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Title: Condensation on all length scales in the dilute Bose gas
Plain: In a very dilute gas the particles rarely meet, so condensation should be easiest to prove there. Current proofs work only in regions somewhat larger than the healing length, the distance over which the condensate recovers from a disturbance.
Precise: Dilute regime $\rho a^3 << 1$ ($a = \text{scattering length of }v$). Show that the condensate fraction in a box of side $L$ tends to $1 - O((\rho a^3)^{1/2})$ uniformly as $L \to \infty$, extending results valid for $L$ up to a power of the healing length $(8 \pi \rho a)^{-1/2}$ times a negative power of $\rho a^3$. Answer: a proof.
Settled by: A proof of a condensate fraction near 1 uniform in $L$ for fixed small $\rho a^{3}$.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Fournais (2021) proved condensation on length scales larger than the healing length; the infinite-volume statement is open.
Posed since: (empty)