A multichannel scheme in smooth scattering theory

Alexander Pushnitski, Dmitri Yafaev

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

In this paper we develop the scattering theory for a pair of self-adjoint operators \mbox{A0=A1⊕⋯⊕AN} and A=A1+⋯+AN under the assumption that all pair products AjAk with j≠k satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products AjAk, j≠k, can be represented as integral operators with smooth kernels in the spectral representation of the operator A0. We show that the absolutely continuous parts of the operators A0 and A are unitarily equivalent. This yields a smooth version of Ismagilov's theorem known earlier in the trace class framework. We also prove that the singular continuous spectrum of the operator A is empty and that its eigenvalues may accumulate only to "thresholds'' of the absolutely continuous spectra of the operators Aj. Our approach relies on a system of resolvent equations which can be considered as a generalization of Faddeev's equations for three particle quantum systems.
Original languageEnglish
Pages (from-to)601–634
JournalJournal of Spectral Theory
Volume3
Issue number4
Early online date23 Oct 2013
DOIs
Publication statusE-pub ahead of print - 23 Oct 2013

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