The uniqueness of the integrated density of states for the Schrödinger operators with magnetic fields

The integrated density of states (IDS) for the Schrödinger operators is defined in two ways: by using the counting function of eigenvalues of the operator restricted to bounded regions with appropriate boundary conditions or by using the spectral projection of the whole space operator. A sufficient condition for the coincidence of the two definitions above is given. Moreover, a sufficient condition for the coincidence of the IDS for the Dirichlet boundary conditions and the IDS for the Neumann boundary conditions is given. The proof is based only on the fundamental items in functional analysis, such as the min-max principle, etc.