Perturbations of the Landau Hamiltonian: Asymptotics of eigenvalue clusters

TL;DR

This paper analyzes the asymptotic distribution of eigenvalues of the Landau Hamiltonian with a decaying potential as the magnetic field strength increases, revealing a semiclassical structure and a Szegő limit theorem involving circular averages of the potential.

Contribution

It establishes a Szegő limit theorem for eigenvalue clusters of the Landau Hamiltonian with a potential, connecting eigenvalue asymptotics to circular Radon transforms.

Findings

Eigenvalue clusters form as magnetic field increases.

Eigenvalue distribution described by a Szegő limit theorem.

Eigenvalues relate to averages of potential over circles.

Abstract

We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a rapidly decaying potential, as the magnetic field strength, $B$, tends to infinity. After a suitable rescaling, this becomes a semiclassical problem where the role of Planck's constant is played by $1/B$. The spectrum of the operator forms eigenvalue clusters. We obtain a Szeg\H{o} limit theorem for the eigenvalues in the clusters as a suitable cluster index and $B$ tend to infinity with a fixed ratio $\calE$. The answer involves the averages of the potential over circles of radius $\sqrt{\calE/2}$ (circular Radon transform). We also discuss related inverse spectral results.