Uniqueness and Existence of the Integrated Density of States for Schrödinger Operators with Magnetic Field and Electric Potential with Singular Negative Part

Abstract

We prove the coincidence of the two definitions of the integrated density of states (IDS) for Schrödinger operators with strongly singular magnetic fields and scalar potentials: the first one using the counting function of eigenvalues of the induced operator on a bounded open set with Dirichlet boundary conditions, the second one using the spectral projections of the whole space operator. Thus we generalize a result of [5], where the scalar potential was non-negative. Moreover, we prove the existence of IDS for the case of periodical magnetic field and scalar potential.