A multichannel scheme in smooth scattering theory

TL;DR

This paper develops a multichannel scattering theory for a sum of self-adjoint operators, showing the equivalence of their absolutely continuous spectra and analyzing spectral properties under smoothness conditions.

Contribution

It introduces a smooth scattering framework for operator sums, extending Ismagilov's theorem and analyzing spectral types using generalized resolvent equations.

Findings

Absolutely continuous spectra are unitarily equivalent.

Singular continuous spectrum is empty.

Eigenvalues only accumulate at spectral thresholds.

Abstract

We develop the scattering theory for a pair of self-adjoint operators $A_{0}=A_{1}\oplus...\oplus A_{N}$ and $A=A_{1}+...+A_{N}$ 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.