A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinking of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Further, we assume that there exists an appropriate $\V$, homogeneous of order $-\rho$ with $\rho \in (0,1)$, such that $V(x) = \V(x) + O(|x|^{-\rho - \epsilon})$, $\epsilon > 0$, as $|x| \to \infty$, and investigate the asymptotic distribution of the eigenvalues within a given cluster, as $q \to \infty$. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of $\V$.