The Second Order Upper Bound for the Ground Energy of a Bose Gas

摘要: Consider N bosons in a finite box Λ=[0,L]3⊂R 3 interacting via two-body smooth repulsive short range potential. We construct variational state which gives the following upper bound on ground energy per particle $$\overline{\lim}_{\rho\to0}\overline{\lim}_{L\to\infty,\,N/L^3\to \rho}\biggl(\frac{e_0(\rho)-4\pi a\rho}{(4\pi a)^{5/2}(\rho)^{3/2}}\biggr )\leq\frac{16}{15\pi^2},$$ where is scattering length of Previously, an form C16/15π 2 for some constant C>1 was obtained (Erdos et al. Phys. Rev. A 78:053627, 2008). Our result proves prediction by Lee and Yang (Phys. 105(3):1119–1120, 1957) 106(6):1135–1145, 1957).