  Consider a one-dimensional Schroedinger operator which is a short range
perturbation of a 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.
