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Reduction and geometric quantization

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A construction is created that makes it possible to geometrically quantize a reduced Hamiltonian system using the procedure of geometric quantization realized for a Hamiltonian system with symmetries (i.e., to find the discrete spectrum and the corresponding eigenfunctions, if these have been found for the initial system). The construction is used to geometrically quantize a system obtained by reduction of a Hamiltonian system that determines the geodesic flow on an n-dimensional sphere.

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  1. 1.

    O. S. Parasyuk, “Horocycle flows on surfaces of constant curvature,” Usp. Mat. Nauk,8, No. 3, 125–126 (1953).

  2. 2.

    S. V. Manakov, “Remark on the integrability of Euler's dynamics equations of an n-dimensional solid,” Funkts. Analiz,10, No. 4, 93–94 (1976).

  3. 3.

    Yu. Mozer, “Certain aspects of integrable Hamiltonian systems,” Usp. Mat. Nauk,36, No. 5, 109–151 (1981).

  4. 4.

    M. A. Ol'shanetskii and A. M. Perelomov, “Toda chain as a reduced system,” Teoret. i Mat. Fiz.,45, No. 1, 3–18 (1980).

  5. 5.

    M. J. Gotay, “Constraints, reduction, and quantization,” J. Math. Phys.,27, No. 8, 2051–2066 (1986).

  6. 6.

    K. Ii, “Geometric quantization for the mechanics on spheres,” Tohöku Math. J.,33, 289–295 (1981).

  7. 7.

    J. Marsden and A. Weinstein, “Reduction of symplectic manifolds with symmetry,” Repts. Math. Phys.,5, No. 1, 121–130 (1974).

  8. 8.

    A. K. Prikarpatskii and I. V. Mikityuk, Algebraic Aspects of Integrability of Nonlinear Dynamic Systems on Manifolds [in Russian], Nauk. Dumka, Kiev (1991), 286 pp.

  9. 9.

    J. Sniatycki, Geometric Quantization and Quantum Mechanics, Springer, New York (1980), 230 pp.

  10. 10.

    S. Sternberg, Lectures on Differential Geometry [Russian translation], Mir, Moscow (1970), 412 pp.

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Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1220–1228, September, 1992.

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Mykytyuk, I.V., Prykarpats'kyy, A.K. Reduction and geometric quantization. Ukr Math J 44, 1116–1122 (1992). https://doi.org/10.1007/BF01058372

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  • Hamiltonian System
  • Discrete Spectrum
  • Initial System
  • Geometric Quantization
  • Geodesic Flow