Control for Schroedinger operators on tori
  A well known result of Jaffard states that an arbitrary region on a torus controls, in the L2 sense, solutions of the free stationary and dynamical Schroedinger equations. In this note we show that the same result is valid in the presence of a smooth time-independent potential. The methods apply to continuous potentials as well and we conjecture that the L2 control is valid for any bounded time dependent potential. 
Nicolas Burq and Maciej Zworski
math.AP cs.SY math-ph math.MP math.OC