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Enhanced binding of an N-particle system interacting with a scalar bose field I

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Abstract

An enhanced binding of an N-particle system linearly coupled to a scalar bose field is investigated, where N ≥  2. It is not assumed that this system has a ground state for a zero coupling. It is shown, however, that there exists a ground state for sufficiently large values of a coupling constant. Basic ingredients of the proof are a weak coupling limit and a modified HVZ theorem.

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References

  1. Arai A. (2001). Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation. Rev. Math. Phys. 13: 1075–1094

    Article  MathSciNet  MATH  Google Scholar 

  2. Arai A. and Hirokawa M. (1997). On the existence and uniqueness of ground states of a generalized spin-boson model. J. Funct. Anal. 151: 455–503

    Article  MathSciNet  MATH  Google Scholar 

  3. Arai A., Hirokawa M. and Hiroshima F. (1999). On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff. J. Funct. Anal. 168: 470–497

    Article  MathSciNet  MATH  Google Scholar 

  4. Arai, A., Hirokawa, M., Hiroshima F.: Regularities of ground states in quantum field models. Kyushu J. Math. (to be published)

  5. Arai A. and Kawano H. (2003). Enhanced binding in a general class of quantum field models. Rev. Math. Phys. 15: 387–423

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bach V., Fröhlich J. and Sigal I.M. (1999). Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207: 249–290

    Article  MATH  Google Scholar 

  8. Barbaroux, J.M., Linde, H., Vugalter, S.: Quaritative estimates on the enhanced binding for the Pauli-Fierz operator. J. Math. Phys. 46, 122103, 11 (2005)

    Google Scholar 

  9. Barbaroux J.M., Chen T. and Vugalter S. (2003). Binding conditions for atomic N-electron systems in non-relativistic QED. Ann. Henri Poincaré 4: 1101–1136

    Article  MathSciNet  Google Scholar 

  10. Barbaroux, J.M., Chen, T., Vugalter, V., Vugalter, S.: On the ground state energy of the translation invariant Pauli-Fierz model, arXiv math-ph/0702098 (2007, preprint)

  11. Betz V., Hiroshima F., Lörinczi J., Minlos R. and Spohn H. (2002). Ground state properties of the Nelson Hamiltonian—a Gibbs measure-based approach. Rev. Math. Phys. 14: 173–198

    Article  MathSciNet  MATH  Google Scholar 

  12. Catto I. and Hainzl C. (2004). Self-energy of one electron in non-relativistic QED. J. Funct. Anal. 207: 68–110

    Article  MathSciNet  MATH  Google Scholar 

  13. Catto I., Exner P. and Hainzl C. (2004). Enhanced binding revisited for a spinless particle in non-relativistic QED. J. Math. Phys. 45: 4174–4185

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen T., Vougalter V. and Vugalter S.A. (2003). The increase of binding energy and enhanced binding in nonrelativistic QED. J. Math. Phys. 44: 1961–1970

    Article  MathSciNet  MATH  Google Scholar 

  15. Cycon H.L., Froese R.G., Kirsch W. and Simon B. (1987). Schrödinger Operators. Springer, Berlin

    MATH  Google Scholar 

  16. Davies E.B. (1977). Asymptotic analysis of some abstract evolution equations. J. Funct. Anal. 25: 81–101

    Article  MATH  Google Scholar 

  17. Davies E.B. (1979). Particle-boson interactions and the weak coupling limit. J. Math. Phys. 20: 345–351

    Article  MathSciNet  Google Scholar 

  18. Dereziński, J., Gérard, C.: Scattering theory of infrared divergent Pauli-Fierz Hamiltonians (2003, preprint)

  19. Gérard, C.: On the existence of ground states for massless Pauli-Fierz Hamiltonians. Anal. Henri Poincare 1, 443–459. A remark on the paper: on the existence of ground states for Hamiltonians, mp-arc 06-146 (2000)

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Hainzl C. and Seiringer R. (2002). Mass renormalization and energy level shift in non-relativistic QED. Adv. Theor. Math. Phys. 6: 847–881

    MathSciNet  Google Scholar 

  22. Hainzl C., Vougalter V. and Vugalter S.A. (2003). Enhanced binding in non-relativistic QED. Commun. Math. Phys. 233: 13–26

    Article  MathSciNet  MATH  Google Scholar 

  23. Hirokawa M. (2007). Infrared catastrophe for Nelson’s model—non-existence of ground state and soft-boson divergence. Publ. RIMS 42: 897–922

    Article  MathSciNet  Google Scholar 

  24. Hirokawa M., Hiroshima F. and Spohn H. (2005). Ground state for point particles interacting through a massless scalar Bose field. Adv. Math. 191: 339–392

    Article  MathSciNet  MATH  Google Scholar 

  25. Hiroshima F. (1997). Functional integral representation of a model in quantum electrodynamics. Rev. Math. Phys. 9: 489–530

    Article  MathSciNet  MATH  Google Scholar 

  26. Hiroshima F. (1998). Weak coupling limit and a removal of an ultraviolet cut-off for a Hamiltonian of particles interacting with a massive scalar field. Inf. Dim. Anal. Quantum Prob. Relat. Topics 1: 407–423

    Article  MathSciNet  MATH  Google Scholar 

  27. Hiroshima F. (1999). Weak coupling limit removing an ultraviolet cut-off for a Hamiltonian of particles interacting with a quantized scalar field. J. Math. Phys. 40: 1215–1236

    Article  MathSciNet  MATH  Google Scholar 

  28. Hiroshima F. (2000). Ground states of a model in nonrelativistic quantum electrodynamics II. J. Math. Phys. 41: 661–674

    Article  MathSciNet  MATH  Google Scholar 

  29. Hiroshima F. (2001). Ground states and spectrum of quantum electrodynamics of non-relativistic particles. Trans. Am. Math. Soc. 353: 4497–4528

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Hiroshima, F.: Fiber Hamiltonians in nonrelativistic quantum electrodynamics, mp-arc 06-194. J. Funct. Anal. (to be published)

  32. Hiroshima F. and Spohn H. (2001). Enhanced binding through coupling to a quantum field. Ann. Henri Poincaré 2: 1159–1187

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  34. Lőrinczi J., Minlos R.A. and Spohn H. (2001). The infrared behaviour in Nelson’s model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincaré 3: 269–295

    Google Scholar 

  35. Nelson E. (1964). Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5: 1190–1197

    Article  Google Scholar 

  36. Reed M. and Simon B. (1980). Methods of Modern Mathematical Physics I. Academic, New York

    MATH  Google Scholar 

  37. Sasaki I. (2005). Ground state of the massless Nelson model in a non-Fock representation. J. Math. Phys. 46: 102–107

    Google Scholar 

  38. Spohn H. (1998). Ground state of quantum particle coupled to a scalar boson field. Lett. Math. Phys. 44: 9–16

    Article  MathSciNet  MATH  Google Scholar 

  39. Spohn H. (2004). Dynamics of Charged Particles and Their Radiation Field. Cambridge University Press, Cambridge

    MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, Fukuoka, 812-8581, Japan

    Fumio Hiroshima

  2. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Itaru Sasaki

Authors
  1. Fumio Hiroshima
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  2. Itaru Sasaki
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Correspondence to Itaru Sasaki.

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Hiroshima, F., Sasaki, I. Enhanced binding of an N-particle system interacting with a scalar bose field I. Math. Z. 259, 657–680 (2008). https://doi.org/10.1007/s00209-007-0243-z

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  • DOI: https://doi.org/10.1007/s00209-007-0243-z

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