{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"e474337a11dc8d67b4a1fed4f3c8f5f719f5cc1ea2400b456bf8be72625ee313","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011","cfa624940ad39870a307945e691d10cf853e0273dc669a3a59e2bd47cf272222"],"salt":"d18d9b9cd0810a4582e63a3555d7d1ac080c9fd33d0a167b2efc057553b4981b","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.lieb-thirring-constants.kinetic-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[{"note":"","parent_revision":"cfa624940ad39870a307945e691d10cf853e0273dc669a3a59e2bd47cf272222","relation":"special_case"}],"plain":"Prove that in three dimensions the total binding energy of all bound states is never larger than the simple semiclassical estimate. Semiclassical means counting states as if each occupies a fixed volume of position-momentum space.","posed_since":"1976","precise":"For $-\\Delta + V$ on $L^2(R^3)$ (units $\\hbar^2/(2m) = 1$) with negative eigenvalues $E_j$, prove or disprove $\\sum_j \\mid E_j \\mid \\le L^{\\mathrm{cl}}_{1,3} \\int V_-^{5/2}\\, dx$ with $L^{\\mathrm{cl}}_{1,3} = 1/(15 \\pi^2)$. Dual form: for orthonormal $\\phi_1..\\phi_N$ in $L^2(R^3)$, sum_j int |grad phi_j|^2 >= (3/5)(6 pi^2)^(2/3) int rho^(5/3), $\\rho = \\sum_j \\mid \\phi_j \\mid^2$. Answer: a proof, or a potential with ratio greater than 1.","problem_ref":null,"references":"","settled_by":"A proof of the inequality with constant $1/(15 \\pi^{2})$, or an explicit potential exceeding it.","status_note":"Best bound $L_{1,3}/L^{\\mathrm{cl}}_{1,3} \\le 1.456$ (Frank, Hundertmark, Jex, Nam, JEMS 2021, valid in every dimension); Frank, Gontier, Lewin (CMP 2021) proved that for $\\gamma \\ge 1$ the optimum is not attained by any potential with finitely many eigenvalues.","title":"Semiclassical constant for the eigenvalue sum in three dimensions","topic_ref":"9b795df8bba67037212e33cc80fcb1b7804f1dc5a26e182277387971a731f808"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"2ff4ec14e0a81f529f9b586e6c446dc012ced89d0408e7f04050ea1468dba1da","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"b02e44d7a4bf072a8e7832443e6545992bb63506e2876439d456d7b827260626","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"HE9gslR2H-sEw84I8gMqEGg8QinzJHUDLH2NFmzXwFI_ae4dBxuWNJCt---2Lg5FsrbozSdO1FZxpbucZeRmAg"},"schema":"pubphys.envelope/1"},"record_hash":"2ff4ec14e0a81f529f9b586e6c446dc012ced89d0408e7f04050ea1468dba1da","leaf_index":1642}