The Time Inversion for Modified Oscillators

Abstract

We discuss a new completely integrable case of the time-dependent Schroedinger equation in $R^n$ with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator recently considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is also found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. Particular solutions of the corresponding Schroedinger equations are also obtained. A Hamiltonian structure of the classical integrable problem and its quantization are also discussed.

33 pages, four figures, and two tables

References in corpus (4)

• The Cauchy Problem for a Forced Harmonic Oscillator
• Semiclassical Nonlinear Schrodinger equations with potential and focusing initial data
• An Integral Form of the Nonlinear Schroedinger Equation with Variable Coefficients
• Hydrodynamic approach to constructing solutions of Hydrodynamic approach to constructing solutions of nonlinear Schrödinger equation in the critical case

Cited by in corpus (1)

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