### Abstract

Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate a quantum-mechanical state as → 0. This idea leads to a family of systems of ordinary differential equations, called Hamilton-Ehrenfest M-systems (M ≤ 0, 1, 2, ...). As noted in the authors' previous works, every M-system is formally equivalent to the semiclassical approximation of order M for the linear Schrödinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Hamilton-Ehrenfest systems, without solving the quantum equation: the semiclassical spectral series are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.

Original language | English |
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Article number | 015 |

Pages (from-to) | 10821-10847 |

Number of pages | 27 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 39 |

Issue number | 34 |

DOIs | |

Publication status | Published - 25 Aug 2006 |

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Physics A: Mathematical and General*,

*39*(34), 10821-10847. [015]. https://doi.org/10.1088/0305-4470/39/34/015