  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.
