Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

TL;DR

This paper analyzes how eigenvalues of a 2D magnetic Schrödinger operator with a decaying electric potential form clusters near Landau levels, providing an explicit asymptotic density description using Radon transforms.

Contribution

It introduces a method to determine the asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian using anti-Wick quantization and Radon transforms.

Findings

Explicit asymptotic density formula for eigenvalue clusters

Connection between eigenvalue distribution and Radon transform of potential

Asymptotic behavior as cluster number tends to infinity

Abstract

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.