2012/09/14 by Alexander Pushnitski, Pushnitski, Alexander, Dmitri Yafaev +1
Computer Science · Mathematics · #47A40 (Primary) 47B25 (Secondary) #FOS: Mathematics #Matrix Theory and Algorithms #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.SP #msc:47A40 #msc:47B25
paper · pdf · doi:10.48550/arxiv.1209.3238
arxiv created 2012/09/14 · openalex publication_date 2012/09/14 · arxiv updated 2012/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≠ k satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products AjAk, j≠ 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.