Gerald Teschl
Algebro-Geometric Constraints on Solitons with Respect to Quasi-Periodic Backgrounds
The paper is published: Bull. London Math. Soc. 39-4 (2007) 677-684

MSC:

30F20 Classification theory of Riemann surfaces

30F30 Differentials on Riemann surfaces

34L25 Scattering theory

47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Abstract: We investigate the algebraic conditions that have to be satisfied by the scattering
data of short-range perturbations of quasi-periodic finite-gap Jacobi operators
in order to allow solvability of the inverse scattering problem.
Our main result provides a Poisson-Jensen-type formula for
the transmission coefficient in terms of Abelian integrals on the underlying
hyperelliptic Riemann surface and an explicit condition for its
single-valuedness. In addition, we establish trace formulas which
relate the scattering data to the conserved quantities in this case.

Keywords: Jacobi operators, scattering theory, periodic, Abelian integrals
Notes: second and final version