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 language | English |
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Pages (from-to) | 601–634 |
Journal | Journal of Spectral Theory |
Volume | 3 |
Issue number | 4 |
Early online date | 23 Oct 2013 |
DOIs | |
Publication status | E-pub ahead of print - 23 Oct 2013 |