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

TL;DR

This paper derives Green functions for specific time-dependent Schrödinger equations using Airy functions, solves initial value problems, and explores their relation to group theory and special functions.

Contribution

It provides explicit solutions and Green functions for a class of time-dependent Schrödinger equations, linking them to Airy functions and group theoretical concepts.

Findings

Explicit Green functions in terms of Airy functions

Solutions to nonlinear Schrödinger equations with variable coefficients

Connection between transition amplitudes and Bargmann's functions

Abstract

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.