In this talk I will try to convince that vessels are good. In the classical theory starting from the
Schrodinger (Strum-Liouville) equation
\[ L(y) = - y'' + q(x) y = \lambda y\]
and assuming that integral of $x |q(x)|$ is bounded from 0 to infinity, one study the spectrum of
the operator L and show then how to
reconstruct the potential from the spectrum (solving Gelfand-Levitan or Marchenko equation).
In the theory of vessels one starts from the spectrum and constructs from this the corresponding
vessel, which in turn uniquely determines the potential, corresponding to this spectrum.
Advantage of this technique is that one can construct and study potentials with different from the
classical one, spectrum. It is enough for the spectrum to be symmetric with respect to the real
line and to be a "contour" in the complex plane(corresponding to the continuous spectrum). In
the new setting Gelfand-Levitan equation is simply derived and moreover, tau function arises as
a determinant of a I+trace class operator. The main feature of vessel construction is Krein space
realizations of symmetric, identity at infinity functions.
Other advantages:
1. Generalizable to a scattering theory of n-th order operator.
2. one can consider scattering when the basic waves are not just exponents, i.e. solutions of $y'' = \lambda y$,
but solutions of the Schrodinger equation for a fixed initial potential.
Among the aims of this research i can mention
1. characterize potentials which have the continuous spectrum on the cut of imaginary axis and
some discrete points.
2. generate new classes of potentials admitting scattering theory, different from the known ones
in the literature.
The project is new, I have just started and need a feedback.