Schr\"odinger operators with random $\delta$ magnetic fields

TL;DR

This paper studies Schr"odinger operators with random delta magnetic fields in two dimensions, proving the spectrum covers all non-negative real numbers and showing the integrated density of states exhibits Lifshitz tail decay at the spectrum's bottom.

Contribution

It establishes the spectrum and Lifshitz tail behavior for Schr"odinger operators with random delta magnetic fields, extending understanding of spectral properties under randomness.

Findings

Spectrum is [0, ∞) under mild conditions.

Integrated density of states decays exponentially at the bottom.

Provides a lower bound for the IDS at the spectrum's bottom.

Abstract

We shall consider the Schr\"odinger operators on $\mathbf{R}^2$ with random $\delta$ magnetic fields. Under some mild conditions on the positions and the fluxes of the $\delta$-fields, we prove the spectrum coincides with $[0,\infty)$ and the integrated density of states (IDS) decays exponentially at the bottom of the spectrum (Lifshitz tail), by using the Hardy type inequality by Laptev-Weidl. We also give a lower bound for IDS at the bottom of the spectrum.