Landau Hamiltonian with Gaussian white noise potential and the asymptotic of its bottom of spectrum

TL;DR

This paper constructs a magnetic Schrödinger operator with Gaussian white noise potential in 2D, extending known spectral asymptotics to include magnetic fields and less regular potentials.

Contribution

It introduces a novel construction method for magnetic Schrödinger operators with low-regularity noise, generalizing spectral asymptotics to magnetic cases and non-translationally invariant potentials.

Findings

Extended asymptotic results to magnetic Schrödinger operators

Constructed operators with minimal regularity assumptions

Covered cases with broken translational invariance

Abstract

We present a simple construction of a random Schr\"odinger operator subject to a magnetic field with a regularity as low as $0^-$-H\"older and a Gaussian white noise electric potential on a two-dimensional bounded box. This construction is based on the exponential Ansatz introduced in [HL15] and leverages the semigroup approach developed in [HL24]. The proposed construction enables us to generalise an asymptotic result for the bottom of the spectrum of the two-dimensional continuous Anderson Hamiltonian, first proved in [CvZ21], to the magnetic case. Our choice of potential not only covers the case of a uniform magnetic field, but also those which would break translational invariance.