Skip to main content
SpringerLink
Log in

L 2-exponential lower bounds to solutions of the Schrödinger equation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study decay properties of solutions ϕ of the Schrödinger equation (−Δ+V)ϕ=Eϕ. Typical of our results is one which shows that ifV=o(|x|−1/2) at infinity or ifV is a homogeneousN-body potential (for example atomic or molecular), then ifE<0 and\(\alpha > \sqrt { - E} ,e^{\alpha \left| x \right|} \psi \notin L^2 \left( {\mathbb{R}^n } \right)\). We also construct examples to show that previous essential spectrum-dependent upper bounds can be far from optimal if ϕ is not the ground state.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agmon, S.: Lectures on exponential decay of solutions of second order elliptic equations. Bounds on eigenfunctions ofN-body Schrödinger operators (to be published)

  2. Agmon, S.: Proc. A. Pleijel Conf., Uppsala 1979

  3. Ahlrichs, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Morgan, J. III: Phys. Rev. A23, 2106 (1981)

    Google Scholar 

  4. Aizenmann, M., Simon, B.: Commun. Pure Appl. Math.35, 209 (1982)

    Google Scholar 

  5. Amrein, W.O., Berthier, A.M., Georgescu, V.: Ann. Inst. Fourier, Grenoble31, 153 (1981)

    Google Scholar 

  6. Bardos, C., Merigot, M.: Proc. R. Soc. (Edinburgh)76A, 323 (1977)

    Google Scholar 

  7. Berthier, A.M.: Comptes rendus290A, 393 (1980)

    Google Scholar 

  8. Carmona, R., Simon, B.: Commun. Math. Phys.80, 59 (1981)

    Google Scholar 

  9. Carmona, R.: Duke Math. J.49, 191 (1982)

    Google Scholar 

  10. Combes, J.M., Hoffmann-Ostenhof, M., Hoffmann-Ostenhoff, T.: J. Math. Phys.22, 1299 (1981)

    Google Scholar 

  11. Combes, J.M., Thomas, L.: Commun. Math. Phys.34, 319 (1973)

    Google Scholar 

  12. Deift, P., Hunziker, W., Simon, B., Vock, E.: Commun. Math. Phys.64, 1 (1978)

    Google Scholar 

  13. Froese, R., Herbst, I.: Exponential bounds and absence of positive eigenvalues forN-body Schrödinger operators. Commun. Math. Phys. (to appear)

  14. Froese, R., Herbst, I., Hoffmann-Ostenhof, M., Hoffmann-Ostenhoff, T.:L 2-lower bounds to solutions of one-body Schrödinger equations (preprint)

  15. Froese, R., Herbst, I., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: On the absence of positive eigenvalues for one-body Schrödinger operators. J. d'Analyse (to appear)

  16. Gårding, L.: Personal communication

  17. Georgescu, V.: Helv. Phys. Acta52, 655 (1979)

    Google Scholar 

  18. Gol'sheid, I.Ja., Molchanov, S.A., Pastur, L.A.: Funct. Anal. Appl.11, 1 (1977)

    Google Scholar 

  19. Hoffmann-Ostenhof, T., Hoffmann-Ostenhof, M., Ahlrichs, R.: Phys. Rev. A18, 328 (1978)

    Google Scholar 

  20. Hoffmann-Ostenhof, T.: Phys. Lett.77A, 140 (1980)

    Google Scholar 

  21. Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  22. Leinfelder, H.: Int. Eqn. and Op. Th.4, 226 (1981)

    Google Scholar 

  23. Lieb, E.M., Simon, B.: Adv. Appl. Math.7, 324 (1980)

    Google Scholar 

  24. Molchanov, S.A.: Math. U.S.S.R. Izvestija12-1, 69 (1978)

    Google Scholar 

  25. O'Connor, A.: Commun. Math. Phys.32, 319 (1973)

    Google Scholar 

  26. Perry, P., Sigal, I.M., Simon, B.: Ann. Math.114, 519 (1981)

    Google Scholar 

  27. Persson, A.: Math. Scand.8, 143 (1960)

    Google Scholar 

  28. Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York: Academic Press 1972

    Google Scholar 

  29. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press 1978

    Google Scholar 

  30. Saut, J.C., Scheurer, B.: Compes rendus290A, 595 (1980)

    Google Scholar 

  31. Schechter, M., Simon, B.: J. Math. Anal. Appl.77, 482 (1980)

    Google Scholar 

  32. Simon, B.: Proc. Am. Math. Soc.42, 395 (1974)

    Google Scholar 

  33. Simon, B.: Personal communication

Download references

Author information

Author notes

  1. Richard Froese & Ira Herbst

    Present address: Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA

Authors and Affiliations

  1. Institut Mittag-Leffler, Auravägen 17, S-182 62, Djursholm, Sweden

    Richard Froese & Ira Herbst

  2. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090, Wien, Austria

    Maria Hoffmann-Ostenhof

  3. Institut für Theoretische Chemie und Strahlenchemie, Universität Wien, Währingerstrasse 17, A-1090, Wien, Austria

    Thomas Hoffmann-Ostenhof

Authors
  1. Richard Froese
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ira Herbst
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maria Hoffmann-Ostenhof
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thomas Hoffmann-Ostenhof
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by B. Simon

Research in partial fulfillment of the requirements for a Ph.D. degree at the University of Virginia

Partially supported by NSF grant MCS-81-01665

Supported by „Fonds zur Förderung der wissenschaftlichen Forschung in Österreich“, Projekt Nr. 4240

Rights and permissions

Reprints and permissions

About this article

Cite this article

Froese, R., Herbst, I., Hoffmann-Ostenhof, M. et al. L 2-exponential lower bounds to solutions of the Schrödinger equation. Commun.Math. Phys. 87, 265–286 (1982). https://doi.org/10.1007/BF01218565

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01218565

Keywords

Search

Navigation