We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations with power-like decay, exponential decay or compact support.
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 shrinki
ng 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 $V$.
