Limitations on the Control of Schrödinger Equations

We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No–go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non– controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control (E(t) ·x)u is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, and we give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.