The Uniqueness of the Integrated Density of States for the Schrödinger Operators for the Robin Boundary Conditions

Abstract

The integrated density of states (IDS) for the Schrödinger operators is defined by using the eigenvalue counting function of the operator restricted to bounded regions with appropriate boundary conditions. Two sufficient conditions for the coincidence of the IDS for the Dirichlet boundary conditions and the IDS for the Robin boundary conditions are given. The proofs of some fundamental formulas, e.g. the change of variables, the chain rule and the divergence formula, for Lipschitz domains are given for the completeness.