{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"82c89c9b307cdfa071a0ee4ba9be2b9ff353f7435d86e81c510da0d26a9d3729","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"3ef58e02283cba0891da59d6713e1aaffa436535ec2a2b5d106f1bbae0ee8c5a","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"mathph.lieb-thirring-constants.lieb-oxford","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"The Lieb-Oxford inequality bounds how far the electrostatic energy of many charges can fall below the value for a smeared-out charge cloud, using only their density. The best constant lies between 1.44 and 1.58 and is unknown.","posed_since":"","precise":"For every $N$-particle probability density on $R^{3N}$ with one-body density $\\rho$, $\\left\\langle \\sum_{i<j} e^2/\\mid x_i - x_j \\mid \\right\\rangle - (e^2/2) \\int \\int \\rho(x) \\rho(y)/\\mid x - y \\mid\\, dx\\, dy \\ge -C_{\\mathrm{LO}} e^2 \\int \\rho^{4/3}\\, dx$. Determine the optimal $C_{\\mathrm{LO}}$, known to satisfy $1.4442 \\le C_{\\mathrm{LO}} \\le 1.58$, and decide whether it equals the uniform-electron-gas constant, the low-density limit of the jellium energy per particle in units $e^2 \\rho^{1/3}$. That constant is $\\ge 1.4442$, the bcc Wigner-crystal value, with equality if mathph.crystallization.jellium-bcc holds.","problem_ref":null,"references":"","settled_by":"A proof of the inequality with the conjectured constant, or a trial state whose ratio exceeds the uniform-electron-gas value.","status_note":"Lewin, Lieb, Seiringer (Lett. Math. Phys. 2022) lowered the upper bound from 1.64 to 1.58 and recorded the lower bound 1.4442.","title":"Sharp Lieb-Oxford constant for the indirect Coulomb energy","topic_ref":"9b795df8bba67037212e33cc80fcb1b7804f1dc5a26e182277387971a731f808"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"c8484e2ab5ce76f7ec0b6150351fb00b3dd689a7a2480461ea9e173a7937f205","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"341bfdb9747b744a2d8b06c4370b5647802a1f3c6843fe19a3df5d09e529a92b","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"M4guB9-7hY01bNG05JL-yX6y7Ufge1XuTdj48Cw_eYslVM1hewTkJf67Y5avXYANmBYD4ROhdSH-6KpbY_N6Cw"},"schema":"pubphys.envelope/1"},"record_hash":"c8484e2ab5ce76f7ec0b6150351fb00b3dd689a7a2480461ea9e173a7937f205","leaf_index":1643}