The time-dependent Schroedinger equation, Riccati equation and Airy functions

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.