A Paley–Wiener theorem for periodic scattering with applications to the Korteweg–de Vries equation

A one-dimensional Schrödinger operator which is a short-range perturbation 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.