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Spectral theory of discontinuous functions of self-adjoint operators and scattering theory
Authors:Alexander Pushnitski  Dmitri Yafaev
Institution:a Department of Mathematics, King's College London, Strand, London, WC2R 2LS, UK
b Department of Mathematics, University of Rennes-1, Campus Beaulieu, 35042, Rennes, France
Abstract:In the smooth scattering theory framework, we consider a pair of self-adjoint operators H0, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H0 and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.
Keywords:Scattering matrix  Carleman operator  Absolutely continuous spectrum  Spectral projections
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