A semiclassical approach to spectral estimates for random Landau Schrodinger operators

TL;DR

This paper develops a semiclassical pseudodifferential calculus approach to analyze spectral properties of random Landau Schrödinger operators, deriving estimates relevant to spectral band analysis.

Contribution

It introduces a novel semiclassical method using the Grushin technique to study spectral estimates for random Landau operators with bounded potentials.

Findings

Proves semiclassical Wegner estimates for the operators.

Establishes Minami estimates in spectral bands.

Analyzes an effective Hamiltonian on L^2(R) using pseudodifferential operators.

Abstract

We prove spectral properties for random Landau Schr\"odinger operators on $L^2(\mathbb{R}^2)$ with bounded, random potentials supported in a square $\Lambda_L \subset \mathbb{R}^2$ of side length $L>0$, using semiclassical pseudodifferential calculus. The semiclassical parameter $h$ is the inverse of the magnetic field strength $B > 0$. By means of the Grushin method, we are led to the analysis of an effective Hamiltonian on $L^2 (\mathbb{R})$, the principal term of which is a sum of certain compact, self-adjoint pseudodifferential operators. By analyzing these operators, we prove semiclassical Wegner and Minami estimates for the random Landau Schrodinger operator in energy intervals in the spectral bands around each Landau level.