Communications in Mathematical Physics

, Volume 323, Issue 2, pp 635–661 | Cite as

On Enhanced Binding and Related Effectsin the Non- and Semi-Relativistic Pauli-Fierz Models



We prove enhanced binding and increase of binding energies in the non- and semi-relativistic Pauli-Fierz models, for arbitrary values of the fine-structure constant and the ultra-violet cut-off, and discuss the resulting improvement of exponential localization of ground state eigenvectors. For the semi-relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self-energy, as the ultra-violet cut-off goes to infinity.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BCVV.
    Barbaroux J.-M., Chen T., Vougalter V., Vugalter S.A.: Quantitative estimates on the binding energy for hydrogen in non-relativistic QED. Ann. Henri Poincaré 11, 1487–1544 (2010)MathSciNetADSCrossRefMATHGoogle Scholar
  2. BLV.
    Barbaroux J.-M., Linde H., Vugalter S.A.: Quantitative estimates on the enhanced binding for the Pauli-Fierz operator. J. Math. Phys. 46, 122103 (2005)MathSciNetADSCrossRefGoogle Scholar
  3. BeVu.
    Benguria R.D., Vugalter S.A.: Binding threshold for the Pauli-Fierz operator. Lett. Math. Phys. 70, 249–257 (2004)MathSciNetADSCrossRefMATHGoogle Scholar
  4. CMS.
    Carmona R., Masters W.C., Simon B.: Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91, 117–142 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. CEH.
    Catto I., Exner P., Hainzl C.: Enhanced binding revisited for a spinless particle in nonrelativistic QED. J. Math. Phys. 45, 4174–4185 (2004)MathSciNetADSCrossRefMATHGoogle Scholar
  6. CaHa.
    Catto I., Hainzl C.: Self-energy of one electron in non-relativistic QED. J. Funct. Anal. 207,68–110 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. Ch.
    Chen T.: Infrared renormalization in non-relativistic QED and scaling criticality. J. Funct. Anal. 254, 2555–2647 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. CFP.
    Chen T., Fröhlich J., Pizzo A.: Infraparticle scattering states in non-relativistic quantum electrodynamics. II. Mass shell properties. J. Math. Phys. 50, 012103 (2009)MathSciNetADSCrossRefGoogle Scholar
  9. CVV.
    Chen T., Vougalter V., Vugalter S.A.: The increase of binding energy and enhanced binding in nonrelativistic QED. J. Math. Phys. 44, 1961–1970 (2003)MathSciNetADSCrossRefMATHGoogle Scholar
  10. Cw.
    Cwikel M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977)MathSciNetCrossRefMATHGoogle Scholar
  11. Da.
    Daubechies I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90, 511–520 (1983)MathSciNetADSCrossRefMATHGoogle Scholar
  12. FGS.
    Fröhlich J., Griesemer M., Schlein B.: Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math. 164, 349–398 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. Gr.
    Griesemer M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210, 321–340 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. GLL.
    Griesemer M., Lieb E.H., Loss M.: Ground states in non-relativistic quantum electrodynamics.Invent. Math. 145, 557–595 (2001)MathSciNetADSCrossRefMATHGoogle Scholar
  15. Ha.
    Hainzl C.: One non-relativistic particle coupled to a photon field. Ann. Henri Poincaré 2, 217–237 (2003)MathSciNetGoogle Scholar
  16. HaSe.
    Hainzl C., Seiringer R.: Mass renormalization and energy level shift in non-relativistic QED.Adv. Theor. Math. Phys. 6, 847–871 (2002)MathSciNetGoogle Scholar
  17. HVV.
    Hainzl C., Vougalter V., Vugalter S.A.: Enhanced binding in non-relativistic QED. Commun. Math. Phys. 233, 13–26 (2003)MathSciNetADSCrossRefMATHGoogle Scholar
  18. Hi1.
    Hiroshima F.: Essential self-adjointness of translation-invariant quantum field models for arbitrary coupling constants. Commun. Math. Phys. 211, 585–613 (2000)MathSciNetADSCrossRefMATHGoogle Scholar
  19. Hi2.
    Hiroshima F.: Fiber Hamiltonians in non-relativistic quantum electrodynamics. J. Funct. Anal. 252, 314–355 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. HiSa1.
    Hiroshima F., Sasaki I.: On the ionization energy of the semi-relativistic Pauli-Fierz model for a single particle. RIMS Kokyuroku Bessatsu 21, 25–34 (2010)MathSciNetGoogle Scholar
  21. HiSa2.
    Hiroshima, F., Sasaki, I.: Enhanced binding of an N-particle system interacting with a scalar field II. Relativistic version. [math-ph], 2012
  22. HiSp.
    Hiroshima F., Spohn H.: Enhanced binding through coupling to a quantum field. Ann. Henri Poincaré 6, 1159–1187 (2001)MathSciNetADSCrossRefGoogle Scholar
  23. HSS.
    Hiroshima F., Spohn H., Suzuki A.: The no-binding regime of the Pauli-Fierz model. J. Math. Phys. 52, 062104 (2011)MathSciNetADSCrossRefGoogle Scholar
  24. KoMa1.
    Könenberg, M., Matte, O.: Ground states of semi-relativistic Pauli-Fierz and no-pair Hamiltonians in QED at critical Coulomb coupling. J. Oper. Theory 70, 211–237 (2013)Google Scholar
  25. KoMa2.
    Könenberg, M., Matte, O.: The mass shell in the semi-relativistic Pauli-Fierz model. Ann. Henri Poincaré. doi:10.1007/s00023-013-0268-2 (2013)
  26. KMS1.
    Könenberg M., Matte O., Stockmeyer E.: Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics I: The semi-relativistic Pauli-Fierz operator. Rev. Math. Phys. 23, 375–407 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. KMS2.
    Könenberg M., Matte O., Stockmeyer E.: Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics II: The no-pair operator. J. Math. Phys. 52, 123501 (2011)MathSciNetADSCrossRefGoogle Scholar
  28. KMS3.
    Könenberg, M., Matte, O., Stockmeyer, E.: Hydrogen-like atoms in relativistic QED. In: Siedentop, H. (ed.) Complex Quantum Systems. (Singapore, 2010). Lecture Note Series of the Institute for Mathematical Sciences, National University of Singapore, vol. 24, pp. 219–290. World Scientific, Singapore (2013)Google Scholar
  29. LiLo1.
    Lieb, E. H., Loss, M.: Self-energy of electrons in non-perturbative QED. In: Differential equations and mathematical physics. (Birmingham, AL, 1999.), AMS/IP Stud. Adv. Math., Vol. 16, Providence, RI: Amer. Math. Soc., 2000, pp. 279–293Google Scholar
  30. LiLo2.
    Lieb E.H., Loss M.: Existence of atoms and molecules in non-relativistic quantum electrodynamics. Adv. Theor. Math. Phys. 7, 667–710 (2003)MathSciNetCrossRefMATHGoogle Scholar
  31. MaSt.
    Matte O., Stockmeyer E.: Exponential localization for a hydrogen-like atom in relativistic quantum electrodynamics. Commun. Math. Phys. 295, 551–583 (2010)MathSciNetADSCrossRefMATHGoogle Scholar
  32. Ma.
    Matte O.: On higher order estimates in quantum electrodynamics. Doc. Math. 15, 207–234 (2010)MathSciNetMATHGoogle Scholar
  33. MiSp.
    Miyao T., Spohn H.: Spectral analysis of the semi-relativistic Pauli-Fierz Hamiltonian. J. Funct. Anal. 256, 2123–2156 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. RRSMS.
    Raynal J.C., Roy S.M., Singh V., Martin A., Stubbe J.: The “Herbst Hamiltonian” and the mass of boson stars. Phys. Lett. B 320, 105–109 (1994)ADSCrossRefGoogle Scholar
  35. ReSi.
    Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press, 1978Google Scholar
  36. SøSt.
    Sørensen T.Ø., Stockmeyer E.: On the convergence of eigenfunctions to threshold energy states.Proc. Roy. Soc. Edinburgh Sect. A 138, 169–187 (2008)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Fakultät für PhysikUniversität WienViennaAustria
  2. 2.Mathematisches InstitutLudwig-Maximilians-UniversitätMünchenGermany
  3. 3.Institut for MatematikÅrhus UniversitetÅrhusDenmark

Personalised recommendations