{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"64bf2808f715911f80e409959fe9c830aa46923277baf3bac94570dc62cd0de3","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"94928650203c28cbe21848928d797097f50c482a1771cf00498e2afc3e946ad6","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.fermi-liquids-bcs.bcs-odlro","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"BCS theory explains superconductivity by electron pairing, but it is an approximation. No one has proven that a full many-electron model with attraction actually forms a pair condensate in an infinite system.","posed_since":"","precise":"$N$ spin-$1/2$ fermions in a box of side $L$ with attractive short-range pair potential, density fixed, $L \\to \\infty$. Prove that the two-body reduced density matrix has an eigenvalue of order $N$ (Yang's off-diagonal long-range order) for $T$ below a $T_{c} > 0$, and compare $T_{c}$ with the BCS functional prediction. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of pair condensation in the thermodynamic limit for the full many-body Hamiltonian.","status_note":"Rigorous results concern the BCS functional (Hainzl, Hamza, Seiringer and Solovej, 2008, and later) and mean-field type Hamiltonians, not the full many-body problem.","title":"Off-diagonal long-range order in a many-body model of attractive 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