The differences of spectral projections and scattering matrix
Seminar Room 1, Newton Institute
Let A and B be self-adjoint operators in a Hilbert space such that the difference B-A is a trace class operator. For an interval I in the real line, we consider the self-adjoint operator D, defined as the difference between the spectral projection of B associated with I and the spectral projection of A associated to I. In his famous 1953 paper on the spectral shift function theory, M.G.Krein gave a simple example which shows that the operator D may fail to belong to the trace class even if the difference between B and A has rank one. Further analysis shows that in Krein's example, D is not even compact. We address the question of the description of the spectrum of D. It appears that under very general assumptions, the essential spectrum and the absolutely continuous spectrum of D can be completely described in terms of the spectrum of the scattering matrix for the pair A, B, evaluated at the endpoints of the interval I. In particular, it follows that D is compact if and only if the scattering matrix coincides with the identity operator.