Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

TL;DR

This paper analyzes a magnetic Schrödinger operator with random delta impurities on a lattice, demonstrating pure point spectrum and localization in the lowest Landau bands under strong magnetic fields.

Contribution

It characterizes the spectral properties and localization phenomena of the Landau Hamiltonian with delta impurities, extending understanding of magnetic random operators.

Findings

Spectrum in lowest N Landau bands is purely point

Energies at Landau levels are infinitely degenerate

Eigenfunctions are localized with bounded localization length

Abstract

We consider a random Schro\"dinger 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.