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

摘要

A one-dimensional Schr?dinger 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.