2011/03/11 by Andrey Melnikov, A. Melnikov, Melnikov, A.
Mathematics · Physics and Astronomy · #34A30 #34A55 #46C20 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #G.1.7 #G.1.9 #J.2 #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Quantum optics and atomic interactions #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #acm:34A30 #acm:34A55 #acm:46C20 #math-ph #math.CA #math.CV #math.MP #math.SP #msc:34A30 #msc:34A55 #msc:46C20
paper · pdf · doi:10.48550/arxiv.1103.2392
37 pages
openalex publication_date 2011/03/11 · arxiv created 2011/08/23 · arxiv updated 2011/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we present a theory of vessels and its application to the classical inverse scattering of the Sturm-Liouville differential equation. The classical inverse scattering theory, including all its ingredients: Jost solutions, the Gelfand-Levitan equation, the tau function, corresponds to regular vessels, defined by bounded operators. A contribution of this work is the construction of models of vessels corresponding to unbounded operators, which is a first step for the inverse scattering for a wider class of potentials. A detailed research of Jost solutions and the corresponding vessel is presented for the unbounded Sturm-Liouville case. Models of vessels on curves, corresponding to unbounded operators are presented as a tool to study Linear Differential equations of finite order with a spectral parameter and as examples, we show how the family of Non Linear Schrodinger equations and Canonical Systems arise.