Journal of Spectral Theory

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Volume 3, Issue 4, 2013, pp. 601–634
DOI: 10.4171/JST/58

Published online: 2013-10-23

A multichannel scheme in smooth scattering theory

Alexander Pushnitski[1] and Dmitri Yafaev[2]

(1) King's College London, UK
(2) Université de Rennes I, France

In this paper we develop the scattering theory for a pair of self-adjoint operators \mbox{$A_{0}=A_{1}\oplus\dots \oplus A_{N}$} and $A=A_{1}+\dots +A_{N}$ under the assumption that all pair products $A_{j}A_{k}$ with $j\neq k$ satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products $A_{j}A_{k}$, $j\neq k$, can be represented as integral operators with smooth kernels in the spectral representation of the operator $A_{0}$. We show that the absolutely continuous parts of the operators $A_{0}$ 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 $A_{j}$. 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.

Keywords: Multichannel problem, Fredholm resolvent equations, smoothness, absolutely continuous and singular spectra, wave operators, scattering matrix

Pushnitski Alexander, Yafaev Dmitri: A multichannel scheme in smooth scattering theory. J. Spectr. Theory 3 (2013), 601-634. doi: 10.4171/JST/58