Weyl law for the Anderson Hamiltonian on a two-dimensional manifold

TL;DR

This paper establishes a Weyl law for the Anderson Hamiltonian on a 2D manifold, using advanced calculus techniques to analyze its spectral properties and eigenvalue bounds.

Contribution

It introduces a rigorous definition of the Anderson Hamiltonian on a 2D manifold and proves a Weyl-type law for its eigenvalues.

Findings

Self-adjoint operator with pure point spectrum

Eigenvalue bounds established

Almost sure Weyl law proven

Abstract

We define the Anderson Hamiltonian H on a two-dimensional manifold using high order paracontrolled calculus. It is a self-adjoint operator with pure point spectrum. We get lower and upper bounds on its eigenvalues which imply an almost sure Weyl-type law for H.