Article

**Bull. London Math. Soc. 39-4, 677-684 (2007)**[DOI: 10.1112/blms/bdm038]

## Algebro-Geometric Constraints on Solitons with Respect to Quasi-Periodic Backgrounds

### Gerald Teschl

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.

** MSC91:** Primary 30E20, 30F30; Secondary 34L25, 47B36

**Keywords:** *Jacobi operators, scattering theory, periodic, Abelian integrals*

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