A trace formula and high energy spectral asymptotics for the perturbed Landau Hamiltonian

A two-dimensional Schrödinger operator with a constant magnetic field perturbed by a smooth compactly supported potential is considered. The spectrum of this operator consists of eigenvalues which accumulate to the Landau levels. We call the set of eigenvalues near the n’th Landau level an n’th eigenvalue cluster, and study the distribution of eigenvalues in the n’th cluster as n → ∞. A complete asymptotic expansion for the eigenvalue moments in the n’th cluster is obtained and some coefficients of this expansion are computed. A trace formula involving the first eigenvalue moments is obtained.