Skip to main content
SpringerLink
Log in

Eigenfunction expansions for singular Schrodinger operators

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We extend Ikebe's theory of eigenfunctions to a class of Schrodinger operators H = -Δ + V on L 2(IR3), where the potential V is replaced by a measure which need not be absolutely continuous with respect to Lebesgue measure. Applications include the proof of the existence and completeness of the wave operators for a free particle which is partly reflected and partly transmitted by a compact 2-dimensional surface.

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. S. Agmon. Spectral properties of Schrodinger operators and scattering theory. Ann. Scu. Norm. Pisa 2 (1975) 151–218.

    Google Scholar 

  2. P. Alsholm and G. Schmidt. Spectral and scattering theory for Schrodinger operators. Arch. Rational Mech. Anal 40 (1971) 281–311.

    Google Scholar 

  3. W.O. Amrein and V. Georgescu. Strong asymptotic completeness of wave operators for highly singular potentials. Helv. Phys. Acta 47 (1974) 517–533.

    Google Scholar 

  4. P. Deift and B. Simon. On the decoupling of finite singularities from the question of asymptotic completeness in two-body quantum systems. Preprint (1976).

  5. K.O. Friedrichs. Perturbation of spectra in Hillbert space. Amer. Math. Soc. Providence, R.I. 1965.

    Google Scholar 

  6. T. Ikebe. Eigenfunction expansions associated with the Schrodinger operator and their applications to scattering theory. Arch. Rational Mech. Anal. 5 (1960) 1–34.

    Google Scholar 

  7. T. Kato. Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966.

    Google Scholar 

  8. T. Kato and S.T. Kuroda. The abstract theory of scattering. Rocky Mountain J. Math. 1 (1971) 127–171.

    Google Scholar 

  9. J. Kupsh and W. Sandhas. Møller operators for scattering on singular potentials. Commun. Math. Phys. 2 (1966) 147–154.

    Google Scholar 

  10. D.B. Pearson. An example in potential scattering illustrating the breakdown in asymptotic completeness. Comm. Math. Phys. 40 (1975) 125–146.

    Google Scholar 

  11. D.B. Pearson. General theory of potential scattering with absorption at local singularities. Preprint (1975).

  12. B. Simon. Quantum mechanics for Hamiltonian defined as quadratic forms. Princeton, N.J.: Princeton Univ. Press 1971.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Oxford

    E. B. Davies

Authors
  1. E. B. Davies
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J. B. McLeod

Rights and permissions

Reprints and permissions

About this article

Cite this article

Davies, E.B. Eigenfunction expansions for singular Schrodinger operators. Arch. Rational Mech. Anal. 63, 261–272 (1977). https://doi.org/10.1007/BF00251583

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00251583

Keywords

Search

Navigation