Perturbations of the Landau Hamiltonian: Asymptotics of Eigenvalue Clusters

Abstract

We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a short-range continuous potential. The spectrum of the operator forms eigenvalue clusters. We obtain a Szegő limit theorem for the eigenvalues in the clusters as the cluster index and the field strength B tend to infinity with a fixed ratio \({\mathcal E}\). The answer involves the averages of the potential over circles of radius \(\sqrt{{\mathcal E}/2}\) (classical orbits). After rescaling, this becomes a semiclassical problem where the role of Planck’s constant is played by 2/B. We also discuss a related inverse spectral result.