Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No 1, pp. 21-33    ( to contents , go back )
 

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

I. Egorova

Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering 47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail: iraegorova@gmail.com

G. Teschl

Faculty of Mathematics, University of Vienna 15 Nordbergstrasse, 1090, Wien, Austria
International Erwin Schrödinger Institute for Mathematical Physics 9 Boltzmanngasse, 1090, Wien, Austria
E-mail: Gerald.Teschl@univie.ac.at

Received November 2, 2009

Abstract

A one-dimensional Schrödinger operator which is a short-range pertur- bation of a finite-gap operator is considered. There are given the 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, these results are applied 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.

Mathematics Subject Classification 2000: 34L25, 35Q53 (primary); 35B60, 37K20 (secondary).
Key words: Inverse scattering, ╥nite-gap background, KdV, nonlinear Paley{Wiener Theorem.

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