  Dynamics of a randomly-perturbed quantum system with 3/2-degrees of freedom
is considered. We introduce a transfer operator being the quantum analogue of
the specific Poincar\'e map. This map was proposed in (Makarov, Uleysky, J.
Phys. A: Math. Gen., 2006) for exploring domains of finite-time stability,
which survive under random excitation. Our attention is concentrated on level
spacing distribution of the transfer operator, averaged over ensemble of
realizations. The problem of sound propagation in an oceanic waveguide is
considered as an example. For the acoustic frequency of 200 Hz, level spacing
distribution undergoes the crossover from Poissonian to Wigner-like statistics
with increasing time, as it is consistent with classical predictions via the
specific Poincar\'e map. For the frequency of 600 Hz, the level spacing
statistics becomes non-universal due to large number of nearly-degenerate
levels whose contribution grows with time. Occurrence of nearly-degenerate
levels is attributed to bifurcations of classical periodic orbits, caused by
fast spatial oscillations of the random perturbation.
