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The Excitation Spectrum for Weakly Interacting Bosons in a Trap

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Abstract

We investigate the low-energy excitation spectrum of a Bose gas confined in a trap, with weak long-range repulsive interactions. In particular, we prove that the spectrum can be described in terms of the eigenvalues of an effective one-particle operator, as predicted by the Bogoliubov approximation.

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References

  1. Dalfovo F., Giorgini S., Pitaevskii L.P., Stringari S.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999)

    Article  ADS  Google Scholar 

  2. Bloch I., Dalibard J., Zwerger W.: Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008)

    Article  ADS  Google Scholar 

  3. Cooper N.R.: Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008)

    Article  ADS  Google Scholar 

  4. Fetter A.L.: Rotating trapped Bose-Einstein condensates. Rev. Mod. Phys. 81, 647–691 (2009)

    Article  ADS  Google Scholar 

  5. Landau L.D.: Theory of the Superfluidity of Helium II. Phys. Rev. 60, 356–358 (1941)

    Article  ADS  MATH  Google Scholar 

  6. Steinhauer J., Ozeri R., Katz N., Davidson N.: Excitation Spectrum of a Bose-Einstein Condensate. Phys. Rev. Lett. 88, 120407 (2002)

    Article  ADS  Google Scholar 

  7. Bogoliubov N.N.: On the theory of superfluidity. J. Phys. (U.S.S.R.) 11, 23–32 (1947)

    Google Scholar 

  8. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars, Vol. 34, Basel: Birkhäuser, 2005. Also available at http://arxiv.org/abs/cond-mat/0610117v1 [cond-mat.stat-mech], 2006

  9. Seiringer, R.: Hot Topics in Cold Gases. In: Proceedings of the XVIth International Congress on Mathematical Physics, P. Exner, ed., River Edge, US: World Scientific, 2010, pp. 231–245

  10. Lieb, E.H., Solovej, J.P.: Ground State Energy of the One-Component Charged Bose Gas. Commun. Math. Phys. 217, 127–163 (2001), Errata 225, 219–221 (2002)

    Google Scholar 

  11. Lieb E.H., Solovej J.P.: Ground State Energy of the Two-Component Charged Bose Gas. Commun. Math. Phys. 252, 485–534 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Solovej J.P.: Upper Bounds to the Ground State Energies of the One- and Two-Component Charged Bose Gases. Commun. Math. Phys. 266, 797–818 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Erdős L., Schlein B., Yau H.-T.: Ground-state energy of a low-density Bose gas: A second-order upper bound. Phys. Rev. A 78, 053627 (2008)

    Article  ADS  Google Scholar 

  14. Giuliani A., Seiringer R.: The Ground State Energy of the Weakly Interacting Bose Gas at High Density. J. Stat. Phys. 135, 915–934 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Yau H.-T., Yin J.: The Second Order Upper Bound for the Ground Energy of a Bose Gas. J. Stat. Phys. 136, 453–503 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Girardeau M.: Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension. J. Math. Phys. 1, 516–523 (1960)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Lieb E.H., Liniger W.: Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. 130, 1605–1616 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Lieb E.H.: Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum. Phys. Rev. 130, 1616–1624 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Calogero F.: Ground State of a One-Dimensional N-Body System. J. Math. Phys. 10, 2197–2200 (1969)

    Article  MathSciNet  ADS  Google Scholar 

  20. Calogero F.: Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials. J. Math. Phys. 12, 419–436 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  21. Sutherland B.: Quantum Many-Body Problem in One Dimension: Ground State. J. Math. Phys. 12, 246–250 (1971)

    Article  ADS  Google Scholar 

  22. Sutherland B.: Quantum Many-Body Problem in One Dimension: Thermodynamics. J. Math. Phys. 12, 251–256 (1971)

    Article  ADS  Google Scholar 

  23. Fröhlich, J., Lenzmann, E.: Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree equation. Séminaire É. D. P. XVIII, 26 p. (2003–2004)

  24. Fröhlich J., Knowles A., Schwarz S.: On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction. Commun. Math. Phys. 288, 1023–1059 (2009)

    Article  ADS  MATH  Google Scholar 

  25. Mottl R., Brennecke F., Baumann K., Landig R., Donner T., Esslinger T.: Roton-Type Mode Softening in a Quantum Gas with Cavity-Mediated Long-Range Interactions. Science 336, 1570–1573 (2012)

    Article  ADS  Google Scholar 

  26. Seiringer R.: The Excitation Spectrum for Weakly Interacting Bosons. Commun. Math. Phys. 306, 565–578 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Reed, M., Simon, B.: Analysis of Operators. New York: Academic Press, 1978

  28. Pitaevskii, L., Stringari, S.: Bose-Einstein Condensation. Oxford: Clarendon Press – Oxford, 2003

  29. Hartmann, P.: Ordinary Differential Equations. New York: Wiley, 1964

  30. Gil’, M.I.: Operator Functions and Localization of Spectra. Lecture Notes in Mathematics, Berlin-Heidelberg-New York: Springer, 2003

  31. Seiringer R., Yin J.: The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions. Commun. Math. Phys. 284, 459–479 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, H3A 0B9, Canada

    Philip Grech & Robert Seiringer

  2. Centre de Recherches Mathématiques, Université de Montréal, 2920 Chemin de la Tour, Montréal, Québec, H3T 1J4, Canada

    Philip Grech

Authors
  1. Philip Grech
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  2. Robert Seiringer
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Correspondence to Robert Seiringer.

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Communicated by H. Spohn

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Grech, P., Seiringer, R. The Excitation Spectrum for Weakly Interacting Bosons in a Trap. Commun. Math. Phys. 322, 559–591 (2013). https://doi.org/10.1007/s00220-013-1736-8

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  • DOI: https://doi.org/10.1007/s00220-013-1736-8

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