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.
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References
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)]
Bogoliubov N.N.: On the theory of superfluidity. J. Phys. (U.S.S.R.) 11, 23–32 (1947)
Bogoliubov N.N.: Energy levels of the imperfect Bose–Einstein gas. Bull. Moscow State Univ. 7, 43–56 (1947)
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)
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)
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)
Giuliani A., Seiringer R.: The ground state energy of the weakly interacting Bose gas at high density. J. Stat. Phys. 135, 915–934 (2009)
Grech P., Seiringer R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 332, 559–591 (2013)
Griffin, A.: Excitations in a Bose-condensed liquid. Cambridge University Press, Cambridge (1993)
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)
Landau L.D.: The theory of superfuidity of Helium II. J. Phys. (USSR) 5, 71 (1941)
Landau L.D.: On the theory of superfuidity of Helium II. J. Phys. (USSR) 11, 91 (1947)
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]
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)
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)
Lieb E.H., Solovej J.P.: Ground state energy of the two-component charged Bose gas. Commun. Math. Phys. 252, 485–534 (2004)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. In: Analysis of Operators, vol. 4. Academic Press, New York (1978)
Seiringer R.: The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306, 565–578 (2011)
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)
Zagrebnov V.A., Bru J.B.: The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350, 291 (2001)
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)
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Communicated by Claude Alain Pillet.
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Dereziński, J., Napiórkowski, M. Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit. Ann. Henri Poincaré 15, 2409–2439 (2014). https://doi.org/10.1007/s00023-013-0302-4
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DOI: https://doi.org/10.1007/s00023-013-0302-4