Bose–Einstein condensation in the thermodynamic limit for dilute Bose gases

Establish Bose–Einstein condensation in the thermodynamic limit for the ground state of the N-boson Hamiltonian with nonnegative, spherically symmetric, finite-range two-body potential v on a d-dimensional torus (d = 2 or 3): prove that, as L tends to infinity with N/L^d tending to a fixed density ρ > 0, the zero-momentum occupation L^{-d}⟨Ψ_L, a_0† a_0 Ψ_L⟩ converges to a strictly positive condensate density ρ_0 > 0.

Background

The paper defines Bose–Einstein condensation (BEC) via macroscopic occupation of the zero-momentum mode in the thermodynamic limit and emphasizes that, while variational (Bogoliubov) trial states exhibit almost complete condensation, proving BEC for the true ground state remains open.

The authors explain that establishing the Lee–Huang–Yang energy formula does not require BEC in the thermodynamic limit; instead, condensation can be proved for gases confined to boxes at the healing-length scale (Gross–Pitaevskii regime). Despite several rigorous results in confined settings, BEC in the thermodynamic limit is only known in a few lattice cases that are not dilute, leaving the continuum dilute gas case unresolved.