{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"e6f54d4413772fe3aa70808c03e94754d1e01b02deca65b40483c5d54c34c657","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["bc0a69bb3f8c97ce6dc407a8acf8401e12e745e0e22fbe4189437cd685d3e0eb","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"fc565626f365b849f06e9a032c4a9f48b3ab5eb6bff1da9a5507cbd434dadf26","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"mathph.lieb-thirring-constants.clr-constant","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"The number of bound states of a 3D potential well is at most a constant times the integral of $\\mid V \\mid^{3/2}$. The best possible constant is unknown.","posed_since":"","precise":"For $-\\Delta + V$ on $L^2(R^3)$, the Cwikel-Lieb-Rozenblum bound reads $N(V) \\le L_{0,3} \\int V_-^{(3/2)} dx$, $N(V)$ = number of negative eigenvalues. Determine the sharp $L_{0,3}$; it exceeds $L^{\\mathrm{cl}}_{0,3} = 1/(6 \\pi^2)$, and it is unknown whether it equals the one-bound-state value $L^1_{0,3}$.","problem_ref":"bc0a69bb3f8c97ce6dc407a8acf8401e12e745e0e22fbe4189437cd685d3e0eb","references":"","settled_by":"A proof that some explicit value is an upper bound on $N(V)/\\int V_{-}^{3/2}$ together with a sequence of potentials attaining it.","status_note":"Known upper bounds exceed the semiclassical value by a factor of several; the 2021 Frank-Gontier-Lewin results do not cover $\\gamma = 0$ in $d = 3$.","title":"Optimal constant for counting bound states in three dimensions","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"15f312f39ece541bebe2fd5508c8d79a6b2863117ac7eda9278424cc27803b63","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"877730c24d8a10b46acdd4265853ab0f5e46971ffd73530b76c11a105a4821a9","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"c5TakZdGmAQVkobCAI9WgHl18Wrqv6bBpxiRzEBl8JWlbeLtzKwSCsrvWKPLC7o0MYkqWOh3ZHpLWRKr8GSIDw"},"schema":"pubphys.envelope/1"},"record_hash":"15f312f39ece541bebe2fd5508c8d79a6b2863117ac7eda9278424cc27803b63","leaf_index":3436}