Letters in Mathematical Physics

, Volume 107, Issue 1, pp 187–200 | Cite as

Degenerate integrability of quantum spin Calogero–Moser systems

Article

Abstract

The main result of this note is the proof of degenerate quantum integrability of quantum spin Calogero–Moser systems and the description of the spectrum of quantum Hamiltonians in terms of the decomposition of tensor products of irreducible representations of corresponding Lie algebra.

Keywords

Quantum Superintegrable Calogero–Moser 

Mathematics Subject Classification

81E12 22E46 22E45 

References

  1. 1.
    Avan, J., Babelon, O., Billey, E.: The Gervais–Neveu–Felder equation and the quantum Calogeor–Moser system. Commun. Math. Phys. 178, 281–299 (1996)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Ayadi, V., Feher, L., Gorbe, T.F.: Superintegrability of rational Ruijsenaars–chneider systems and their action-angle duals. J. Geom. Symmetry Phys. 27, 27–44 (2012)MATHGoogle Scholar
  3. 3.
    Calogero, F.: Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Etingof, P., Kirillov Jr., A.: On a unified representatiuon theoretical approach to special functions. Funk. Anal. i Prilozh. 28, 91–94 (1994)Google Scholar
  5. 5.
    Etingof, P., Frenkel, I., Kirillov, A.: Spherical functions on affine Lie groups. Duke Math. J. 80, 79–90 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Etingof, P.I.: Quantum integrable systems and representations of Lie alge bras. J. Math. Phys. 36, 2636–2651 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Etingof, P., Varchenko, A.: Traces of intertwiners for quantum groups and difference equations. I. Duke Math. J. 104(3), 391–432 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Etingof, P., Schiffmann, O., Varchenko, A.: Traces of intertwiners for quantum groups and difference equations. Lett. Math. Phys. 62(2), 143–158 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Etingof, P., Varchenko, A.: Orthogonality and the QKZB-heat equation for traces of Uq(g)-intertwiners. Duke Math. J. 128(1), 83–117 (2005)CrossRefMATHGoogle Scholar
  10. 10.
    Feher, L., Pusztai, B.G.: Twisted spin Sutherland models from quantum Hamiltonian reduction. J. Phys. A 41(19), 194009 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fomenko, A.T.: Symplectic Geometry, Advanced Studies in Contemporary Mathematics, vol. 5. Gordon and Breach Science Publishers, New York (1988)Google Scholar
  12. 12.
    Frish, J., Mandrosov, V., Smorodinsky, Y.A., Uhlir, M., Winternitz, P.: On higher symmetries in quantum mechanics. Phys. Lett. 16, 354–356 (1965)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gibbons, J., Hermsen, T.: A generalization of the Calogero–Moser system. Physica 11D, 337 (1984)ADSMATHGoogle Scholar
  14. 14.
    Hikami, K., Wadati, M.: Integrability of Calogero–Moser spin system. J. Phys. Soc. Jpn. 62(2), 469–472 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. 31(4), 481–507 (1978)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Krichever, I., Babelon, O., Billey, E., Talon, M.: Spin generalization of Calogero–Moser system and the matrix KP equation. Transl AMS Ser 2 Adv Math Sci Topics Topol Math. Phys. 170 (1975). arXiv:hep-th/9411160
  17. 17.
    Kuznetsov, V.B.: Hidden symmetry of the quantum Calogero–Moser system. Phys. Lett. A 218(3–6), 212–222 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Li, L.C., Xu, P.: Spin Calogero–Moser systems associated with simple Lie algebras. Acad. Sci. Paris Serie I 331(1), 55–61 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220 (1975)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nakai, Ryota, Kato, Yusuke: Particle propagator of spin Calogero–Sutherland model. J. Phys. A Math. Theor. 47, 305205 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nekhoroshev, N.N.: Action-angle variables and their generalizations. Trans. Moscow Math. Soc. 26, 180–197 (1972)MathSciNetMATHGoogle Scholar
  22. 22.
    Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rept. 94, 313–404 (1983)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Pauli, W.: On the hydrogen spectrum from the standpoint of the new quantum mechanics. Zeitschrift fur Physik 36, 336–363 (1926)ADSCrossRefGoogle Scholar
  24. 24.
    Reshetikhin, N.: Degenerate integrability of the spin Calogero–Moser systems and the duality with the spin Ruijsenaars systems. Lett. Math. Phys. 63(1), 55–71 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Reshetikhin, N.: Degenerately integrable systems. Preprint, arXiv:1509.00730
  26. 26.
    Sutherland, B.: Exact results for a many-body problem in one dimension. II. Phys. Rev. A 5(3), 1372–1376 (1972)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Tempesta, P., Winternitz, P., Harnad, J., Miller Jr., W., Pogosyan, G., Rodriguez, M. (eds.).: Superintegrability in classical and quantum systems. CRM Proc. Lect. Notes 37 (2004)Google Scholar
  28. 28.
    Wojciechowski, S.: Superintegrability of the Calogero–Moser system. Phys. Lett. A 95, 279 (1983)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.ITMO UniversitySaint PetersburgRussia
  3. 3.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations