Coherent State Maps for Kummer Shapes

  • Anatol Odzijewicz
  • Ewa Wawreniuk
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 205)


In the paper we investigate a system of nonlinearly coupled harmonic oscillators in the quantum as well as classical mechanics frameworks. Combining the Marsden-Weinstein reduction procedure with the quantum reduction procedure we construct coherent states map for this system and find explicit expression for the suitable reproducing measure.


  1. 1.
    T. Goliński, A. Odzijewicz, Hierarchy of integrable Hamiltonians describing the nonlinear n-wave interaction. J. Phys. A Math. Theor. 45(4), 045204 (2012)Google Scholar
  2. 2.
    T. Goliński, M. Horowski, A. Odzijewicz, A. Sliżewska, \(sl(2,R)\) symmetry and solvable multiboson system. J. Math. Phys. 48(2), 023508 (1–19) (2007)Google Scholar
  3. 3.
    D.D. Holm, Geometric Mechanics, Part I: Dynamics and Symmetry (Imperial College Press, London, 2008)zbMATHGoogle Scholar
  4. 4.
    M. Horowski, A. Odzijewicz, Positive kernels and quantization. J. Geom. Phys. 63, 80–98 (2013)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Horowski, A. Odzijewicz, A. Tereszkiewicz, Some integrate system in nonlinear quantum optics. J. Math. Phys. 44(2), 480–506 (2003)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Kummer, On resonant classical Hamiltonians with \(n\) frequencies. J. Diff. Eq. 83, 220–243 (1990)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Odzijewicz, Coherent states and geometric quantization. Commun. Math. Phys. 150, 385–413 (1992)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Odzijewicz, E. Wawreniuk, Classical and quantum Kummer shape algebras. J. Phys. A: Math. Gen. 49, 265202 (2016)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Odzijewicz, T.S. Ratiu, Banach lie-poisson spaces and reduction. Commun. Math. Phys. 243(1), 1–54 (2003)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Odzijewicz, M. Świȩtochowski, Coherent states map for MIC-Kepler system. J. Math. Phys. 38(10), 5010–5030 (1997)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    V. Sunilkumar, B.A. Bambah, R. Jagannathan, P.K. Panigrahi, V. Srinivasan, Coherent states of nonlinear algebras: applications to quantum optics. J. Opt. B: Quantum Semiclass. Opt. 2, 126–132 (2000)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Tereszkiewicz, A. Odzijewicz, M. Horowski, I. Jex, G. Chadzitaskos, Explicitly solvable models of two-mode coupler in Kerr media. Phys. Rev. A 75, 063817(1–10) (2007)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity in BiałystokBiałystokPoland

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