Communications in Mathematical Physics

, Volume 309, Issue 1, pp 123–154 | Cite as

Hamiltonian Dynamics and Spectral Theory for Spin–Oscillators

Article

Abstract

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah M.F.: Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14(1), 1–15 (1982)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Babelon, O., Cantini, L., Douç, B.: A semi-classical study of the Jaynes–Cummings model. J. Stat. Mech. Theory Exp. (2009). doi:10.1088/1742-5468/2009/07/P07011
  3. 3.
    Bargmann V.: On a Hilbert space of analytic functions and an associated integral transform I. Comm. Pure Appl. Math. 19, 187–214 (1961)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Boutetde Monvel L., Guillemin V.: spectral theory of Toeplitz operators Number 99 in Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1981)Google Scholar
  5. 5.
    Charles L.: Berezin-toeplitz operators, a semi-classical approach. Commun. Math. Phys. 239(1-2), 1–28 (2003)CrossRefMATHADSMathSciNetGoogle Scholar
  6. 6.
    Cushman R., Duistermaat J.J.: The quantum spherical pendulum. Bull. Amer. Math. Soc. (N.S.) 19, 475–479 (1988)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Delzant T.: Hamiltoniens périodiques et image convexe de l’application moment. Bull. Soc. Math. France 116, 315–339 (1988)MATHMathSciNetGoogle Scholar
  8. 8.
    Dimassi M., Sjöstrand J.: Spectral asymptotics in the semi-classical limit Volume 268 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1999)Google Scholar
  9. 9.
    Duistermaat J.J.: On global action-angle variables. Comm. Pure Appl. Math. 33, 687–706 (1980)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Duistermaat J.J., Heckman G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69, 259–268 (1982)CrossRefMATHADSMathSciNetGoogle Scholar
  11. 11.
    Eliasson, L.H.: Hamiltonian systems with Poisson commuting integrals. PhD thesis, University of Stockholm, 1984Google Scholar
  12. 12.
    Garay M., van Straten D.: Classical and quantum integrability. Mosc. Math. J. 10, 519–545 (2010)MATHMathSciNetGoogle Scholar
  13. 13.
    Groenewold H.J.: On the principles of elementary quantum mechanics. Physica 12, 405–460 (1946)CrossRefMATHADSMathSciNetGoogle Scholar
  14. 14.
    Gross M., Siebert B.: Mirror symmetry via logarithmic degeneration data. I. J. Diff. Geom. 72(2), 169–338 (2006)MATHMathSciNetGoogle Scholar
  15. 15.
    Guillemin V., Sternberg S.: Convexity properties of the moment mapping. Invent. Math. 67(3), 491–513 (1982)CrossRefMATHADSMathSciNetGoogle Scholar
  16. 16.
    Kostant, B., Pelayo, Á.: Geometric Quantization. Monograph to appear in Springer.Google Scholar
  17. 17.
    Leung N.C., Symington M.: Almost toric symplectic four-manifolds. J. Sympl. Geom. 8, 143–187 (2011)MathSciNetGoogle Scholar
  18. 18.
    Moyal J.E.: Quantum mechanics as a statistical theory. Proc. Cambridge Philos. Soc. 45, 99–124 (1949)CrossRefMATHADSMathSciNetGoogle Scholar
  19. 19.
    Pelayo Á., Vũ Ngoc S.: Semitoric integrable systems on symplectic 4-manifolds. Invent. Math. 177, 571–597 (2009)CrossRefMATHADSMathSciNetGoogle Scholar
  20. 20.
    Pelayo Á., Vũ Ngọc S.: Constructing integrable systems of semitoric type. Acta Math. 206, 93–125 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Pelayo Á., Vũ Ngoc S.: Symplectic theory of completely integrable Hamiltonian systems. Bull. Amer. Math. Soc 48, 409–455 (2011)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Symington, M.: Four dimensions from two in symplectic topology. In: Topology and geometry of manifolds (Athens, GA, 2001), Volume 71 of Proc. Sympos. Pure Math., Providence, RI: Amer. Math. Soc., 2003, pp. 153–208Google Scholar
  23. 23.
    Vũ Ngoc, S.: Symplectic inverse spectral theory for pseudodifferential operators. In: Geometric aspects of analysis and mechanics. Progress in Mathematics, vol. 292, pp. 353–372. Birkhäuser, Boston (2011)Google Scholar
  24. 24.
    Vũ Ngoc S.: Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type. Comm. Pure Appl. Math. 53(2), 143–217 (2000)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Vũ Ngoc S.: On semi-global invariants for focus-focus singularities. Topology 42(2), 365–380 (2003)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Vũ Ngoc, S.: Systèmes intégrables semi-classiques: du local au global. Number 22 in Panoramas et Syhthèses. Paris: SMF, 2006Google Scholar
  27. 27.
    Vũ Ngoc S.: Moment polytopes for symplectic manifolds with monodromy. Adv. in Math. 208, 909–934 (2007)CrossRefMATHGoogle Scholar
  28. 28.
    Vũ Ngọc, S., Wacheux, C.: Smooth normal forms for integrable hamiltonian systems near a focus-focus singularity. http://arXiv.org/abs/1103.3282v1 [math.SG], 2011
  29. 29.
    Weyl, H.: The theory of groups and quantum mechanics. Newyork: Dover, 1950, translated from the (second) German editionGoogle Scholar
  30. 30.
    Williamson J.: On the algebraic problem concerning the normal form of linear dynamical systems. Amer. J. Math. 58(1), 141–163 (1936)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Zung, N.T.: A topological classification of integrable hamiltonian systems. In: R. Brouzet, editor, Séminaire Gaston Darboux de géometrie et topologie différentielle, Université Montpellier II, 1994–1995, pp. 43–54Google Scholar
  32. 32.
    Zung N.T.: Symplectic topology of integrable hamiltonian systems, I: Arnold-Liouville with singularities. Compositio Math. 101, 179–215 (1996)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Mathematics DepartmentWashington UniversitySt. LouisUSA
  3. 3.Institut de Recherches Mathématiques de RennesUniversité de Rennes 1Rennes CedexFrance

Personalised recommendations