The Spectrum of a Magnetic Schr\"odinger Operator with Randomly Located Delta Impurities

TL;DR

This paper extends the analysis of the spectrum of a magnetic Schrödinger operator with delta impurities by allowing random positions and strengths, showing that at high magnetic fields, the spectrum is pure point with localized eigenfunctions.

Contribution

It generalizes previous lattice-based models to include randomly located impurities with arbitrary bounded distributions, characterizing the spectrum under high magnetic fields.

Findings

Spectrum is pure point at high magnetic fields.

Eigenfunctions are exponentially localized.

First Landau level is infinitely degenerate.

Abstract

We consider a single band approximation to the random Schroedinger operator in an external magnetic field. The spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice. In this paper we generalize these results by letting the delta impurites have random positions as well as strengths; they are located in squares of a lattice with a general bounded distribution. We characterize the entire spectrum of this operator when the magnetic field is sufficiently high. We show that the whole spectrum is pure point, the energy coinciding with the first Landau level is infinitely degenerate and that the eigenfunctions corresponding to other Landau band energies are exponentially localized.