2008/07/25 by L. Golinskii, Leonid Golinskiĭ, A. Kheifets +9 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP
paper · pdf · doi:10.48550/arxiv.0807.4017
arxiv created 2008/07/25 · openalex publication_date 2008/07/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
B. Simon proved the existence of the wave operators for the CMV matrices with Szego class Verblunsky coefficients, and therefore the existence of the scattering function. Generally, there is no hope to restore a CMV matrix when we start from the scattering function, in particular, because it does not contain any information about the (possible) singular measure. Our main point of interest is the solution of the inverse scattering problem (the heart of the Faddeev--Marchenko theory), that is, to give necessary and sufficient conditions on a certain class of CMV matrices such that the restriction of this correspondence (from a matrix to the scattering function) is one to one. In this paper we show that the main questions on inverse scattering can be solved with the help of three important classical results: Adamyan-Arov-Krein (AAK) Theory, Helson-Szego Theorem and Strong Szego Limit Theorem. Each of these theorem states the equivalence of certain conditions. Actually, to each theorem we add one more equivalent condition related to the CMV inverse scattering problem.