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Algebro-geometric constraints on solitons with respect to quasi-periodic backgrounds. (English) Zbl 1131.34012

The authors examine the solvability of the inverse scattering problem for quasi-periodic finite-gap Jacobi operators. There are algebraic conditions that must be imposed on the scattering data of short range perturbations. These constraints are related to the fact that the resolvent set of the background operator is not simply connected in the quasi-periodic case – the transmission coefficient must be reconstructed from its boundary values on this nonsimply connected domain.
The main result provides a Poisson-Jensen-type formula forthe transmission coefficient. An Abelian integral is involved on the underlying hyperelliptic Riemann surface for the reconstrution. An explicit condition is given for its single-valuedness. Trace formulas are established relating the scattering data to the conserved quantities. The reconstruction is made explicit for the case of Jacobi operators or for the Toda equations. However similar results apply to the one-dimensional Schrödinger operator or to the KdV equation.

MSC:

34A55 Inverse problems involving ordinary differential equations
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
34L25 Scattering theory, inverse scattering involving ordinary differential operators
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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