{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b19a6df157e9dc5c60f8592380189c35cacd127e52d73b9e2f376aa655690286","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["2ff4ec14e0a81f529f9b586e6c446dc012ced89d0408e7f04050ea1468dba1da","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"fa6e907fcf74600bcfd09daf0454ad5be7f2813c0a4caf7480251735ac86d583","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.lieb-thirring-constants.kinetic-3d","kind":"well-posed","literature_status":"open","n":"2","parents":[],"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 $L^2(R^3)$, $\\sum_j \\int \\mid \\mathrm{grad}\\, \\phi_j \\mid^2 \\ge (3/5)(6 \\pi^2)^{2/3} \\int \\rho^{5/3}$, $\\rho = \\sum_j \\mid \\phi_j \\mid^2$. Answer: a proof, or a potential with ratio greater than 1.","problem_ref":"2ff4ec14e0a81f529f9b586e6c446dc012ced89d0408e7f04050ea1468dba1da","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":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"b1937a5c91046695890745bf15f7b5abe0854e7dc9d59ba78875b4f28adbeec6","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"a8fe4b3e103bfcefc634bdc834153455b05f63002fc454ade0998c59ea46f03c","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"asR1xFWKfNFxcNMpizSaGAkbrMhYkz-89TMMtaVOdFJNQUuGTOb7bi-_mZ2OUJnTyHQfP5DYp_k3hbwbM3-gCg"},"schema":"pubphys.envelope/1"},"record_hash":"b1937a5c91046695890745bf15f7b5abe0854e7dc9d59ba78875b4f28adbeec6","leaf_index":3440}