Interaction-dominated ground state with nearly ideal condensation temperature ($N^{-1}\ll a/L\ll N^{-1/3}$)

In a box of size $L$, confinement energy scales as ${\hslash }^2/(m{L}^2)$, transition temperature as that scale times $N^{2/3}$, and mean interaction energy as ${\hslash }^{2}Na/(m{L}^3)$. The displayed window therefore makes interaction energy large compared with confinement but small compared with $k_B{T}_c$. It simultaneously implies dilute-gas parameter $n{a}^3\ll 1$. Ground-state interaction dominance does not require a large relative shift of the condensation temperature.