  We construct the Green functions (or Feynman's propagators) for the
Schroedinger equations of the form $i\psi_{t}+{1/4}\psi_{xx}\pm tx^{2}\psi =0$
in terms of Airy functions and solve the Cauchy initial value problem in the
coordinate and momentum representations. Particular solutions of the
corresponding nonlinear Schroedinger equations with variable coefficients are
also found. A special case of the quantum parametric oscillator is studied in
detail first. The Green function is explicitly given in terms of Airy functions
and the corresponding transition amplitudes are found in terms of a
hypergeometric function. The general case of quantum parametric oscillator is
considered then in a similar fashion. A group theoretical meaning of the
transition amplitudes and their relation with Bargmann's functions is
stablished.
