A semiclassical approach to spectral estimates for random Landau Schrodinger operators

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.