  A Lyapunov-based approach for the trajectory generation of an $N$-dimensional
Schr{\"o}dinger equation in whole $\RR^N$ is proposed. For the case of a
quantum particle in an $N$-dimensional decaying potential the convergence is
precisely analyzed. The free system admitting a mixed spectrum, the dispersion
through the absolutely continuous part is the main obstacle to ensure such a
stabilization result. Whenever, the system is completely initialized in the
discrete part of the spectrum, a Lyapunov strategy encoding both the distance
with respect to the target state and the penalization of the passage through
the continuous part of the spectrum, ensures the approximate stabilization.
