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.

We study the spectral properties of a two-dimensional magnetic Schrödinger operatorThe magnetic field is given by rot . . . , N) and the points {z j } N j=1 are uniformly separated. We give an upper bound for the number of eigenvalues of H N between two Landau levels or below the lowest Landau level, when N is finite. We prove the spectral localization of H N near the spectrum of the single solenoid operator, when {z j } N j=1 are far from each other, all the values {α j } N j=1 are the same, and the boundary conditions at z j are uniform. We determine the deficiency indices of the minimal operator and give a characterization of self-adjoint extensions of the minimal operator.

We consider the magnetic Schrödinger operator on R2. The magnetic field is the sum of a homogeneous magnetic field and periodically varying pointlike magnetic fields on a lattice. We shall give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. This condition is also necessary for the lowest Landau level. In the threshold case, we see that the spectrum near the lowest Landau level is purely absolutely continuous. Moreover, we shall give an estimate for the density of states for Landau levels and their gaps. The proof is based on the method of Geyler and Šťovíček, the magnetic Bloch theory, and canonical commutation relations.

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. §1. Introduction §1. Definition of the integrated density of states and resultsIn this paper we shall consider the Schrödinger operators with magnetic fields in d-dimensional space:

No abstract