Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1435–1449 | Cite as

Rellich’s theorem for spherically symmetric repulsive Hamiltonians

  • Kyohei ItakuraEmail author


For spherically symmetric repulsive Hamiltonians we prove Rellich’s theorem, or identify the largest weighted space of Agmon–Hörmander type where the generalized eigenfunctions are absent. The proof is intensively dependent on commutator arguments. Our novelty here is a use of conjugate operator associated with some radial flow, not with dilations and not with translations. Our method is simple and elementary, and does not employ any advanced tools such as the operational calculus or the Fourier analysis.


Repulsive Hamiltonians Rellich’s theorem 

Mathematics Subject Classification

81Q05 35J10 35P05 



The author would like to thank Kenichi Ito and Erik Skibsted for informative advice regarding this work.


  1. 1.
    Agmon, S.: Lower bounds for solutions of Schrödinger equations. J. Anal. Math. 23, 1–25 (1970)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bony, J.F., Carles, R., Häfner, D., Michel, L.: Scattering theory for the Schrödinger equation with repulsive potential. J. Math. Pures Appl. 84, 509–579 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Froese, R., Herbst, I.: Exponential bounds and absence of positive eigenvalues for \(N\)-body Schrödinger operators. Comm. Math. Phys. 87(3), 429-447 (1982/83)Google Scholar
  4. 4.
    Froese, R., Herbst, I., Hoffmann-Ostenhof, M., Hoffman-Ostenhof, T.: On the absence of positive eigenvalues for one-body Schrödinger operators. J. Anal. Math. 41, 272–284 (1982)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. vol. II, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1985)Google Scholar
  6. 6.
    Inoescu, A.D., Jerison, D.: On the absence of positive eigenvalues of Schrödin-ger operators with rough potentials. Geom. Funct. Anal. 13(5), 1029–1081 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Isozaki, H., Morioka, H.: A Rellich type theorem for discrete Schrödinger operators. Inverse Probl. Imaging 8(2), 475–489 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ito, K., Skibsted, E.: Stationary scattering theory on manifolds, I. Preprint, (2016)Google Scholar
  9. 9.
    Ishida, A.: On inverse scattering problem for the Schrödinger equation with repulsive potentials. J. Math. Phys. 55(8), 082101 (2014). 12 ppMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Isozaki, H.: A uniqueness theorem for the \(N\)-body Schrödinger equation and its applications. Spectral and scattering theory (Sanda, 1992), pp. 63–84, Lecture Notes in Pure and Appl. Math., 161, Dekkaer, New York (1994)Google Scholar
  11. 11.
    Kreh, M.: Bessel functions. Lecture notes, Penn State-Göttingen Summer School on Number Theory (2012)Google Scholar
  12. 12.
    Matsumoto, S., Kakazu, K., Nagamine, T.: Eigenvalue problem for Schrödin-ger’s equation with repulsive potential. J. Math. Phys. 27(1), 232–237 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Reed, M., Simon, B.: Methods of modern mathematical physics II and IV, New York: Academic Press (1975 and 1978)Google Scholar
  14. 14.
    Skibsted, E.: Sommerfeld radiation condition at threshold. Comm. Partial Diff. Equ. 38, 1601–1625 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sigal, I.M.: Stark effect in multielectron systems: non-existence of bound states. Commun. Math. Phys. 122, 1–22 (1989)CrossRefzbMATHGoogle Scholar
  16. 16.
    Wolff, Thomas H.: Recent work on sharp estimates in second-order elliptic unique continuation problems. J. Geom. Anal. 3(6), 621–650 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of ScienceKobe UniversityKobeJapan

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