Functional Analysis and Its Applications

, Volume 35, Issue 4, pp 247–256 | Cite as

The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation

  • A. A. Akhmetshin
  • Yu. S. Volvovsky


We study a system of particles on a Riemann surface with a puncture. This system describes the behavior of zeros of finite-gap solutions of the Schrödinger equation corresponding to a degenerate hyperelliptic curve. We show that this system is Hamiltonian and integrable by constructing action-angle type coordinates.


Functional Analysis Riemann Surface Hyperelliptic Curve 
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  1. 1.
    J. F. Van Diejen and H. Puschmann, “Reflectionless Schrüdinger operators, the dynamics of zeros, and the solitonic Sato formula,” Duke Math. J., 104, No. 2, 269–318 (2000).Google Scholar
  2. 2.
    S. N. M. Ruijsenaars and H. Schneider, “A new class of integrable systems and its relation to solitons,” Ann. Physics, 170, No. 2, 370–405 (1986).Google Scholar
  3. 3.
    B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties,” Usp. Mat. Nauk, 31, No. 1 (187), 55–136 (1976).Google Scholar
  4. 4.
    B. A. Dubrovin, “Periodic problems for the Korteweg-de Vries equation in the class of finiteband potentials,” Funkts. Anal. Prilozhen., 9, No. 3, 41–52 (1975); English transl. Functional Anal. Appl., 9, No. 3, 215–223 (1975).Google Scholar
  5. 5.
    I. M. Krichever, “Potentals with zero coefficient of reflection on a background of finite-zone potentials,” Funkts. Anal. Prilozhen., 9, No. 2, 77–79 (1975); English transl. Functional Anal. Appl., 9, No. 2, 161–163 (1975).Google Scholar
  6. 6.
    I. M. Krichever, “Spectral theory of finite-zone nonstationary Schrüdinger operators. A nonstationary Peierls model,” Funkts. Anal. Prilozhen., 20, No. 3, 42–54 (1986); English transl. Functional Anal. Appl.,20, No. 3, 203–213 (1986).Google Scholar
  7. 7.
    H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. II, McGraw-Hill, 1953.Google Scholar
  8. 8.
    S. P. Novikov and A. P. Veselov, “Poisson brackets that are compatible with the algebraic geometry and the dynamics of the Korteweg-de Vries equation on the set of finite-gap potentials,” Dokl. Akad. Nauk SSSR, 266, No. 3, 533–537 (1982).Google Scholar
  9. 9.
    I. M. Krichever and D. Phong, “Symplectic forms in the theory of solitons,” In: Surv. Differ. Geom. IV, Suppl. J. Differ. Geom., 239–313 (1998).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • A. A. Akhmetshin
    • 1
  • Yu. S. Volvovsky
    • 2
  1. 1.Landau Institute for Theoretical PhysicsColumbia UniversityUSA
  2. 2.Landau Institute for Theoretical PhysicsColumbia UniversityUSA

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