The initial value problem for Schrödinger equations on the torus

We discuss well-posedness for the initial-boundary value problem for linear and nonlinear Schrödinger equations with periodic boundary condition. We, first, show that the initial value problem for linear equationsis L²-well-posed if and only if the imaginary part of magnetic field has a scalar potential. Secondly, we present local existence theorem for nonlinear equations under a certain condition corresponding to that for linear equations. Lastly, we give examples of ill-posed initial value problem for nonlinear equations which do not satisfy this condition.