Annales Henri Poincaré

, Volume 11, Issue 8, pp 1487–1544 | Cite as

Quantitative Estimates on the Binding Energy for Hydrogen in Non-Relativistic QED

  • Jean-Marie Barbaroux
  • Thomas Chen
  • Vitali Vougalter
  • Semjon Vugalter
Article

Abstract

In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli–Fierz model up to the order α5 log α−1, where α denotes the fine structure constant, and prove rigorous bounds on the remainder term of the order o(α5 log α−1). As a consequence, we prove that the binding energy is not a real analytic function of α, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of α, as conjectured in the recent literature.

References

  1. 1.
    Bach V., Chen T., Fröhlich J., Sigal I.M.: The renormalized electron mass in non-relativistic QED. J. Funct. Anal. 243(2), 426–535 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bach V., Fröhlich J., Pizzo A.: Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field. Commun. Math. Phys. 264(1), 145–165 (2006)MATHCrossRefADSGoogle Scholar
  3. 3.
    Bach V., Fröhlich J., Sigal I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999)MATHCrossRefADSGoogle Scholar
  4. 4.
    Bach V., Fröhlich J., Sigal I.M.: Quantum electrodynamics of confined relativistic particles. Adv. Math. 137(2), 299–395 (1998)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bach V., Fröhlich J., Sigal I.M.: Mathematical theory of radiation. Found. Phys. 27(2), 227–237 (1997)CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Barbaroux J.-M., Vugalter S.: Non-analyticity of the ground state energy of the Hydrogen atom in nonrelativistic QED. J. Phys. A: Math. Theor. 43, 474004 (2010)CrossRefADSGoogle Scholar
  7. 7.
    Barbaroux J.-M., Chen T., Vougalter V., Vugalter S.A.: On the ground state energy of the translation invariant Pauli-Fierz model. Proc. Am. Math. Soc. 136, 1057–1064 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen T.: Infrared renormalization in non-relativistic QED and scaling criticality. J. Funct. Anal. 354(10), 2555–2647 (2008)CrossRefGoogle Scholar
  9. 9.
    Chen, T., Fröhlich, J.: Coherent infrared representations in non-relativistic QED. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proc. Symp. Pure Math., vol. I. AMS (2007)Google Scholar
  10. 10.
    Chen T., Vougalter V., Vugalter S.A.: The increase of binding energy and enhanced binding in non-relativistic QED. J. Math. Phys. 44(5), 1961–1970 (2003)MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Fröhlich J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré Sect. A (N.S.) 19, 1–103 (1973)Google Scholar
  12. 12.
    Griesemer M., Hasler D.: Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation. Ann. Henri Poincaré 10(3), 577–621 (2009)CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Griesemer M., Lieb E.H., Loss M.: Ground states in non-relativistic quantum electrodnamics. Inv. Math 145, 557–595 (2001)MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Hainzl, C.: Enhanced binding through coupling to a photon field. Mathematical results in quantum mechanics (Taxco, 2001). Contemp. Math., vol. 307, pp. 149–154. Amer. Math. Soc., Providence (2002)Google Scholar
  15. 15.
    Hainzl C., Hirokawa M., Spohn H.: Binding energy for hydrogen-like atoms in the Nelson model without cutoffs. J. Funct. Anal. 220(2), 424–459 (2005)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hainzl C., Vougalter V., Vugalter S.A.: Enhanced binding in non-relativistic QED. Commun. Math. Phys. 233, 13–26 (2003)MATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Jean-Marie Barbaroux
    • 1
    • 2
  • Thomas Chen
    • 3
  • Vitali Vougalter
    • 4
  • Semjon Vugalter
    • 5
  1. 1.Centre de Physique ThéoriqueMarseille Cedex 9France
  2. 2.Département de MathématiquesUniversité du Sud Toulon-VarLa Garde CedexFrance
  3. 3.Department of MathematicsUniversity of Texas at AustinAustinUSA
  4. 4.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa
  5. 5.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMunichGermany

Personalised recommendations