### 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 |
---|---|

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 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Physics A: Mathematical and General*,

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

**Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton-Ehrenfest system.** / Belov, V. V.; Kondratieva, M. F.; Trifonov, A. Yu.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 39, no. 34, 015, pp. 10821-10847. https://doi.org/10.1088/0305-4470/39/34/015

}

TY - JOUR

T1 - Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton-Ehrenfest system

AU - Belov, V. V.

AU - Kondratieva, M. F.

AU - Trifonov, A. Yu

PY - 2006/8/25

Y1 - 2006/8/25

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33747175767&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747175767&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/39/34/015

DO - 10.1088/0305-4470/39/34/015

M3 - Article

VL - 39

SP - 10821

EP - 10847

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

SN - 0305-4470

IS - 34

M1 - 015

ER -