Skip to main content
SpringerLink
Log in

Mathematical theory of nonrelativistic matter and radiation

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

Abstract

We consider a system of finitely many nonrelativistic electrons bound in an atom or molecule which are coupled to the electromagnetic field via minimal coupling or the dipole approximation. Among a variety or results, we give sufficient conditions for the existence of a ground state (an eigenvalue at the bottom of the spectrum) and resonances (eigenvalues of a complex dilated Hamiltonian) of such a system. We give a brief outline of the proofs of these statements which will appear at full length in a later work.

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. Aguilar, J. and Combes, J. M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians,Comm. Math. Phys. 22 (1971), 269–279.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Arai, A.: Spectral analysis of a quantum harmonic oscillator coupled to infinitely many scalar bosons.J. Math. Anal. Appl. 140 (1989), 270–288.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bach, V., Fröhlich, J., and Sigal, I. M.: Mathematical theory of radiation in systems of atoms or molecules, in preparation, 1995.

  4. Balslev, E. and Combes, J. M.: Spectral properties of Schrödinger operators with dilatation analytic potentials,Comm. Math. Phys. 22 (1971), 280–294.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bethe, H. A. and Salpeter, E.:Quantum Mechanics of One and Two Electron Atoms, Springer, Heidelberg, 1957.

    Book  MATH  Google Scholar 

  6. Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G.:Photons and Atoms — Introduction to Quantum Electrodynamics, Wiley. New York, 1991.

    Google Scholar 

  7. Cycon, H.: Resonances defined by modified dilations,Helv. Phys. Acta 53 (1985), 969–981.

    MathSciNet  Google Scholar 

  8. Cycon, H., Froese, R., Kirsch, W., and Simon, B.:Schrödinger Operators, Springer, Berlin, Heidelberg, New York, 1987.

    MATH  Google Scholar 

  9. Fefferman, C. L.: Stability of coulomb systems in a magnetic field. In preparation, 1995.

  10. Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless scalar bosons.Ann. Inst. H. Poincaré 19 (1973), 1–103.

    MathSciNet  MATH  Google Scholar 

  11. Fröhlich, J., Lieb, E. H., and Loss, M.: Stability of Coulomb systems with magnetic fields 1: The one-electron atom,Comm. Math. Phys. 104 (1986), 251–270.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Glimm, J. and Jaffe, A.: The λ(ϕ4)2 quantum field theory without cutoffs: Ii. The field operators and the approximate vacuum.Ann. Math. 91 (1970), 362–401.

    Article  MathSciNet  MATH  Google Scholar 

  13. Hübner, M. and Spohn, H.: Atom interacting with photons: ann-body Schrödinger problem, Preprint, 1994.

  14. Hübner, M. and Spohn, H.: Radiative decay: Nonperturbative approaches,Rev. Math. Phys., to be published, 1994.

  15. Hunziker, W.: Distortion analyticity and molecular resonance curves,Ann. Inst. H. Poincaré 45 (1986), 339–358.

    MathSciNet  MATH  Google Scholar 

  16. Hunziker, W.: Resonances, metastable states and exponential decay laws in perturbation theory,Comm. Math. Phys. 132 (1990), 177–188.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Hunziker, W. and Sigal, I. M.: The general theory ofn-body quantum systems, in J. Feldman et al. (eds),Mathematical Quantum Theory: II. Schrödinger Operators. AMS-Publ., Montreal, 1994.

    Google Scholar 

  18. Kato, T.: Smooth operators and commutators.Stud. Math. Appl. 31 (1968), 535–546.

    MathSciNet  MATH  Google Scholar 

  19. King, C.: Exponential decay near resonance, without analyticity,Lett. Math. Phys. 23 (1991), 215–222.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Lavine, R.: Absolute continuity of Hamiltonian operators with repulsive potentials,Proc. Amer. Math. Soc. 22 (1969), 55–60.

    MathSciNet  MATH  Google Scholar 

  21. Lieb, E. H. and Loss, M.: Stability of Coulomb systems with magnetic fields: II. The many-electron atom and the one-electron molecule,Comm. Math. Phys. 104 (1986), 271–282.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Loss, M. and Yau, H. T.: Stability of Coulomb systems with magnetic fields: III. Zero energy bound states of the Pauli operator,Comm. Math. Phys. 104 (1986), 283–290.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators,Comm. Math. Phys. 78 (1981), 391–408.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Okamoto, T. and Yajima, K.: Complex scaling technique in non-relativistic QED,Ann. Inst. H. Poincaré 42 (1985), 311–327.

    MathSciNet  MATH  Google Scholar 

  25. Perry, P., Sigal, I. M. and Simon, B.: Spectral analysis ofn-body Schrödinger operators.Annals Math. 114 (1981), 519–567.

    Article  MathSciNet  MATH  Google Scholar 

  26. Reed, M. and Simon, B.:Methods of Modern Mathematical Physics: Analysis of Operators, vol. 4. Academic Press, San Diego, 1978.

    MATH  Google Scholar 

  27. Sigal, I. M.: Complex transformation method and resonances in one-body quantum systems.Ann. Inst. H. Poincaré 41 (1984), 333.

    MathSciNet  MATH  Google Scholar 

  28. Simon, B.: Resonances inn-body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory,Ann. Math. 97 (1973), 247–274.

    Article  MATH  Google Scholar 

  29. Simon, B.: The definition of molecular resonance curves by the method of exterior complex scaling.Phys. Lett. A 71 (1979), 211–214.

    Article  ADS  Google Scholar 

  30. Spohn, H.: Ground state(s) of the spin-boson Hamiltonian.Comm. Math. Phys. 123 (1989), 277–304.

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB Mathematik MA 7-2, Technische Universität Berlin, D-10623, Berlin, Germany

    V. Bach

  2. Theoretische Physik, ETH-Hönggerberg, CH-8093, Zürich, Switzerland

    J. Fröhlich

  3. Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Canada

    I. M. Sigal

Authors
  1. V. Bach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Fröhlich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. M. Sigal
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to the memory of J. Schwinger, whose understanding of Quantum Electrodynamics was profound

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bach, V., Fröhlich, J. & Sigal, I.M. Mathematical theory of nonrelativistic matter and radiation. Lett Math Phys 34, 183–201 (1995). https://doi.org/10.1007/BF01872776

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01872776

Mathematics Subject Classifications (1991)

Search

Navigation