Mathematical Physics, Analysis and Geometry

, Volume 10, Issue 4, pp 359–373 | Cite as

The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators



This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper (Remling, The absolutely continuous spectrum of Jacobi matrices,, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas.


Absolutely continuous spectrum Schrödinger operator Reflectionless potential 

Mathematics Subject Classifications (2000)

Primary 34L40 81Q10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, Texts and Monographs in Physics. Springer, New York (1988)Google Scholar
  2. 2.
    Atkinson, F.V.: On the location of the Weyl circles. Proc. R. Soc. Edinb. Sect. A Math 88, 345–356 (1981)MATHGoogle Scholar
  3. 3.
    Ben Amor, A., Remling, C.: Direct and inverse spectral theory of Schrödinger operators with measures. Integr. Equ. Oper. Theory 52, 395–417 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bessaga, C., Pelczynski, A.: Selected Topics in Infinite-Dimensional Topology, Mathematical Monographs, vol. 58. Polish Scientific, Warsaw (1975)Google Scholar
  5. 5.
    Brasche, J.F., Exner, P., Kuperin, Y.A., Seba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brasche, J.F., Figari, R., Teta, A.: Singular Schrödinger operators as limits of point interaction Hamiltonians. Potential Anal. 8, 163–178 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Breimesser, S.V., Pearson, D.B.: Asymptotic value distribution for solutions of the Schrödinger equation. Math. Phys. Anal. Geom. 3, 385–403 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Breimesser, S.V., Pearson, D.B.: Geometrical aspects of spectral theory and value distribution for Herglotz functions. Math. Phys. Anal. Geom. 6, 29–57 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Clark, S., Gesztesy, F., Holden, H., Levitan, B.M.: Borg-type theorems for matrix-valued Schrödinger operators. J. Differ. Equ. 167, 181–210 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)MATHGoogle Scholar
  11. 11.
    Denisov, S.: On the continuous analog of Rakhmanov’s theorem for orthogonal polynomials. J. Funct. Anal. 198, 465–480 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Denisov, S.: On Rakhmanov’s theorem for Jacobi matrices. Proc. Am. Math. Soc. 132, 847–852 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gesztesy, F., Simon, B.: A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure. Ann. Math. 152, 593–643 (2000)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hinton, D.B., Klaus, M., Shaw, J.K.: Series representation and asymptotics for Titchmarsh-Weyl m-functions. Differ. Integral Equ. 2, 419–429 (1989)MATHMathSciNetGoogle Scholar
  15. 15.
    Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Last, Y., Simon, B.: The essential spectrum of Schrödinger, Jacobi, and CMV operators. J. Anal. Math. 98, 183–220 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rakhmanov, E.A.: The asymptotic behavior of the ratio of orthogonal polynomials II (Russian). Mat. Sb. (N.S.) 118(160), 104–117 (1982)MathSciNetGoogle Scholar
  18. 18.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I. Functional Analysis. Academic, New York (1980)MATHGoogle Scholar
  19. 19.
    Remling, C.: The absolutely continuous spectrum of Jacobi matrices. (2007)
  20. 20.
    Rybkin, A.: Some new and old asymptotic representations of the Jost solution and the Weyl m-function for Schrödinger operators on the line. Bull. Lond. Math. Soc. 34, 61–72 (2002)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sims, R., Stolz, G.: Localization in one-dimensional random media: a scattering theoretic approach. Commun. Math. Phys. 213, 575–597 (2000)MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of OklahomaNormanUSA

Personalised recommendations