### Abstract

Based on the ideology of the Maslov complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated functions. The nonlinear evolution operator has been obtained in explicit form in the class of semiclassically concentrated functions. Parametric families of symmetry operators have been found for the Hartree-type equation. With the help of symmetry operators, families of exact solutions of the equation have been constructed. Exact expressions are obtained for the quasi-energies and their respective states. The Aharonov-Anandan geometric phases are found in explicit form for the quasi-energy states.

Original language | English |
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Pages (from-to) | 4535-4556 |

Number of pages | 22 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 37 |

Issue number | 16 |

DOIs | |

Publication status | Published - 23 Apr 2004 |

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### ASJC Scopus subject areas

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

### Cite this

**The evolution operator of the Hartree-type equation with a quadratic potential.** / Lisok, Alexander Leonidovich; Trifonov, A. Yu; Shapovalov, Aleksandr Vasilievich.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The evolution operator of the Hartree-type equation with a quadratic potential

AU - Lisok, Alexander Leonidovich

AU - Trifonov, A. Yu

AU - Shapovalov, Aleksandr Vasilievich

PY - 2004/4/23

Y1 - 2004/4/23

N2 - Based on the ideology of the Maslov complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated functions. The nonlinear evolution operator has been obtained in explicit form in the class of semiclassically concentrated functions. Parametric families of symmetry operators have been found for the Hartree-type equation. With the help of symmetry operators, families of exact solutions of the equation have been constructed. Exact expressions are obtained for the quasi-energies and their respective states. The Aharonov-Anandan geometric phases are found in explicit form for the quasi-energy states.

AB - Based on the ideology of the Maslov complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated functions. The nonlinear evolution operator has been obtained in explicit form in the class of semiclassically concentrated functions. Parametric families of symmetry operators have been found for the Hartree-type equation. With the help of symmetry operators, families of exact solutions of the equation have been constructed. Exact expressions are obtained for the quasi-energies and their respective states. The Aharonov-Anandan geometric phases are found in explicit form for the quasi-energy states.

UR - http://www.scopus.com/inward/record.url?scp=2442449392&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2442449392&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/37/16/005

DO - 10.1088/0305-4470/37/16/005

M3 - Article

VL - 37

SP - 4535

EP - 4556

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

SN - 0305-4470

IS - 16

ER -