A Simple 2nd Order Lower Bound to the Energy of Dilute Bose Gases

Abstract

For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound $e(\rho ) \ge 4\pi a \rho ^2 (1- C \sqrt{\rho a^3} \,)$.

A Bogoliubov Method

In this section we recall a simple consequence of the Bogoliubov method. In [6] we use the following version (and allow ${\mathcal {B}}=-{\mathcal {A}}$ if $\kappa =0$)—see also [14, Theorem 6.3].

Theorem A.1

(Simple case of Bogoliubov’s method). For arbitrary ${\mathcal {A}},{\mathcal {B}}\in {\mathbb {R}}$ satisfying ${\mathcal {A}}>0$, $-{\mathcal {A}}<{\mathcal {B}}\le {\mathcal {A}}$ and $\kappa \in {\mathbb {C}}$ we have the operator inequality

$$\begin{aligned}&{\mathcal {A}}(b^*_+b^{}_+ +b^*_{-}b^{}_{-})+{\mathcal {B}}(b^*_+b^*_{-}+b^{}_+b^{}_{-})+ \kappa (b^*_++b^{}_{-})+{\overline{\kappa }}(b^{}_++b^*_{-})\\&\qquad \ge -\frac{1}{2}({\mathcal {A}}-\sqrt{{\mathcal {A}}^2-{\mathcal {B}}^2}) ([b^{}_{+},b^*_{+}]+[b^{}_{-},b^*_{-}])-\frac{2|\kappa |^2}{{\mathcal {A}}+{\mathcal {B}}}, \end{aligned}$$

where $b_\pm$ are operators on a Hilbert space satisfying $[b_+,b_-]=0$.