{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"547ca03d6a848c2ea7d47eeaebf2934cbcbd8076a3aef9350c4cd7805fd42c17","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"7f794ebd04d379f77a343cf90b21d91be22351c33d880311c384deb568b1096e","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":"1","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 $N(V) \\le L_{0,3} \\operatorname{int} V_-^{(3/2)} dx$, $N(V)$ = 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":null,"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":"9b795df8bba67037212e33cc80fcb1b7804f1dc5a26e182277387971a731f808"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"bc0a69bb3f8c97ce6dc407a8acf8401e12e745e0e22fbe4189437cd685d3e0eb","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"0ac6aac98b99cdf4b9242c069fbb980fc8839a81e3e8e11141f35642313905b1","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"c8jvnD5L-NV6I1UIEcr72o51-t9zCDaKhGC_jpQut9GYJCE2KVeD27NJP_2-U-Y4ossEtmY2QgK9sCpJmCAdBQ"},"schema":"pubphys.envelope/1"},"record_hash":"bc0a69bb3f8c97ce6dc407a8acf8401e12e745e0e22fbe4189437cd685d3e0eb","leaf_index":1640}