The Lee-Huang-Yang Formula in the Hard-Core Simplified Approach to the Bose gas

We study the hard-core variant of the Simplified Approach to the ground state of the dilute Bose gas. The hard-core potential has been the subject of renewed attention, and is physically relevant to realistic atomic Bose gases. The hard-core Simple Equation is an integro-differential equation, and we construct its solution via a monotone iteration scheme and prove existence and uniqueness (in a restricted sense) of the solution. The energy per particle $e$ and the density $ρ$ of the solution are shown to satisfy the Lee-Huang-Yang formula $e = 2πρ(1 + \frac{128}{15\sqrtπ}\sqrtρ + o(\sqrtρ))$ as $ρ\to 0$, and the condensate fraction $η$ predicted by the hard-core Simple Equation is shown to satisfy $η= 1 - \frac{8}{3\sqrtπ}\sqrtρ + o(\sqrtρ)$ as $ρ\to 0$, in agreement with Bogolyubov's theory. We thus extend results of Carlen, Jauslin, Lieb to the hard-core case. The hard-core equation is defined on its own terms, and the proofs proceed by approximating the hard-core potential with the soft-core potentials $m\,χ_{\{|x| \le 1\}}$ of increasing height $m$: at fixed energy, we establish the $L^1$ convergence of the corresponding solutions and the convergence of their densities at an $O(m^{-1/2})$ rate, and we show that the Lee-Huang-Yang expansions of the energy and of the condensate fraction for the soft-core equations hold with errors uniform in $m$, so that they survive the hard-core limit.