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
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.