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

Abstract

We construct the Green functions (or Feynman's propagators) for the Schroedinger equations of the form $i{\psi }_t+1/4{\psi }_{xx}\pm t{x}^2\psi =0$ 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.

28 pages, one figure

References in corpus (3)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• An Integral Form of the Nonlinear Schroedinger Equation with Variable Coefficients
• The Time Inversion for Modified Oscillators

Cited by in corpus (13)

• Quantum Integrals of Motion for Variable Quadratic Hamiltonians
• The Degenerate Parametric Oscillator and Ince's Equation
• The Minimum-Uncertainty Squeezed States for for Atoms and Photons in a Cavity
• Soliton-like Solutions for Nonlinear Schroedinger Equation with Variable Quadratic Hamiltonians
• A generalization of Clausen's identity
• On Integrability of Nonautonomous Nonlinear Schroedinger Equations
• Dynamical Invariants and Berry's Phase for Generalized Driven Harmonic Oscillators
• On Schroedinger Equation with Time-Dependent Quadratic Hamiltonian in $R^d$
• A Graphical Approach to a Model of Neuronal Tree with Variable Diameter
• On a Hidden Symmetry of Quantum Harmonic Oscillators
• Exact Wave Functions for Generalized Harmonic Oscillators
• The Riccati System and a Diffusion-Type Equation
• On persistence of superoscillations for Schrödinger equation with time-dependent quadratic Hamiltonians