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

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.