Advertisement

Mathematical Notes

, Volume 104, Issue 5–6, pp 833–847 | Cite as

Algebra of Symmetries of Three-Frequency Resonance: Reduction of a Reducible Case to an Irreducible Case

  • M. V. KarasevEmail author
  • E. M. Novikova
Article
  • 10 Downloads

Abstract

For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

Keywords

frequency resonance algebra of symmetries nonlinear commutation relations coherent states 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in Itogi Nauki i Tekhniki [Progress in Science and Technology, Current Problems in Mathematics. Fundamental Directions], Vol. 3: Dynamical Systems (VINITI, Moscow, 1985), pp. 5–290 [in Russian].Google Scholar
  2. 2.
    A. S. Egilsson, “On embedding the 1: 1: 2 resonance space in a Poissonmanifold,” Electron. Res. Announc. Amer.Math. Soc. 1 (2), 48–56 (1995).MathSciNetCrossRefGoogle Scholar
  3. 3.
    H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts Monogr. Phys. (Springer-Verlag, Berlin, 1987; Mir, Moscow, 1990).zbMATHGoogle Scholar
  4. 4.
    V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Nauka, Moscow, 1972) [in Russian].Google Scholar
  5. 5.
    B. Simon, “Semiclassical analysis of low lying eigenvalues. I,” Ann. Inst. H. Poincaré. Phys. Théor. 38 (3), 295–307 (1983); Erratum in 40,224.zbMATHGoogle Scholar
  6. 6.
    B. Helffer and J. Sjöstrand, “Multiple wells in the semiclassical limit. I,” Comm. Partial Diff. Equ. 9 (4), 337–408 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Sjöstrand, “Semi-excited states in nondegenerate potential wells,” Asymptot. Anal. 6, 29–43 (1992).MathSciNetzbMATHGoogle Scholar
  8. 8.
    F. G. Gustavson, “On constructing formal integrals of a Hamiltonian system near an equilibrium point,” Astron. J. 71, 670–686 (1966).CrossRefGoogle Scholar
  9. 9.
    R. T. Swimm and J. B. Delos, “Semiclassical calculations of vibrational energy levels for nonseparable Systems using the Birkhoff–Gustavson normal form,” J. Chem. Phys. 71 (4), 1706–1717 (1979).MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. K. Ali, “The quantum normal form and its equivalents,” J. Math. Phys. 26 (10), 2565–2572 (1985).MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Eckhardt, “Birkhoff–Gustavson normal form in classical and quantum mechanics,” J. Phys. A 19 (1986), 2961–2972.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. V. Kozlov, Symmetries, Topology, and Resonances in Hamiltonian Mechanics, in Ergeb. Math. Grenzgeb. (Springer-Verlag, Berlin, 1996; Izd. UdGU, Izhevsk, 1995), Vol.31.Google Scholar
  13. 13.
    J. Robert and M. Joyeux, “Canonical Perturbation Theory versus Born–Oppenheimer-Type separation of motions: the vibrational dynamics of C3,” J. Chem. Phys. 119, 8761–8762 (2003).CrossRefGoogle Scholar
  14. 14.
    M. V. Karasev, “Noncommutative algebras, nano-structures, and quantum dynamics generated by resonances. II,” Adv. Stud. Contemp.Math. 11, 33–56 (2005).MathSciNetzbMATHGoogle Scholar
  15. 15.
    M. V. Karasev and E. M. Novikova, “Algebra and quantum geometry ofmultifrequency resonance,” Izv. Ross. Akad. Nauk Ser.Mat. 74 (6), 55–106 (2010).MathSciNetCrossRefGoogle Scholar
  16. 16.
    E. M. Novikova, “Minimal basis of the symmetry algebra for three-frequency resonance,” Russ. J. Math. Phys. 16 (4), 518–528 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    M. Karasev, “Advances in quantization: quantum tensors, explicit star-products, and restriction to irreducible leaves,” Diff. Geom. Appl. 9 (1–2), 89–134 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Karasev and E. Novikova, “Non-Lie permutation relations, coherent states, and quantum embedding,” in Coherent Transform, Quantization, and Poisson Geometry, Ed. by M.V. Karasev (AMS, Providence, RI, 1998), Vol. 187, pp. 1–202.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

Personalised recommendations