Absolutely continuous and singular spectral shift functions

Nurulla Azamov

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    Abstract

    Given a self-adjoint operator H 0 ; a self-adjoint trace-class operator V and a fixed Hilbert- Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ (H 0 ; F) ∨ R of full Lebesgue measure is defined, such that for all λ ∈ Λ(H 0 + rV; F) ∩ Λ(H 0 ; F); where r ∈ R; the wave w±(λ;H 0 + rV, H 0 ) and the scattering matrices S(λ;H 0 + rV, H 0 ) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This version of abstract scattering theory allows us, in particular, to prove that, where is the so called absolutely continuous part of the spectral shift function defined by, and where, denotes the absolutely continuous part of the spectral projection. Combined with the Birman-Kreǐn formula, this implies that the singular part of the spectral shift function, is an almost everywhere integer-valued function, where, denotes the singular part of the spectral projection. It is also shown that eigenvalues of the scattering matrix S(λ;H 0 +V;H 0 ) can be connected to 1 in two natural ways, and that the singular spectral shift function measures the difference of the spectral ows of eigenvalues of the scattering matrix.

    Original languageEnglish
    Pages (from-to)3-102
    Number of pages100
    JournalDissertationes Mathematicae
    Volume480
    Issue number480
    DOIs
    Publication statusPublished - 2011

    Keywords

    • Absolutely continuous spectral shift function
    • Birman-Kreǐn formula
    • Infinitesimal scattering matrix
    • Infinitesimal spectral flow
    • Pushnitski μ-invariant
    • Scattering matrix
    • Singular spectral shift function
    • Spectral shift function

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