Letters in Mathematical Physics

, Volume 34, Issue 3, pp 183–201 | Cite as

Mathematical theory of nonrelativistic matter and radiation

  • V. Bach
  • J. Fröhlich
  • I. M. Sigal


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.

Mathematics Subject Classifications (1991)

81C12 81E15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aguilar, J. and Combes, J. M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians,Comm. Math. Phys. 22 (1971), 269–279.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arai, A.: Spectral analysis of a quantum harmonic oscillator coupled to infinitely many scalar bosons.J. Math. Anal. Appl. 140 (1989), 270–288.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bach, V., Fröhlich, J., and Sigal, I. M.: Mathematical theory of radiation in systems of atoms or molecules, in preparation, 1995.Google Scholar
  4. 4.
    Balslev, E. and Combes, J. M.: Spectral properties of Schrödinger operators with dilatation analytic potentials,Comm. Math. Phys. 22 (1971), 280–294.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bethe, H. A. and Salpeter, E.:Quantum Mechanics of One and Two Electron Atoms, Springer, Heidelberg, 1957.CrossRefzbMATHGoogle Scholar
  6. 6.
    Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G.:Photons and Atoms — Introduction to Quantum Electrodynamics, Wiley. New York, 1991.Google Scholar
  7. 7.
    Cycon, H.: Resonances defined by modified dilations,Helv. Phys. Acta 53 (1985), 969–981.MathSciNetGoogle Scholar
  8. 8.
    Cycon, H., Froese, R., Kirsch, W., and Simon, B.:Schrödinger Operators, Springer, Berlin, Heidelberg, New York, 1987.zbMATHGoogle Scholar
  9. 9.
    Fefferman, C. L.: Stability of coulomb systems in a magnetic field. In preparation, 1995.Google Scholar
  10. 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.MathSciNetzbMATHGoogle Scholar
  11. 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.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hübner, M. and Spohn, H.: Atom interacting with photons: ann-body Schrödinger problem, Preprint, 1994.Google Scholar
  14. 14.
    Hübner, M. and Spohn, H.: Radiative decay: Nonperturbative approaches,Rev. Math. Phys., to be published, 1994.Google Scholar
  15. 15.
    Hunziker, W.: Distortion analyticity and molecular resonance curves,Ann. Inst. H. Poincaré 45 (1986), 339–358.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hunziker, W.: Resonances, metastable states and exponential decay laws in perturbation theory,Comm. Math. Phys. 132 (1990), 177–188.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 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. 18.
    Kato, T.: Smooth operators and commutators.Stud. Math. Appl. 31 (1968), 535–546.MathSciNetzbMATHGoogle Scholar
  19. 19.
    King, C.: Exponential decay near resonance, without analyticity,Lett. Math. Phys. 23 (1991), 215–222.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lavine, R.: Absolute continuity of Hamiltonian operators with repulsive potentials,Proc. Amer. Math. Soc. 22 (1969), 55–60.MathSciNetzbMATHGoogle Scholar
  21. 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.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 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.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators,Comm. Math. Phys. 78 (1981), 391–408.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Okamoto, T. and Yajima, K.: Complex scaling technique in non-relativistic QED,Ann. Inst. H. Poincaré 42 (1985), 311–327.MathSciNetzbMATHGoogle Scholar
  25. 25.
    Perry, P., Sigal, I. M. and Simon, B.: Spectral analysis ofn-body Schrödinger operators.Annals Math. 114 (1981), 519–567.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Reed, M. and Simon, B.:Methods of Modern Mathematical Physics: Analysis of Operators, vol. 4. Academic Press, San Diego, 1978.zbMATHGoogle Scholar
  27. 27.
    Sigal, I. M.: Complex transformation method and resonances in one-body quantum systems.Ann. Inst. H. Poincaré 41 (1984), 333.MathSciNetzbMATHGoogle Scholar
  28. 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.CrossRefzbMATHGoogle Scholar
  29. 29.
    Simon, B.: The definition of molecular resonance curves by the method of exterior complex scaling.Phys. Lett. A 71 (1979), 211–214.ADSCrossRefGoogle Scholar
  30. 30.
    Spohn, H.: Ground state(s) of the spin-boson Hamiltonian.Comm. Math. Phys. 123 (1989), 277–304.ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • V. Bach
    • 1
  • J. Fröhlich
    • 2
  • I. M. Sigal
    • 3
  1. 1.FB Mathematik MA 7-2Technische Universität BerlinBerlinGermany
  2. 2.Theoretische PhysikETH-HönggerbergZürichSwitzerland
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations