Comment to Shor (apologies for the answer, I cannot yet write comments):

Maybe you are referring to Quantum controllability theorems.

Basically quantum controllability tells you what are the requirements needed for any state of the system to be accessible from any other state by means of an external electromagnetic field at a finite time. The problems are of course related to degeneracies in the spectra of many Hamiltonians. The first papers addressing this problem are J. Math. Phys. 24, 2608, (1983) and Phys. Rev. A, 51, 960 (1995). There are many works after this, particularly due to its importance in Quantum Control and its connection with Quantum Computation.

By the way: the harmonic oscillator is a well known uncontrollable system. However any truncation of the Hamiltonian makes the system controllable again.

Comment to Shor (apologies for the answer, I cannot yet write comments):

Maybe you are referring to Quantum controllability theorems.

Basically quantum controllability tells you what are the requirements needed for any state of the system to be accessible from any other state by means of an external electromagnetic field at a finite time. The problems are of course related to degeneracies in the spectra of many Hamiltonians. The first papers addressing this problem are J. Math. Phys. 24, 2608, (1983) and Phys. Rev. A, 51, 960 (1995). There are many works after this, particularly due to its importance in Quantum Control and its connection with Quantum Computation.

To Emilio Pisanty: By the way, the harmonic oscillator is a well known uncontrollable system. However any truncation of the Hamiltonian makes the system controllable again.