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 $(E(t)\cdot x) u$ 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.