  The quantum mechanical equivalent of parametric resonance is studied. A
simple model of a periodically kicked harmonic oscillator is introduced which
can be solved exactly. Classically stable and unstable regions in parameter
space are shown to correspond to Floquet operators with qualitatively different
properties. Their eigenfunctions, which are calculated exactly, exhibit a
transition: for parameter values with classically stable solutions the
eigenstates are normalizable while they cannot be normalized for parameter
values with classically instable solutions. Similarly, the spectrum of quasi
energies undergoes a specific transition. These observations remain valid
qualitatively for arbitrary linear systems exhibiting classically parametric
resonance such as the paradigm example of a frequency modulated pendulum
described by Mathieu's equation.
