Skip to main content

Connectedness of the Isospectral Manifold for One-Dimensional Half-Line Schrödinger Operators

Abstract

Let V 0be a real-valued function on [0,∞) and VL 1([0,R]) for all R>0 so that H(V 0)=−\(\frac{{d^2 }}{{dx^2 }}\)+V 0in L 2([0,∞)) with u(0)=0 boundary conditions has discrete spectrum bounded from below. Let \(M\)(V 0) be the set of Vso that H(V) and H(V 0) have the same spectrum. We prove that \(M\)(V 0) is connected.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. D. Chelkak, P. Kargaev, and E. Korotyaev, The inverse problem for the harmonic oscillator perturbed by a potential. Characterization, Comm. Math. Phys., to appear.

  2. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations(Krieger, Malabar, FL, 1985).

    Google Scholar 

  3. N. Dunford and J. T. Schwartz, Linear Operators Part II: Spectral Theory(Interscience, New York, 1988).

    Google Scholar 

  4. F. Gesztesy and B. Simon, A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure, Ann. of Math. (2) 152:593–643 (2000).

    Google Scholar 

  5. P. Hartman, Differential equations with non-oscillatory eigenfunctions, Duke Math. J. 15:697–709 (1948).

    Google Scholar 

  6. B. M. Levitan, Inverse Sturm-Liouville Problems(VNU Science Press, Utrecht, 1987).

    Google Scholar 

  7. B. Levitan, Sturm-Liouville operators on the entire real axis with the same discrete spectrum, Math. USSR-Sb. 60:77–106 (1988).

    Google Scholar 

  8. V. A. Marchenko, Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications, Vol. 22 (Birkhäuser, Basel, 1986).

    Google Scholar 

  9. H. P. McKean and E. Trubowitz, The spectral class of the quantum-mechanical harmonic oscillator, Comm. Math. Phys. 82:471–495 (1981/82).

    Google Scholar 

  10. C. Remling, Schrödinger operators and de Branges spaces, J. Funct. Anal. 196:323–394 (2002).

    Google Scholar 

  11. F. S. Rofe-Beketov, The spectral matrix and the inverse Sturm-Liouville problem on the axis (-∞,∞).Teor. Funktsii Funkts. Analiz Prilozh.4:189–197 (1967). [Russian]

    Google Scholar 

  12. F. S. Rofe-Beketov, The inverse Sturm-Liouville problem for the spectral matrix on the whole axis and associated problems, in Integral Equations and Inverse Problems, V. Petkov and R. Lazarov, eds. (Longman, New York, 1991), pp. 234–238.

    Google Scholar 

  13. B. Simon, A new approach to inverse spectral theory. I. Fundamental formalism, Ann. of Math. (2) 150:1029–1057 (1999).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gesztesy, F., Simon, B. Connectedness of the Isospectral Manifold for One-Dimensional Half-Line Schrödinger Operators. Journal of Statistical Physics 116, 361–365 (2004). https://doi.org/10.1023/B:JOSS.0000037217.89500.b3

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOSS.0000037217.89500.b3

  • isospectral sets of potentials
  • half-line Schrödinger operators
  • inverse problems