Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 061, 50 pages      math-ph/0502028      http://dx.doi.org/10.3842/SIGMA.2007.061
Contribution to the Vadim Kuznetsov Memorial Issue

Completely Integrable Systems Associated with Classical Root Systems

Toshio Oshima
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8914, Japan

Received December 14, 2006, in final form March 19, 2007; Published online April 25, 2007

Abstract
We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants. We review conditions supporting the conjecture and give a new condition assuring it.

Key words: completely integrable systems; Calogero-Moser systems; Toda lattices with boundary conditions.

pdf (554 kb)   ps (337 kb)   tex (45 kb)

References

  1. Adler M., Some finite-dimensional integrable systems and their behaviour, Comm. Math. Phys. 55 (1977), 195-230.
  2. Bogoyavlensky O.I., On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201-209.
  3. Braden H.W., Byatt-Smith J.G.B., On a functional differential equation of determinantal types, Bull. London Math. Soc. 31 (1999), 463-470, math.CA/9804082.
  4. Buchstaber V.M., Perelomov A.M., On the functional equation related to the quantum three-body problem, in Berezin Memorial Volume, Amer. Math. Soc. Transl. (2) 175 (1996), 15-34, math-ph/0205032.
  5. Calogero F., Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair-potentials, J. Math. Phys. 12 (1971), 419-436.
  6. Chalykh O.A., Feigin M., Veselov A., New integrable generalizations of Calogero-Moser quantum problems, J. Math. Phys. 39 (1998), 695-703.
  7. Chalykh O.A., Veselov A., Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126 (1990), 597-611.
  8. van Diejen J.F., Difference Calogero-Moser systems and finite Toda chains, J. Math. Phys. 36 (1995), 1299-1323.
  9. Goodman R., Wallach N.R., Classical and quantum-mechanical systems of Toda lattice type. I, Comm. Math. Phys. 83 (1982), 355-386.
  10. Goodman R., Wallach N.R., Classical and quantum-mechanical systems of Toda lattice type. III. Joint eigenfunctions of the quantized systems, Comm. Math. Phys. 105 (1986), 473-509.
  11. Heckman G.J., Opdam E.M., Root system and hypergeometric functions. I, Comp. Math. 64 (1987), 329-352.
  12. Inozemtsev V.I., Lax representation with spectral parameter on a torus for integrable particle systems, Lett. Math. Phys. 17 (1989), 11-17.
  13. Inozemtsev V.I., The finite Toda lattices, Comm. Math. Phys. 121 (1989), 629-638.
  14. Levi D., Wojciechowski S., On the Olshanetsky-Perelomov many-body system in an external field, Phys. Lett. A 103 (1984), 11-14.
  15. Helgason S., Groups and geometric analysis, Academic Press, 1984.
  16. Kashiwara M., Oshima T., Systems of differential equations with regular singularities and their boundary value problems, Ann. Math. 106 (1977), 145-200.
  17. Kostant B., The solution to a generalized Toda lattices and representation theory, Adv. Math. 34 (1979), 195-338.
  18. Kuznetsov V.B., Separation of variables for the Dn-type periodic Toda lattice, J. Phys. A: Math. Gen. 30 (1997), 2127-2138, solv-int/9701009.
  19. Kuznetsov V.B., Jørgensen J., Christiansen P.L., New boundary conditions for integrable lattices, J. Phys. A: Math. Gen. 28 (1995), 4639-4654, hep-th/9503168.
  20. Kuznetsov V.B., Tsyganov A.V., Infinite series of Lie algebras and boundary conditions for integrable systems, J. Soviet Math. 59 (1992), 1085-1092.
  21. Kuznetsov V.B., Tsyganov A.V., Separation of variables for the quantum relativistic Toda lattices, Mathematical Preprint Series, University of Amsterdam, hep-th/9402111.
  22. Moser J., Three integrable Hamiltonian systems connected with isospectral deformation, Adv. Math. 16 (1975), 197-220.
  23. Ochiai H., Commuting differential operators of rank two, Indag. Math. (N.S.) 7 (1996), 243-255.
  24. Ochiai H., Oshima T., Commuting differential operators with B2 symmetry, Funkcial. Ekvac. 46 (2003), 297-336.
  25. Ochiai H., Oshima T., Sekiguchi H., Commuting families of symmetric differential operators, Proc. Japan Acad. A 70 (1994), 62-66.
  26. Olshanetsky M.A., Perelomov A.M., Explicit solutions of the periodic Toda lattices, Invent. Math. 54 (1979), 261-269.
  27. Olshanetsky M.A., Perelomov A.M., Classical integrable finite dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), 313-400.
  28. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  29. Oshima T., Completely integrable systems with a symmetry in coordinates, Asian J. Math. 2 (1998), 935-956, math-ph/0502028.
  30. Oshima T., A class of completely integrable quantum systems associated with classical root systems, Indag. Mathem. 48 (2005), 655-677, math-ph/0502019.
  31. Oshima T., Commuting differential operators with regular singularities, in Algebraic Analysis of Differential Equations, Springer-Verlag, Tokyo, to appear, math.AP/0611899.
  32. Oshima T., Sekiguchi H., Commuting families of differential operators invariant under the action of a Weyl group, J. Math. Sci. Univ. Tokyo 2 (1995), 1-75.
  33. Ruijsenaars S.N.M., Systems of Calogero-Moser type, Proceedings of the 1994 CRM Banff Summer School 'Particles and Fields', CRM Ser. Math. Phys. (1999), 251-352.
  34. Sekiguchi J., Zonal spherical functions on some symmetric spaces, Publ. Res. Inst. Math. Sci. 12 (1977), suppl., 455-459.
  35. Sergeev A.N., Veselov A.P., Deformed quantum Calogero-Moser problems and Lie superalgebras, Comm. Math. Phys. 245 (2004), 249-278, math-ph/0303025.
  36. Sutherland B., Exact results for a quantum many-body problems in one dimension II, Phys. Rev. A 5 (1972), 1372-1376.
  37. Taniguchi K., On the symmetry of commuting differential operators with singularities along hyperplanes, Int. Math. Res. Not. 36 (2004), 1845-1867, math-ph/0309011.
  38. Toda M., Wave propagation in anharmonic lattice, IPSJ J. 23 (1967), 501-506.
  39. Veselov A., Stykas K.L., Chalykh O.A., Algebraic integrability for the Schrödinger equations and groups generated by reflections, Theor. Math. Phys. 94 (1993), 573-574.
  40. Wakida S., Quantum integrable systems associated with classical Weyl groups, Master Thesis, University of Tokyo, 2004.
  41. Whittaker E.T., Watson G.N., A course of modern analysis, 4th ed., Cambridge University Press, 1927.

Previous article   Next article   Contents of Volume 3 (2007)