A multichannel scheme in smooth scattering theory

Abstract

In this paper we develop the scattering theory for a pair of self-adjoint operators \mbox{$A_0=A_1\oplus \cdots \oplus A_N$} and $A=A_1+\cdots +A_N$ under the assumption that all pair products $A_j{A}_k$ with $j\ne k$ satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products $A_j{A}_k$, $j\ne k$, 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.