rotating rigid symmetric $3$D molecule this application of this method is obstructed by the fact that eigenspaces may have arbitrarily large dimension, since at every higher energy level new degeneracies appear. In some sense the strategy of [@ancoranoi Section 8] does not fully exploit the potentialities of the geometric method based on the controllability of the Galerkin approximations.

In this paper, we present a sufficient condition for controllability of the discrete-spectrum bilinear Schrödinger equation which can be applied to cases in which the spectrum of the internal Hamiltonian $H_{0}$ is very degenerate. The results fully exploit the use of more than one control and extend to simultaneous controllability, approximate controllability in $H^s$, and stalking. Proving that a system is a stalker (i.e. it permits to track in modulus any given trajectory; for precise definitions see Section \[sec:mainresults\]) is a crucial issue when describing systems containing dissipative levels (and the dissipation is not taken into account in the mathematical model). In this case, a strategy is to keep the population of the dissipative levels as low as possible during the transitions in order to minimize the effects of the dissipation (see for instance the STIRAP model [@CH90; @VHSB01; @Gauthier_Jauslin]).

The result 