  We consider the Schr{\"o}dinger equation in $\mathbf{R}^d$, $d \ge 1$, with a
confining potential growing at most quadratically. Our main theorem
characterizes open sets from which observability holds, provided they are
sufficiently regular in a certain sense. The observability condition involves
the Hamiltonian flow associated with the Schr{\"o}dinger operator under
consideration. It is obtained using semiclassical analysis techniques. It
allows to provide with an accurate estimation of the optimal observation time.
We illustrate this result with several examples. In the case of two-dimensional
harmonic potentials, focusing on conical or rotation-invariant observation
sets, we express our observability condition in terms of arithmetical
properties of the characteristic frequencies of the oscillator.
