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 + 128/15√π√ρ + o(√ρ)) as ρ→ 0, and the condensate fraction $η$ predicted by the hard-core Simple Equation is shown to satisfy η= 1 - 8/3√π√ρ + o(√ρ) as ρ→ 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| ≤ 1\ of increasing height m: at fixed energy, we establish the L¹ 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.