Abstract
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.