{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"7a9a20eedc7258d98605954c3c4e897e70b28b4a9b2fe9b36a83bf5e8ce622c4","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"20d8efbfc6abda35606159014f75ccdbae3004350362fd851e971d34b9f3ef8b","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"mathph.lieb-thirring-constants.critical-exponent","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"For large enough exponent $\\gamma$ the best constant equals the simple semiclassical value; for small $\\gamma$ it is larger. Find the crossover exponent in two and three dimensions.","posed_since":"1976","precise":"Define gamma_c(d) = inf{gamma : L_{gamma,d} = L^cl_{gamma,d}}, where $L^{\\mathrm{cl}}_{\\gamma,d} = \\Gamma(\\gamma+1)/((4 \\pi)^{d/2} \\Gamma(\\gamma+1+d/2))$; $L_{\\gamma,d}/L^{\\mathrm{cl}}_{\\gamma,d}$ is nonincreasing in $\\gamma$ (Aizenman, Lieb 1978). Known: $1 \\le \\gamma_c(d) \\le 3/2$ for all d (lower bound Helffer, Robert 1990; upper bound Laptev, Weidl, Acta Math 2000), $\\gamma_c(1) = 3/2$, and $\\gamma_c(2)$ >= about 1.165, the exponent at which the one-bound-state and semiclassical constants cross. Determine $\\gamma_c(2)$ and $\\gamma_c(3)$; the 1976 conjecture $\\gamma_c(3) = 1$ follows from a proof of mathph.lieb-thirring-constants.kinetic-3d.","problem_ref":null,"references":"","settled_by":"Proofs of $L_{\\gamma,d} = L^{\\mathrm{cl}}_{\\gamma,d}$ for $\\gamma \\ge \\gamma_{*}$ together with explicit potentials beating $L^{\\mathrm{cl}}$ for $\\gamma < \\gamma_{*}$, in $d = 2$ and $d = 3$.","status_note":"Frank, Gontier, Lewin (CMP 2021) showed $L_{\\gamma,d}$ exceeds the one-bound-state value for $\\gamma > \\max(0, 2 - d/2)$, so in $d = 2$ for $1 < \\gamma < \\text{about } 1.165$ the sharp constant is neither the one-bound-state nor the semiclassical value.","title":"Exponent at which the semiclassical constant becomes optimal","topic_ref":"9b795df8bba67037212e33cc80fcb1b7804f1dc5a26e182277387971a731f808"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"cfa624940ad39870a307945e691d10cf853e0273dc669a3a59e2bd47cf272222","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"8005e45123b45d4325cc527750e76e6715de2c54831f8055930ba0ad14f9e3a1","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"UexdqtnWuPikBq-JuVvJZn0IYrupaV4aOcJCm5ZObMXalZCNNYqgl3inpoPKKaizH2eDrIlj_p_EtyrgLIhzDA"},"schema":"pubphys.envelope/1"},"record_hash":"cfa624940ad39870a307945e691d10cf853e0273dc669a3a59e2bd47cf272222","leaf_index":1641}