{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"d60a3a45e0f678f4611fd747e115e3385fda110623292bef9b71ce9379921ee6","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"b51beea9b5b2c12628f7c1e7c4031d05fe92fedbcd92d9cec4c75c9154b7ee44","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.bose-condensation.third-order-energy","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"The energy of a dilute repulsive Bose gas is known rigorously to two terms in powers of the density. The next predicted term, with a logarithm, is proven only from above.","posed_since":"1959","precise":"With $\\hbar^{2}/(2m) = 1$, the ground-state energy density is predicted to be $e(\\rho) = 4 \\pi a \\rho^2 [1 + (128/(15 \\sqrt{\\pi})) (\\rho a^3)^{1/2} + 8 (4 \\pi/3 - \\sqrt{3}) \\rho a^3 \\operatorname{log}(\\rho a^3) + o(\\rho a^3 \\operatorname{log}(\\rho a^3))]$ (Wu, 1959). Prove the matching lower bound for general short-range $v \\ge 0$. Answer: a proof of the two-sided expansion.","problem_ref":null,"references":"","settled_by":"A lower bound on $e(\\rho)$ matching the Wu term.","status_note":"The two-term Lee-Huang-Yang formula is proven (Yau and Yin 2009, upper bound; Fournais and Solovej 2020, lower bound); a third-order upper bound in the thermodynamic limit was proven by Brooks, Oldenburg, Saint Aubin and Schlein (arXiv 2506.04153, 2025).","title":"Third-order term in the dilute Bose gas ground-state 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