Letters in Mathematical Physics

, Volume 109, Issue 12, pp 2625–2648 | Cite as

A note on the Schrödinger operator with a long-range potential

  • D. R. YafaevEmail author


Our goal is to develop spectral and scattering theories for the one-dimensional Schrödinger operator with a long-range potential q(x), \(x\ge 0\). Traditionally, this problem is studied with a help of the Green–Liouville approximation. This requires conditions on the first two derivatives \(q' (x)\) and \(q'' (x)\). We suggest a new Ansatz that allows us to develop a consistent theory under the only assumption \(q' \in L^1\).


Schrödinger equation Dimension one Modified Green–Liouville Ansatz Limiting absorption principle Eigenfunction expansion 

Mathematics Subject Classification

34E20 34L10 34L15 47A40 81U05 



  1. 1.
    Birman, M. Sh. Solomyak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Reidel, Doldrecht (1987)Google Scholar
  2. 2.
    Buslaev, V.S., Matveev, V.B.: Wave operators for the Schrödinger equation with a slowly decreasing potential. Theor. Math. Phys. 2, 266–274 (1970)CrossRefGoogle Scholar
  3. 3.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  4. 4.
    Gilbert, D., Pearson, D.B.: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128(2), 30–56 (1987)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hörmander, L.: The existence of wave operators in scattering theory. Math. Z. 146, 69–91 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jost, R.: Über die falschen Nullstellen der Eigenwerte des \(S\)-matrix. Helv. Phys. Acta 20, 250–266 (1947)Google Scholar
  7. 7.
    Levinson, N.: On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase. Danske Videnskab. Selskab, Mat.-Fyz. Medd. 25(9), 29 (1949)zbMATHGoogle Scholar
  8. 8.
    Matveev, V.B., Skriganov, M.M.: Scattering problem for radial Schrödinger equation with a slowly decreasing potential. Theor. Math. Phys. 10(2), 156–164 (1972)CrossRefGoogle Scholar
  9. 9.
    Mourre, E.: Absence of singular spectrum for certain self-adjoint operators. Commun. Math. Phys. 78, 391–400 (1981)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Naboko, S.N.: On the dense point spectrum of Schrödinger and Dirac operators. Theor. Math. Phys. 68(1), 18–28 (1986)CrossRefGoogle Scholar
  11. 11.
    Newton, R.: Scattering Theory of Waves and Particles. Springer, Berlin (1982)CrossRefGoogle Scholar
  12. 12.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)zbMATHGoogle Scholar
  13. 13.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Academic Press, New York (1978)zbMATHGoogle Scholar
  14. 14.
    Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations, vol. 1. Oxford, (1946)Google Scholar
  15. 15.
    Weidmann, J.: Lineare Operatoren in Hilberträumen, Teil II: Anwendungen. Teubner Verlag, Stuttgart (2003)CrossRefGoogle Scholar
  16. 16.
    Weyl, H.: Über gewöhnliche lineare Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen, Nachr. Ges. Wiss. Göttingen, 37-64 (1909)Google Scholar
  17. 17.
    Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitatten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68, 220–269 (1910)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yafaev, D.R.: Wave operators for the Schrödinger equation. Theor. Math. Phys. 45(2), 992–998 (1981)CrossRefGoogle Scholar
  19. 19.
    Yafaev, D.R.: The low energy scattering for slowly decreasing potentials. Commun. Math. Phys. 85, 177–196 (1982)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Yafaev, D.R.: Mathematical Scattering Theory: Analytic Theory. American Mathematical Society, Providence, Rhode Island (2010)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CNRS, IRMAR-UMR 6625Univ RennesRennesFrance
  2. 2.SPGUSaint PetersburgRussia

Personalised recommendations