  In finite dimensions, controllability of bilinear quantum control systems can
be decided quite easily in terms of the "Lie algebra rank condition" (LARC),
such that only the systems Lie algebra has to be determined from a set of
generators. In this paper we study how this idea can be lifted to infinite
dimensions. To this end we look at control systems on an infinite dimensional
Hilbert space which are given by an unbounded drift Hamiltonian and bounded
control Hamiltonians. The drift is assumed to have empty continuous spectrum.
We use recurrence methods and the theory of Abelian von Neumann algebras to
develop a scheme, which allows us to use an approximate version of LARC, in
order to check approximate controllability of the control system in question.
Its power is demonstrated by looking at some examples. We recover in particular
previous genericity results with a much easier proof. Finally several possible
generalizations are outlined.
