Advertisement

Singular soliton operators and indefinite metrics

  • Petr G. Grinevich
  • Sergey P. Novikov
Article

Abstract

We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebrogeometrical potentials (i.e. “singular finite-gap” and “singular solitons”) satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functionswith singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics ug(g + 1)/x 2. Here g is the genus of spectral curve.

Keywords

singular Schrodinger operator singular finite-gap potentials indefinite inner product Pontryagin-Sobolev spaces 

Mathematical subject classification

47B50 46C20 34L10 34L40 33E10 47E05 47B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P.G. Grinevich and S.P. Novikov. Singular finite-gap operators and indefinite metrics. Russian Mathematical Survey, 64(4) (2009), 625–650.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    P.G. Grinevich and S.P. Novikov. Singular Solitons and Indefinite Metrics. Doklady Mathematics, 83(3) (2011), 56–58.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    B.A. Dubrovin, V.B. Matveev and S.P. Novikov. Nonlinear equations of the Korteweg-de-Vries type, finite-zone linear operators and Abelian Varieties. Russiam Mathematical Surveys, 31(1) (1976), 55–136.zbMATHMathSciNetGoogle Scholar
  4. [4]
    I.M. Krichever. Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model. Functional Analysis and Its Applications, 20(3) (1986), 203–214.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    I.M. Krichever. Spectral theory of two-dimensional periodic operators and its applications. Russian Mathematical Surveys, 44:2(266) (1989), 121–184.MathSciNetGoogle Scholar
  6. [6]
    I.M. Krichever and S.P. Novikov. Riemann Surfaces, Operator Fields, Strings. Analogues of the Laurent-Fourier Bases. Memorial Volume for Vadim Kniznik, “Physics and Mathematics of Strings”, eds. L. Brink, E. Friedan, A.M. Polyakov, World Scientific Singapore, (1990), 356–388.CrossRefGoogle Scholar
  7. [7]
    V.A. Arkad’ev, A.K. Pogrebkov and M.K. Polivanov. Singular solutions of the KdV equation and the inverse scattering method. Journal of Soviet Mathematics, 31(6) (1985), 3264–3279.CrossRefzbMATHGoogle Scholar
  8. [8]
    P.A. Clarkson and E.L. Mansfield. The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity, 16 (2003), R1–R26.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    A.A. Shkalikov and O.A. Veliev. On the Riesz basis property of the eigenand associated functions of periodic and antiperiodic Sturm-Liouville problems. Mathematical Notes, 85(5–6) (2009), 647–660.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    F. Gesztesy and V. Tkachenko. A criterion for Hill operators to be spectral operators of scalar type. J. d’Analyse Math., 107 (2009), 287–353.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    P. Djakov and B. Mityagin. Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials. Math. Ann., 351(3) (2011), 509–540.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    P. Djakov and D. Mityagin. Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators. J. Funct. Anal., 263(8) (2012), 2300–2332.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    F. Gesztezy and R. Weikard. Picard potentials and Hill’s equation on a torus. Acta Math., 176 (1996), 73–107.CrossRefMathSciNetGoogle Scholar
  14. [14]
    B. Deconinck and H. Segur. Pole Dynamics for Elliptic Solutions of the Kortewegde Vries Equation. Mathematical Physics, Analysis and Geometry, 3(1) (2000), 49–74.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2013

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.University of MarylandCollege ParkUSA

Personalised recommendations