  We investigate the correspondence between classical and quantum mechanics for
periodically time dependent Hamiltonian systems, using the example of a
periodically forced particle in a one-dimensional triangular well potential. In
particular, we consider quantum mechanical Floquet states associated with
resonances in the classical phase space. When the classical motion exhibits
{\it sub}harmonic resonances, the corresponding Floquet states maintain the
driving field's periodicity through dynamical tunneling. This principle applies
both to Floquet states associated with classical invariant vortex tubes
surrounding stable, elliptic periodic orbits and to Floquet states that are
associated with unstable, hyperbolic periodic orbits. The triangular well model
also poses a yet unsolved mathematical problem, related to perturbation theory
for systems with a dense pure point spectrum. The present approximate
analytical and numerical results indicate that quantum tunneling between
different resonance zones is of crucial importance for the question whether the
driven triangular well has a dense point or an absolutely continuous
quasienergy spectrum, or whether there is a transition from the one to the
other.
