2D random magnetic Laplacian with white noise magnetic field

TL;DR

This paper constructs a 2D random magnetic Laplacian with white noise magnetic field on a torus, establishing its spectral properties and eigenvalue bounds using advanced calculus techniques.

Contribution

It introduces a novel definition of the random magnetic Laplacian with white noise magnetic field and analyzes its spectral characteristics.

Findings

The operator is self-adjoint with pure point spectrum.

Eigenvalue bounds are established, leading to a Weyl-type law.

The domain consists of nonsmooth functions in L2.

Abstract

We define the random magnetic Laplacien with spatial white noise as magnetic field on the two-dimensional torus using paracontrolled calculus. It yields a random self-adjoint operator with pure point spectrum and domain a random subspace of nonsmooth functions in L 2. We give sharp bounds on the eigenvalues which imply an almost sure Weyl-type law.