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**Integrability in the mesoscopic dynamics.**
*(English)*
Zbl 1078.81520

Summary: The Mesoscopic Mechanics (MeM), as introduced in [the author, J. Chem. Solids 65, No. 8/9, 1507–1515 (2004)], is relevant to the electron gas confined to two spatial dimensions. It predicts a special way of collective response of correlated electrons to the external magnetic field. The dynamic variable of this theory is a finite-dimensional operator, which is required to satisfy the mesoscopic Schrödinger equation [cf. V. I. Fal’ko, Phys. Rev. B 50, No. 23, 17406–17410 (1994)].

In this article, we describe general solutions of the mesoscopic Schrödinger equation. Our approach is specific to the problem at hand. It relies on the unique structure of the equation and makes no reference to any other techniques, with the exception of the geometry of unitary groups. In conclusion, a surprising fact comes to light. Namely, the mesoscopic dynamics “filters” through the (microscopic) Schrödinger dynamics as the latter turns out to be a clearly separable part, in fact an autonomous factor, of the evolution. This is a desirable result also from the physical standpoint.

In this article, we describe general solutions of the mesoscopic Schrödinger equation. Our approach is specific to the problem at hand. It relies on the unique structure of the equation and makes no reference to any other techniques, with the exception of the geometry of unitary groups. In conclusion, a surprising fact comes to light. Namely, the mesoscopic dynamics “filters” through the (microscopic) Schrödinger dynamics as the latter turns out to be a clearly separable part, in fact an autonomous factor, of the evolution. This is a desirable result also from the physical standpoint.

### MSC:

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

### Keywords:

mesoscopic dynamics; Schrödinger equation; integrable; nonlinear differential equation; operator evolution equation; correlated electrons### References:

[1] | V.I. Arnold, Mathematical Methods of Classical Mechanics, Nauka, Moscow, 1974 [Polish Transl.: PWN, Warszawa, 1981; English Transl.: Springer-Verlag, New York, 1978].; V.I. Arnold, Mathematical Methods of Classical Mechanics, Nauka, Moscow, 1974 [Polish Transl.: PWN, Warszawa, 1981; English Transl.: Springer-Verlag, New York, 1978]. · Zbl 0647.70001 |

[2] | Fal’ko, V. I., Phys. Rev. B, 50, 23, 17406-17410 (1994) |

[3] | W. Greiner, J. Reinhardt, Field Quantization, Springer-Verlag, Berlin, 1996.; W. Greiner, J. Reinhardt, Field Quantization, Springer-Verlag, Berlin, 1996. · Zbl 0844.00006 |

[4] | Milnor, J., Morse Theory (1963), Princeton University Press: Princeton University Press Princeton · Zbl 0108.10401 |

[5] | Sowa, A., J. Phys. Chem. Solids, 65, 8/9, 1507-1515 (2004) |

[6] | Sowa, A., J. Geom. Phys., 45, 54-74 (2003) · Zbl 1012.78003 |

[7] | Sowa, A., Commun. Math. Phys., 226, 559-566 (2002) · Zbl 1005.78001 |

[8] | Vladimirov, V. S., Equations of Mathematical Physics (1988), Nauka: Nauka Moscow · Zbl 0652.35002 |

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