Article

**Zh. Mat. Fiz. Anal. Geom. 6:1, 21-33 (2010)**[url]

## A Paley-Wiener Theorem for Periodic Scattering with Applications to the Korteweg-de Vries Equation

### Iryna Egorova and Gerald Teschl

Consider a one-dimensional Schrödinger operator which is a short-range perturbation
of a quasi-periodic, finite-gap operator. We give necessary and sufficient conditions on the left, right reflection
coefficient such that the difference of the potentials has finite support to the left, right, respectively.
Moreover, we apply these results to show a unique continuation type result for solutions
of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an
analogous result for the modified Korteweg-de Vries equation is also obtained.

** MSC2000:** Primary 34L25, 35Q53; Secondary 35B60, 37K20

**Keywords:** *Inverse scattering, finite-gap background, KdV, nonlinear Paley-Wiener Theorem*

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