Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials

We study a nonrelativistic charged particle on the Euclidean plane ℝ² subject to a perpendicular constant magnetic field and an ℝ²-homogeneous random potential in the approximation that the corresponding random Landau Hamiltonian on the Hilbert space L²(ℝ²) is restricted to the eigenspace of a single but arbitrary Landau level. For a wide class of ℝ²-homogeneous Gaussian random potentials we rigorously prove that the associated restricted integrated density of states is absolutely continuous with respect to the Lebesgue measure. We construct explicit upper bounds on the resulting derivative, the restricted density of states. As a consequence, any given energy is seen to be almost surely not an eigenvalue of the restricted random Landau Hamiltonian.