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Hyperfine Splitting in Non-relativistic QED: Uniqueness of the Dressed Hydrogen Atom Ground State

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Abstract

We consider a free hydrogen atom composed of a spin-\({\frac{1}{2}}\) nucleus and a spin-\({\frac{1}{2}}\) electron in the standard model of non-relativistic QED. We study the Pauli-Fierz Hamiltonian associated with this system at a fixed total momentum. For small enough values of the fine-structure constant, we prove that the ground state is unique. This result reflects the hyperfine structure of the hydrogen atom ground state.

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References

  1. Amour L., Faupin J.: Hyperfine splitting of the dressed hydrogen atom ground state in non-relativistic QED. Rev. Math. Phys. 23, 553–574 (2011)

    Article  MATH  Google Scholar 

  2. Amour, L., Faupin, J., Grébert, B., Guillot, J.-C.: On the infrared problem for the dressed non-relativistic electron in a magnetic field. In: Spectral and Scattering Theory for Quantum Magnetic Systems, Vol. 500 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2009, pp 1–24

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

    MathSciNet  MATH  Google Scholar 

  4. Bach V., Chen T., Fröhlich J., Sigal I.M.: Smooth Feshbach map and operator-theoretic renormalization group methods. J. Funct. Anal. 203, 44–92 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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, 145–165 (2006)

    Article  ADS  MATH  Google Scholar 

  6. Bach V., Fröhlich J., Sigal I.M.: Quantum electrodynamics of confined non-relativistic particles. Adv. in Math. 137, 299–395 (1998)

    Article  MATH  Google Scholar 

  7. 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, 249–290 (1999)

    Article  ADS  MATH  Google Scholar 

  8. Barbaroux, J.-M., Guillot, J.-C.: Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W±. I. Adv. in Math. Phys. 978903 (2009)

  9. Chen T., Fröhlich J., Pizzo A.: Infraparticle scattering states in non-relativistic QED. II. Mass shell properties. J. Math. Phys. 50, 012103 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  10. Cohen-Tannoudji, C., Diu, B., Laloë, F.: Mécanique quantique II. Paris: Hermann, 1977

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

    Google Scholar 

  12. Fröhlich J., Pizzo A.: Renormalized Electron Mass in Nonrelativistic QED. Commun. Math. Phys. 294, 439–470 (2010)

    Article  ADS  MATH  Google Scholar 

  13. Griesemer M., Hasler D.: On the smooth Feshbach-Schur map. J. Funct. Anal. 254, 2329–2335 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Hasler, D., Herbst, I.: Uniqueness of the ground state in the Feshbach renormalization analysis. Lett. Math. Phys. 100, 171–180 (2012)

    Google Scholar 

  16. Hiroshima F.: Multiplicity of ground states in quantum field models: application of asymptotic fields. J. Funct. Anal. 224, 431–470 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hiroshima F., Lorinczi J.: Functional integral representations of nonrelativistic quantum electrodynamics with spin 1/2. J. Funct. Anal. 254, 2127–2185 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hiroshima F., Spohn H.: Ground state degeneracy of the Pauli-Fierz Hamiltonian with spin. Adv. Theor. Math. Phys. 5, 1091–1104 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Kato, T.: Perturbation Theory for Linear Operators. (Second edition), Berlin: Springer-Verlag, 1976

  20. Loss M., Miyao T., Spohn H.: Lowest energy states in nonrelativistic QED: atoms and ions in motion. J. Funct. Anal. 243, 353–393 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Loss M., Miyao T., Spohn H.: Kramers degeneracy theorem in nonrelativistic QED. Lett. Math. Phys. 89, 21–31 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Reed, M., Simon, B.: Methods of modern mathematical physics I-IV. New York: Academic Press, 1972–1978

  24. Spohn, H.: Dynamics of charged particles and their radiation field. Cambridge: Cambridge University Press, 2004

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Authors and Affiliations

  1. Laboratoire de Mathématiques, FR-CNRS 3399, Université de Reims, Moulin de la Housse, BP 1039, 51687, Reims Cedex 2, France

    Laurent Amour

  2. Institut de Mathématiques de Bordeaux, UMR-CNRS 5251, Université de Bordeaux 1, 351 Cours de la Libération, 33405, Talence Cedex, France

    Jérémy Faupin

Authors
  1. Laurent Amour
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  2. Jérémy Faupin
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Correspondence to Jérémy Faupin.

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Communicated by H. Spohn

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Amour, L., Faupin, J. Hyperfine Splitting in Non-relativistic QED: Uniqueness of the Dressed Hydrogen Atom Ground State. Commun. Math. Phys. 319, 425–450 (2013). https://doi.org/10.1007/s00220-012-1625-6

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  • DOI: https://doi.org/10.1007/s00220-012-1625-6

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