0$. This holds for all small enough $Y$, provided $\alpha+5\beta<2$. \item $\alpha>0$ in order that $\varepsilon\to 0$ for $Y\to 0$. \item $3\beta-1>0$ in order that $Y^{-1}(a/\ell)^{3}\to 0$ for for $Y\to 0$. \item $1-\alpha-2\beta-\gamma>0$ to control the second factor in (\ref{Kformula2}). \end{itemize} Taking \begin{equation}\label{exponents} \alpha=1/17,\quad \beta=6/17,\quad \gamma=3/17 \end{equation} all these conditions are satisfied, and \begin{equation} \alpha= 3\beta-1=1-\alpha-2\beta-\gamma=1/17. \end{equation} This completes the proof of Theorems 3.1 and 3.2, for the case of potentials with finite range. By optimizing the proportionality constants in (\ref{ans}) one can show that $C=8.9$ is possible in Theorem 1.1 \cite{S1999}. The extension to potentials of infinite range decreasing faster than $1/r^3$ at infinity is obtained by approximation by finite range potentials, controlling the change of the scattering length as the cut-off is removed. See Appendix B in \cite{LSY1999} for details. A slower decrease than $1/r^3$ implies infinite scattering length. \hfill$$\Box$$ The exponents (\ref{exponents}) mean in particular that \begin{equation}a\ll R\ll \rho^{-1/3}\ll \ell \ll(\rho a)^{-1/2},\end{equation} whereas Dyson's method required $R\sim \rho^{-1/3}$ as already explained. The condition $\rho^{-1/3}\ll \ell$ is required in order to have many particles in each box and thus $n(n-1)\approx n^2$. The condition $\ell \ll(\rho a)^{-1/2}$ is necessary for a spectral gap gap $\gg e_{0}(\rho)$ in Temple's inequality. It is also clear that this choice of $\ell$ would lead to a far too big energy and no bound for $ e_{0}(\rho)$ if we had chosen Dirichlet instead of Neumann boundary conditions for the cells. But with the latter the method works! %%%%%%%%%% \begin{thebibliography}{99} \bibitem{TRAP} W. Ketterle, N. J. van Druten, in B. Bederson, H. Walther, eds., Advances in Atomic, Molecular and Optical Physics, {\bf 37}, 181, Academic Press (1996). \bibitem{DGPS} F. Dalfovo, S.\ Giorgini, L.P.\ Pitaevskii, and S.\ Stringari, {\it Theory of Bose-Einstein condensation in trapped gases}, Rev. Mod. Phys. \textbf{71}, 463--512 (1999) \bibitem{LY1998} E.H. Lieb, J. Yngvason, {\it Ground State Energy of the low density Bose Gas}, Phys. Rev. Lett. \textbf{80}, 2504--2507 (1998) \bibitem{LSY1999} E.H. Lieb, R. Seiringer, and J. Yngvason, {\it Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional}, mp\_arc 99-312, xxx e-print archive math-ph/9908027 (1999). \bibitem{S1999} R. Seiringer, Diplom thesis, University of Vienna, 1999. \bibitem{BO} N.N. Bogoliubov, J. Phys. (U.S.S.R.) {\bf 11}, 23 (1947); N.N. Bogoliubov and D.N. Zubarev, Sov. Phys.-JETP {\bf 1}, 83 (1955). \bibitem{Lee-Huang-YangEtc}K.~Huang, and C.N.~Yang, Phys. Rev. {\bf 105}, 767-775 (1957); T.D.~Lee, K.~Huang, and C.N.~Yang, Phys. Rev. {\bf 106}, 1135-1145 (1957); K.A. Brueckner, K. Sawada, Phys. Rev. {\bf 106}, 1117-1127, 1128-1135 (1957).; S.T. Beliaev, Sov. Phys.-JETP {\bf 7}, 299-307 (1958); T.T. Wu, Phys. Rev. {\bf 115}, 1390 (1959); N. Hugenholtz, D. Pines, Phys. Rev. {\bf 116}, 489 (1959); M. Girardeau, R. Arnowitt, Phys. Rev. {\bf 113}, 755 (1959); T.D. Lee, C.N. Yang, Phys. Rev. {\bf 117}, 12 (1960). %% \bibitem{Lieb63} E.H. Lieb, {\it Simplified Approach to the Ground State Energy of an Imperfect Bose Gas}, Phys. Rev. {\bf 130}, 2518--2528 (1963). See also Phys. Rev. {\bf 133}, A899-A906 (1964) (with A.Y. Sakakura) and Phys. Rev. {\bf 134}, A312-A315 (1964) (with W. Liniger). \bibitem{EL2} E.H.~Lieb, {\it The Bose fluid,} in Lecture Notes in Theoretical Physics VIIC, W.E.~Brittin, ed., Univ. of Colorado Press, pp. 175 (1964). \bibitem{dyson} F.J. Dyson, {\it Ground-State Energy of a Hard-Sphere Gas}, Phys. Rev. \textbf{106}, 20--24 (1957) \bibitem{BA} B. Baumgartner,{\it The existence of many-particle bound states despite a pair interaction with positive scattering length}, J. Phys. A {\bf 30}, L741--L747 (1997). \bibitem{LL} E.H. Lieb, W. Liniger, {\it Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State}, Phys. Rev. \textbf{130}, 1605--1616 (1963). \bibitem{TE} G. Temple, {\it The theory of Rayleigh's principle as applied to continuous systems}, Proc. Roy. Soc. London A \textbf{119}, 276-293 (1928). \end{thebibliography} \end{document} %----------------------------------------------------------------------- % End of article.top %-----------------------------------------------------------------------