Advertisement

Annales Henri Poincaré

, Volume 15, Issue 12, pp 2409–2439 | Cite as

Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

  • Jan DerezińskiEmail author
  • Marcin Napiórkowski
Open Access
Article

Abstract

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys. 306:565–578, 2011) to large volumes.

Keywords

Excitation Spectrum Zero Mode Partial Isometry Extended Space Physical Hilbert Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Beliaev, S.T.: Energy spectrum of a non-ideal Bose gas. Sov. Phys. JETP 7, 299 (1958) [reprinted in D. Pines “The Many-Body Problem” (W.A. Benjamin, New York, 1962)]Google Scholar
  2. 2.
    Bogoliubov N.N.: On the theory of superfluidity. J. Phys. (U.S.S.R.) 11, 23–32 (1947)Google Scholar
  3. 3.
    Bogoliubov N.N.: Energy levels of the imperfect Bose–Einstein gas. Bull. Moscow State Univ. 7, 43–56 (1947)Google Scholar
  4. 4.
    Cornean H.D., Dereziński J., Ziń P.: On the infimum of the energy–momentum spectrum of a homogeneous Bose gas. J. Math. Phys. 50, 062103 (2009)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dereziński J., Napiórkowski M., Meissner K.A.: On the infimum of the energy–momentum spectrum of a homogeneous Fermi gas. Ann. Henri Poincaré 14, 1–36 (2013)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Erdös 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)ADSCrossRefGoogle Scholar
  7. 7.
    Giuliani A., Seiringer R.: The ground state energy of the weakly interacting Bose gas at high density. J. Stat. Phys. 135, 915–934 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Grech P., Seiringer R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 332, 559–591 (2013)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Griffin, A.: Excitations in a Bose-condensed liquid. Cambridge University Press, Cambridge (1993)Google Scholar
  10. 10.
    Hodby E., Maragó O.M., Hechenblaikner G., Foot C.J.: Experimental observation of beliaev coupling in a Bose–Einstein condensate. Phys. Rev. Lett. 86, 2196–2199 (2001)ADSCrossRefGoogle Scholar
  11. 11.
    Landau L.D.: The theory of superfuidity of Helium II. J. Phys. (USSR) 5, 71 (1941)Google Scholar
  12. 12.
    Landau L.D.: On the theory of superfuidity of Helium II. J. Phys. (USSR) 11, 91 (1947)Google Scholar
  13. 13.
    Lewin, M., Nam, P.T., Serfaty, S., Solovej, J.P.: Bogoliubov spectrum of interacting Bose gases. Commun. Pure Appl. Math. arXiv:1211.2778 [math-ph]Google Scholar
  14. 14.
    Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The mathematics of the Bose gas and its condensation. In: Oberwolfach Seminars, vol. 34. Birkhäuser, Basel (2005)Google Scholar
  15. 15.
    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
  16. 16.
    Lieb E.H., Solovej J.P.: Ground state energy of the two-component charged Bose gas. Commun. Math. Phys. 252, 485–534 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. In: Analysis of Operators, vol. 4. Academic Press, New York (1978)Google Scholar
  18. 18.
    Seiringer R.: The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306, 565–578 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Zagrebnov V.A., Bru J.B.: The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350, 291 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Department of Mathematical Methods in Physics, Faculty of PhysicsUniversity of WarsawWarsawPoland

Personalised recommendations