Skip to main content
SpringerLink
Log in

Rayleigh Scattering at Atoms with Dynamical Nuclei

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

Abstract

Scattering of photons at an atom with a dynamical nucleus is studied on the subspace of states of the system with a total energy below the threshold for ionization of the atom (Rayleigh scattering). The kinematics of the electron and the nucleus is chosen to be non-relativistic, and their spins are neglected. In a simplified model of a hydrogen atom or a one-electron ion interacting with the quantized radiation field in which the helicity of photons is neglected and the interactions between photons and the electron and nucleus are turned off at very high photon energies and at photon energies below an arbitrarily small, but fixed energy (infrared cutoff), asymptotic completeness of Rayleigh scattering is established rigorously. On the way towards proving this result, it is shown that, after coupling the electron and the nucleus to the photons, the atom still has a stable ground state, provided its center of mass velocity is smaller than the velocity of light; but its excited states are turned into resonances. The proof of asymptotic completeness then follows from extensions of a positive commutator method and of propagation estimates for the atom and the photons developed in previous papers.

The methods developed in this paper can be extended to more realistic models. It is, however, not known, at present, how to remove the infrared cutoff.

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. Amour L., Grébert B., Guillot J.C. (2006) The dressed mobile atoms and ions. J. Math. Pures Appl. 86(3): 177–200

    MathSciNet  Google Scholar 

  2. Bach V., Fröhlich J., Pizzo A. (2006) Infrared-finite algorithms in QED: I. The groundstate of an atom interacting with the quantized radiation field. Commun. Math. Phys. 264(1): 145–165

    Article  MATH  ADS  Google Scholar 

  3. Bach V., Fröhlich J., Sigal I.M. (1998) Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2): 299–395

    Article  MATH  MathSciNet  Google Scholar 

  4. Bach V., Fröhlich J., Sigal I.M., Soffer A. (1999) Positive commutators and spectrum of nonrelativistic QED. Commun. Math. Phys. 207(3): 557–587

    Article  MATH  ADS  Google Scholar 

  5. Dereziński J., Gérard C. (1999) Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11(4): 383–450

    Article  MathSciNet  Google Scholar 

  6. Fröhlich J. (1973) On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré, Sect. A XIX(1): 1–103

    Google Scholar 

  7. Fröhlich J. (1974) Existence of dressed one-electron states in a class of persistent models. Fortschr. Phys. 22, 159–198

    Google Scholar 

  8. Fröhlich J., Griesemer M., Schlein B. (2001) Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math. 164(2): 349–398

    Article  MATH  MathSciNet  Google Scholar 

  9. Fröhlich J., Griesemer M., Schlein B. (2002) Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré 3, 107–170

    Article  MATH  Google Scholar 

  10. Fröhlich J., Griesemer M., Schlein B. (2004) Asymptotic completeness for Compton Scattering. Commun. Math. Phys. 252, 415–176

    Article  MATH  ADS  Google Scholar 

  11. Fröhlich, J., Griesemer, M., Sigal, I.M. Mourre estimate and spectral theory for the standard model of non-relativistic QED. Preprint arXiv: math-ph/0611013

  12. Griesemer M. (2004) Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210(3): 321–340

    Article  MATH  MathSciNet  Google Scholar 

  13. Griesemer M., Lieb E.H., Loss M. (2001) Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3): 557–595

    Article  MATH  MathSciNet  Google Scholar 

  14. Lieb E.H., Loss M. (2003) Existence of atoms and molecules in non-relativistic quantum electrodynamics. Adv. Theor. Math. Phys. 7(4): 1–54

    MathSciNet  Google Scholar 

  15. Hübner M., Spohn H. (1995) Spectral properties of the spin-boson Hamiltonian. Ann. Inst. H. Poincaré 62(3): 289–323

    MATH  Google Scholar 

  16. Pizzo A. (2003) One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré 4(3): 439–486

    Article  MATH  MathSciNet  Google Scholar 

  17. Reed M., Simon B. (1979). Methods of modern mathematical physics: Scattering Theory. Volume 3, Academic Press, New York

    MATH  Google Scholar 

  18. Reed M., Simon B. (1978) Methods of modern mathematical physics: Analysis of Operators. Volume 4, Academic Press, New York

    MATH  Google Scholar 

  19. Skibsted E. (1998) Spectral analysis of N-body systems coupled to a bosonic field. Rev. Math. Phys. 10(7): 989–1026

    Article  MATH  MathSciNet  Google Scholar 

  20. Spohn H. (1997) Asymptotic completeness for Rayleigh scattering. J. Math. Phys. 38(5): 2281–2296

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Physics, ETH-Hönggerberg, CH-8093, Zürich, Switzerland

    J. Fröhlich

  2. Fachbereich Mathematik, Universität Stuttgart, D-70569, Stuttgart, Germany

    M. Griesemer

  3. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    B. Schlein

Authors
  1. J. Fröhlich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Griesemer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Schlein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Fröhlich.

Additional information

Communicated by H.-T. Yau

Activities supported, in part, by a grant from the Swiss National Foundation.

Work partially supported by U.S. National Science Foundation grant DMS 01-00160.

Supported by a NSF postdoctoral fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fröhlich, J., Griesemer, M. & Schlein, B. Rayleigh Scattering at Atoms with Dynamical Nuclei. Commun. Math. Phys. 271, 387–430 (2007). https://doi.org/10.1007/s00220-006-0134-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-006-0134-x

Keywords

Search

Navigation