A multichannel scheme in smooth scattering theory

Abstract

We develop the scattering theory for a pair of self-adjoint operators A0 = A1 ⊕ · · · ⊕AN and A = A1 + · · ·+ AN under the assumption that all pair products AjAk with j 6= k satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products AjAk, j 6= 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.