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KdV ’95 pp 435-443 | Cite as

Moment Problem of Hamburger, Hierarchies of Integrable Systems, and the Positivity of Tau-Functions

  • Yoshimasa Nakamura
  • Yuji Kodama

Abstract

A moment problem of Hamburger is studied to find a parametric Stieltjes measure from given moments. It is shown that if a deformation, or a dynamics, of moments is governed by a hierarchy of a Kac—van Moerbeke system, then the Stieltjes measure can be constructed explicitly by integrating a hierarchy of Moser’s nonlinear dynamical system. The positivity of tau-functions is related to the existence of the Stieltjes measure at a deep level.

Key words

Hamburger moment problem Moser’s system Kac-van Moerbeke system tau functions 

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References

  1. 1.
    Ahiezer, N. I. and Kreǐn, M: Some Questions inthe Theory of Moments, Transl. Math. Monographs, Vol. 2, Amer. Math. Soc.,Providence, 1962.Google Scholar
  2. 2.
    Bloch, A. M., Brockett, R. W., andRatiu, T.: Completely integrable gradient flows, Commun.Math. Phys.147(1992), 57–74.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Case, K. M.: Orthogonal polynomialsfrom the viewpoint of scattering theory, J. Math. Phys.15 (1974), 2166–2174.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Gantmacher, F. R.: The Theory of Matrices, Vol. 2, Chelsea, NewYork, 1959.zbMATHGoogle Scholar
  5. 5.
    Harada, H.: New subhierarchies ofthe KP hierarchy in the Sato theory. I. Analysis of theBurgers-Hopf hierarchy by the Sato theory, J. Phys. Soc. Japan 12(1985), 4507–4512.MathSciNetGoogle Scholar
  6. 6.
    Hirota, R., Ohta, Y., and Satsuma,J.: Solutions of the Kadomtsev-Petviashvili equation andthe two-dimensional Toda equations, J. Phys. Soc. Japan 57 (1988),1901–1904.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kac, M. and van Moerbeke, P.: On anexplicitly soluble system of nonlinear differentialequations related to certain Toda lattice, Adv. Math.16 (1975), 160–169.zbMATHCrossRefGoogle Scholar
  8. 8.
    Kato, Y. and Aomoto, K.:Jacobi-Perron algorithms, bi-orthogonal polynomials and inversescattering problems, Publ. RIMS, Kyoto Univ.20 (1984), 635–658.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kodama, Y.:Solutions of the dispersionless Toda equation, Phys. Lett.A 147(1990),477–482.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Moser, J.: Finitely many points on the line under theinfluence of an exponential potential An integrable system, in: J. Moser (Ed.), Dynamical Systems, Theory and Applications, Lee.Notes Phys., Vol. 38, Springer-Verlag,Berlin, New York, 1975, pp. 467–497.CrossRefGoogle Scholar
  11. 11.
    Nakamura, Y.:Geometry of rational functions and nonlinear integrable systems, SIAM J.Math.Anal.22 (1991), 1744–1754.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 11a.
    Nakamura, Y.:The level manifolds of ageneralized Toda equation hierarchy,Trans. Amer. Math. Soc.333(1992), 83–94.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 12.
    Nakamura, Y.: A tau-function forthe finite Toda molecule, and information spaces, in: YMaeda, H. Omori and A. Weinstein (Eds), Symplectic Geometry andQuantization, Contemp.Math. Vol. 179, Amer. Math. Soc,Providence, 1994.Google Scholar
  14. 13.
    Szegö, G.: Orthogonal Polynomials, 4th edn, Colloq. Publ., Vol. 23,Amer. Math. Soc, Providence, 1975.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Yoshimasa Nakamura
    • 1
  • Yuji Kodama
    • 2
  1. 1.Applied Mathematics LaboratoryDoshisha UniversityTanabe, KyotoJapan
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

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