On a theory of vessels and the inverse scattering

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.