Perturbations of the Landau Hamiltonian: Asymptotics of eigenvalue clusters

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 Szegő limit theorem for the eigenvalues in the clusters as a suitable cluster index and B tend to infinity with a fixed ratio E. The answer involves the averages of the potential over circles of radius E/2 (circular Radon transform). We also discuss related inverse spectral results. Contents 1. Introduction 2. The main lemma 3. Reduction to a one-dimensional pseudo-differential operator 3.1. A preliminary rotation 3.2. Averaging 4. Analysis of the reduced operator 4.1. The Weyl symbol of T n 4.2. Localization 4.3. Estimates on Φ 1 5. Proof of Theorem 1.2 6. An inverse spectral result Appendix A. The Weyl quantization of radial functions Appendix B. The Remainder in Taylor's theorem References G.