The Bispectral Involution as a Linearizing Map

  • Alex Kasman
Part of the CRM Series in Mathematical Physics book series (CRM)


In 1975, Moser introduced the map σ to linearize the motion of the Calogero—Moser—Sutherland model with rational inverse square potential. This linearizing map was unusual in that it turned out to be an involution on the phase space. In 1990, Wilson introduced an involution on a part of the Sato Grassmannian to demonstrate the bispectrality of certain Kadomtsev—Petviashvili (KP) solutions. Using the correspondence between particle systems and the poles of KP solutions, one may then also view Wilson’s bispectral involution as a map on the phase space of the particle system and compare them. As a result, we find that the bispectral involution is the linearizing map of the Calogero—Moser particle system and that it generates nonisospectral master symmetries of the KP flows. This chapter will review this material and then update it with reference to recent papers of Wilson, Shiota, Kasman—Rothstein, and Rothstein that simplify the proof and extend the results to include other integrable Hamiltonian systems of particle systems with linearizing involutions.


Particle System Young Diagram Darboux Transformation Integrable Hamiltonian System Phase Space Variable 
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  1. 1.
    H. Airault, H. P. McKean, and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Commun. Pure Appl. Math. 30 (1977), No. 1, 95–148.MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    B. Bakalov, E. Horozov, and M. Yakimov, Bäcklund-Darboux transformations in Sato’s Grassmannian, Serdica Math. J. 22 (1996), No. 4, 571–588.MathSciNetMATHGoogle Scholar
  3. 3.
    B. Bakalov, E. Horozov, and M. Yakimov, Highest weight modules of W1+∞, Darboux transformations and the bispectral problem, Serdica Math. J. 23 (1997), No. 2, 95–112.MathSciNetMATHGoogle Scholar
  4. 4.
    J. F. van Diejen, Deformations of Calogero-Moser systems, Theoret, and Math. Phys. (1994), No. 2, 549–554.Google Scholar
  5. 5.
    J. J. Duistermaat and F. Grünbaum, Differential equations in the spectral parameter, Commun. Math. Phys. 103 (1986), No. 2, 177–240.ADSMATHCrossRefGoogle Scholar
  6. 6.
    F. A. Grünbaum, The limited angle problem in tomography and some related mathematical problems, Bifurcation Theory, Mechanics and Physics (C. P. Bruter, A. Aragnol, and A. Lichnerowicz, eds.), Math. Appl., Reidei, Dordrecht, 1983, pp. 317–329.Google Scholar
  7. 7.
    W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry. II, Cambridge Univ. Press, Cambridge, 1952.MATHGoogle Scholar
  8. 8.
    A. Kasman, Bispectral KP solutions and linearization of Calo-gero-Moser particle systems, Commun. Math. Phys. 172 (1995), No. 2, 427–448.MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    A. Kasman, Darboux transformations from n-KdV to KP, Acta Appl. Math. 49 (1997), No. 2, 179–197.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    A. Kasman and M. Rothstein, Bispectral Darboux transformations, Physica 102D (1997), No. 3-4, 159–177.MathSciNetADSGoogle Scholar
  11. 11.
    I. M. Krichever, Rational solutions of the Kadomtsev-Petviašhvili equation and integrable systems of N particles on a line, Funct. Anal. Appl. 12 (1978), No. 1, 59–61.MATHGoogle Scholar
  12. 12.
    W. Oevel and M. Falck, Master symmetries for finite-dimensional integrable systems: the Calogero-Moser system, Progr. Theor. Phys. 75 (1986), No. 6, 1328–1341.MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    M. Rothstein, Explicit formulas for the Airy and Bessel bispectral involutions in terms of Calogero-Moser pairs, The Bispectral Problem (Montréal, QC, 1997) (J. Hamad and A. Kasman, eds.), CRM Proc. Lecture Notes, Vol. 14, Amer. Math. Soc, Providence, RI, 1998, pp. 105–110, q-alg/9611027.Google Scholar
  14. 14.
    M. Sato and Y. Sato, Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982) (H. Fujita, P. D. Lax, and G. Strang, eds.), Lecture Notes Numer. Appl. Anal., Vol. 5, Ki-nokuniya, Tokyo, 1983, pp. 259–271.Google Scholar
  15. 15.
    G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. (1985), No. 61, 5–65.Google Scholar
  16. 16.
    T. Shiota, Calogero-Moser hierarchy and Kp hierarchy, J. Math. Phys. 35 (1994), No. 11, 5844–5849.MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    M. Stone (ed.), Bozonization, World Sci. Publ., Singapore, 1994.Google Scholar
  18. 18.
    A. P. Veselov, Rational solutions of the Kadomtsev-Petviashvili equation and Hamiltonian systems, Russian Math. Surveys 35 (1980), No. 1, 239–240.MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    G. Wilson, Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177–204.MathSciNetMATHGoogle Scholar
  20. 20.
    G. Wilson, Collisions of Calogero-Moser particles and an adelic Grass-mannian, Invent. Math. 133 (1998), No. 1, 1–41.MathSciNetADSMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2000

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  • Alex Kasman

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