Eigenvalue Asymptotics for Weakly Perturbed Dirac and Schrödinger Operators with Constant Magnetic Fields of Full Rank

The even-dimensional Dirac and Schrödinger 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 Vwhich tends to zero at infinity, not too fast, it is known for the Schrödinger 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 Schrödinger and Dirac operators, that, for anysign-definite, bounded Vwhich 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-classicalformula, not depending on V, describing how the eigenvalues converge to the Landau levels asymptotically.