Abstract
The leading term of the ground state energy/particle of a dilute gas of bosons with mass m in the thermodynamic limit is \({2\pi \hbar^2 a \varrho/m}\) when the density of the gas is \({\varrho}\), the interaction potential is non-negative and the scattering length a is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, a > 0 and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.
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Ruelle D.: Statistical Mechanics: Rigorous Results. Addison Wesley, Reading, MA (1989)
Erdos L., Schlein B., Yau H.T.: The ground state energy of a low density Bose gas: a second order upper bound. Phys. Rev. A 78, 053627 (2008)
Lieb E.H., Seiringer R.: Derivation of the Gross-Pitaevskii equation for rotating bose gases. Commun. Math. Phys. Volume 264(2), 505–537 (2005)
Lieb, E.H., Seiringer, R., Solovej, J.P.: Ground state energy of the low density fermi gas. In: Recent Advances in Differential Equations and Mathematical Physics, N. Chernov, Y. Karpeshina, I. Knowles, R. Lewis, R. Weikard, eds., Contemporary Math. series 412, Providence, RI: Amer. Math. Soc., 2006, pp. 239–248
Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The ground state of the Bose gas. In: Current Developments in Mathematics, (2001), Cambridge, MA: International Press, (2002), pp. 131–178
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-Verlag, 2005
Lieb E.H., Seiringer R., Yngvason J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2000)
Lieb E.H., Yngvason J.: Ground state energy of the low density Bose gas. Phys. Rev. Lett. 80, 2504–2507 (1998)
Lieb E.H., Yngvason J.: The ground state energy of a dilute two-Dimensional Bose gas. J. Stat. Phys. 103, 509 (2001)
Dyson F.J.: Ground state energy of a hard-sphere gas. Phys. Rev. 106, 20–26 (1957)
Temple G.: The theory of Rayleigh’s principle as applied to continuous systems. Proc. Roy. Soc. London A 119, 276–293 (1928)
Lee J.O.: Ground state energy of dilute Bose gas in small negative potential case. J. Stat. Phys. 134, 1–18 (2009)
Yau H.T., Yin J.: The second order upper bound for the ground energy of a Bose gas. J. Stat. Phys. 136(3), 453–503 (2009)
Lee, J.O., Yin, J.: A lower bound on the ground state energy of dilute Bose gas. http://arxiv.org/abs/0908.0109v1[math-ph], 2009
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Communicated by H.-T. Yau
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Yin, J. The Ground State Energy of Dilute Bose Gas in Potentials with Positive Scattering Length. Commun. Math. Phys. 295, 1–27 (2010). https://doi.org/10.1007/s00220-009-0977-z
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DOI: https://doi.org/10.1007/s00220-009-0977-z