A multichannel scheme in smooth scattering theory

Abstract

In this paper we develop the scattering theory for a pair of self-adjoint operators \mbox{$A_{0}=A_{1}\oplus\dots \oplus A_{N}$} 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.