S. Doi et al., The uniqueness of the integrated density of states for the Schrodinger operators with magnetic fields, MATH Z, 237(2), 2001, pp. 335-371
The integrated density of states (IDS) for the Schrodinger operators is def
ined in two ways: by using the counting function of eigenvalues of the oper
ator 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. Moreov
er, a sufficient condition for the coincidence of the IDS for the Dirichlet
boundary conditions and the IDS for the Neumann boundary conditions is giv
en. The proof is based only on the fundamental items in functional analysis
, such as the min-max principle, etc.