Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

We consider a random Schrödinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N . We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman-Molchanov method to prove localization. Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales, U.K.e-mail: [email protected] Institut de Physique Théorique, Ecole Polytechnique Fédérale de Lausanne, CH 1015 Lausanne, Switzerland. e-mail: [email protected] Department of Mathematical Physics, National University of Ireland, Dublin, (University College Dublin), Belfield, Dublin 4, Ireland. e-mail: [email protected] Research Associate, School of Theoretical Physics, Dublin Institute for Advanced Studies. This work was partially supported by the Forbairt (Ireland) International Collaboration Programme 1997. Landau Hamiltonian with Delta Impurities 2