between two eigenstates, no such a restriction is imposed in the [Lie–Galerkin Control Condition]{} (see Section \[sec:buca\]).

The [Lie–Galerkin Control Condition]{} ensures strong controllability properties for the Galerkin approximations. Indeed it provides controllability for a fixed Galerkin approximation while avoiding the transfer of population to higher energy levels. This allows estimates on the difference between the dynamics of the finite-dimensional Galerkin approximation and the ones of the original infinite-dimensional system. The [Lie–Galerkin Control Condition]{}  also ensures a bound on the $L^{1}$ norm of the control achieving controllability which is uniform with respect to the prescribed tolerance, when the required transfer is between finite combinations of eigenstates.

Under the [Lie–Galerkin Stalking Condition]{}, a slight modification of the [Lie–Galerkin Control Condition]{}, we can prove that any trajectory can be tracked in modulus (see Theorem \[metalemma-tracking\]).

The [Lie–Galerkin Control Condition]{} under the additional assumption that the system is $s$-weakly coupled (see Definition \[def:wc\]) as introduced in [@weakly], allows to conclude that the system is controllable in $H^{s/2}$ (see Theorem \[thm:controlHs\]).

Application to the quantum angular momentum
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Rotational molecular dynamics is one of the most important examples of quantum systems with an infinite-dimensional Hilbert space and a discrete spectrum. Molecular orientation and alignment 