Eigenvalue asymptotics for weakly perturbed Dirac and Schrodinger operators with constant magnetic fields of full rank

The even-dimensional Dirac and Schrodinger operators with a constant magnetic field of full rank have purely essential spectrum consisting of isolated eigenvalues, so-called Landau levels. For a sign-definite electric potential V which tends to zero at infinity, not too fast, it is known for the Schrodinger operator that the number of eigenvalues near each Landau level is infinite and their leading (quasi-classical) asymptotics depends on the rate of decay for V. We show, both for Schrodinger and Dirac operators, that, for any sign-definite, bounded V which tends to zero at infinity, there still are an infinite number of eigenvalues near each Landau level. For compactly supported V, we establish the non-classical formula, not depending on V, describing how the eigenvalues converge to the Landau levels asymptotically.