{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"a0cfe79db66964040a2019c343a9cce171dd1a1ca6b3dae4d98ee3c11747ad82","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b","cfa624940ad39870a307945e691d10cf853e0273dc669a3a59e2bd47cf272222"],"salt":"2f488564d663506726c5f8ef184b7cac29d72dd05bb23a152f81db66fe736dc8","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.lieb-thirring-constants.critical-exponent","kind":"well-posed","literature_status":"open","n":"2","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 $\\gamma_c(d) = \\mathrm{inf}\\{\\gamma : L_{\\gamma,d} = L^{\\mathrm{cl}}_{\\gamma,d}\\}$, 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 $\\gamma_c(2)$ $\\ge\\ \\mathrm{about}\\ 1.165$, 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":"cfa624940ad39870a307945e691d10cf853e0273dc669a3a59e2bd47cf272222","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":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"db0717888d15382e66ef8ddd24b4d83cd967033f43da6e3a52a41c75d1db8792","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"79e45eadc0641b13df4cc12eb6b69291097dc26017007a8cbdfde06d6084834a","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"93Y4iYx8nD2s2CUDHmFwCn3mh99mjMYL_0BO1GM6ou2cpn-sfsIHDm7-fPWntTcd8sC8H2jO4po6b_rBvV01DA"},"schema":"pubphys.envelope/1"},"record_hash":"db0717888d15382e66ef8ddd24b4d83cd967033f43da6e3a52a41c75d1db8792","leaf_index":3438}