## Abstract

We consider a new completely integrable case of the time-dependent Schrödinger equation in ℝ^{n} with variable coefficients for a modified oscillator that is dual (with respect to time reversal) to a model of the quantum oscillator. We find a second pair of dual Hamiltonians in the momentum representation. The examples considered show that in mathematical physics and quantum mechanics, a change in the time direction may require a total change of the system dynamics to return the system to its original quantum state. We obtain particular solutions of the corresponding nonlinear Schrödinger equations. We also consider a Hamiltonian structure of the classical integrable problem and its quantization.

Original language | English (US) |
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Pages (from-to) | 286-316 |

Number of pages | 31 |

Journal | Theoretical and Mathematical Physics |

Volume | 162 |

Issue number | 3 |

DOIs | |

State | Published - 2010 |

## Keywords

- Cauchy initial value problem
- Green's function
- Hyperspherical harmonic
- Nonlinear Schrödinger equation
- Propagator
- Schrödinger equation with variable coefficients
- Time reversal

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics