JournalsjstVol. 3, No. 4pp. 601–634

A multichannel scheme in smooth scattering theory

  • Alexander Pushnitski

    King's College London, UK
  • Dmitri Yafaev

    Université de Rennes I, France
A multichannel scheme in smooth scattering theory cover
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Abstract

In this paper we develop the scattering theory for a pair of self-adjoint operators \mbox{A0=A1ANA_{0}=A_{1}\oplus\dots \oplus A_{N}} and A=A1++ANA=A_{1}+\dots +A_{N} under the assumption that all pair products AjAkA_{j}A_{k} with jkj\neq k satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products AjAkA_{j}A_{k}, jkj\neq k, can be represented as integral operators with smooth kernels in the spectral representation of the operator A0A_{0}. We show that the absolutely continuous parts of the operators A0A_{0} and AA 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 AA is empty and that its eigenvalues may accumulate only to "thresholds'' of the absolutely continuous spectra of the operators AjA_{j}. 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.

Cite this article

Alexander Pushnitski, Dmitri Yafaev, A multichannel scheme in smooth scattering theory. J. Spectr. Theory 3 (2013), no. 4, pp. 601–634

DOI 10.4171/JST/58