established pure point spectrum for the operator

Derived tail estimates showing eigenvalues lack exponential moments in $d=3$

Abstract

We construct the continuous Anderson hamiltonian on $(-L,L)^d$ driven by a white noise and endowed with either Dirichlet or periodic boundary conditions. Our construction holds in any dimension $d\le 3$ and relies on the theory of regularity structures: it yields a self-adjoint operator in $L^2\big((-L,L)^d\big)$ with pure point spectrum. In $d\ge 2$, a renormalisation of the operator by means of infinite constants is required to compensate for ill-defined products involving functionals of the white noise. We also obtain left tail estimates on the distributions of the eigenvalues: in particular, for $d=3$ these estimates show that the eigenvalues do not have exponential moments.