Integral Equations and Operator Theory

, Volume 40, Issue 3, pp 268–277 | Cite as

Bound states of a canonical system with a pseudo-exponential potential

  • I. Gohberg
  • M. A. Kaashoek
  • A. L. Sakhnovich
Article
Received:

Abstract

Explicit formulas are given for the bound states (theL2-eigenfunctions) and the corresponding eigenvalues of a self-adjoint operator defined by a canonical system with a pseudo-exponential potential. The formulas are expressed in terms of three matrices determining the potential. Both the half line and the full line case are considered.

AMS Classification

Primary: 34L05 34A55 Secondary: 34A05 34B20 47N20 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N.I. Achiezer, and I.M. Glazman,Theory of liner operators in Hilbert space, vol. I, Pitman, London, 1980.Google Scholar
  2. [2]
    P.A. Deift, Applications of a commutation formula,Duke Math. J. 45 (1987), 267–310.Google Scholar
  3. [3]
    P. Deift and E. Trubowitz, Inverse scattering on the line,Commun. Pure Appl. Math. 32 (1979), 121–251.Google Scholar
  4. [4]
    I. Gohberg, S. Goldberg, and M.A. Kaashoek,Classes of linear operators, vol. II, Birkhäuser Verlag, Basel, 1993Google Scholar
  5. [5]
    I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Canonical systems with rational spectral densities: explicit formulas and applications,Math. Nachr. 194 (1998), 93–125.Google Scholar
  6. [6]
    I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Canonical systems on the line with rational spectral densities: explicit formulas, in:Operator Theory and Related Topics, Volume I of the Proceedings of the Mark Krein International Conference, Odessa, Ukraine, August 1–22, 1997, Birkhäuser Verlag, Basel,2000, pp. 127–139.Google Scholar
  7. [7]
    F. Gesztesy and G. Teschl, On the double commutation method,Proc. Amer. Math. Soc. 124 (1996), 1831–1840.Google Scholar
  8. [8]
    F.E. Melik-Adamjan, Canonical differential operators in a Hilbert space,Izv. Akad. Nauk Arm. SSR Math. 12 (1977), 10–31.Google Scholar
  9. [9]
    B.N. Zakhariev and V.M. Chabanov, New situation in quantum mechanics (wonderful potentials from the inverse problem),Inverse Problems 13 (1997), R47-R79.Google Scholar

Copyright information

© Birkhäuser Verlag 2001

Authors and Affiliations

  • I. Gohberg
    • 1
  • M. A. Kaashoek
    • 2
  • A. L. Sakhnovich
    • 3
  1. 1.School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityRamat AvivIsrael
  2. 2.Divisie Wiskunde en Informatica Faculteit der Exacte WetenschappenVrije UniversiteitAmsterdamThe Netherlands
  3. 3.Branch of Hydroacoustics Marine Institute of HydrophysicsNASUOdessaUkraine

Personalised recommendations