{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"5a96e03fc467f433466859c95715d79e48c8074737294b7d68f1ff62e1a809c6","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["4488e6f955ebf7c16bc621b98839387488a20933b532c363830bee662d9a397b","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"438fe5d8651ee4635b7136fb08d9e6742e52bd05078d4eaa65afdc571957de15","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.lieb-thirring-constants.one-dim-intermediate","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"In one dimension the conjecture says the worst-case potential well, the one with the most binding for its size, has only a single bound state when the exponent $\\gamma$ is between $1/2$ and $3/2$. Prove it.","posed_since":"1976","precise":"For $-\\mathrm{d}^2/\\mathrm{d}x^2 + V$ on $L^2(R)$, with $L_{\\gamma,1}$ the sharp constant in $\\sum_j \\left|E_j\\right|^\\gamma \\le L_{\\gamma,1} \\int V_-^{(\\gamma + 1/2)} dx$, prove $L_{\\gamma,1} = L^1_{\\gamma,1}$ for $1/2 < \\gamma < 3/2$, where $L^1_{\\gamma,1}$ is the sharp constant over potentials with exactly one negative eigenvalue. Known endpoints: equality at $\\gamma = 1/2$ (Hundertmark, Lieb, Thomas 1998) and $L_{\\gamma,1} = L^{\\mathrm{cl}}_{\\gamma,1}$ for $\\gamma \\ge 3/2$.","problem_ref":"4488e6f955ebf7c16bc621b98839387488a20933b532c363830bee662d9a397b","references":"","settled_by":"A proof that multi-bound-state potentials never beat the single-bound-state value for $1/2 < \\gamma < 3/2$, or a numerical-plus-rigorous counterexample.","status_note":"The 2021 Frank-Gontier-Lewin disproof applies in $d = 1$ only for $\\gamma > 3/2$, so the conjecture remains consistent with current results; at $\\gamma = 1$ the best bound is $L_{1,1}/L^{\\mathrm{cl}}_{1,1} \\le 1.456$ against the conjectured $2/\\sqrt{3}$, about 1.155 (Frank, Hundertmark, Jex, Nam 2021).","title":"One-bound-state optimality in one dimension for $1/2 < \\gamma < 3/2$","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"e8a936689185e2a498c500728465d7d6e220c459635079634ee3c872eda23a56","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"687a81c62815adcfe9092cbb14a83c7a7d307e4ad32e67a0f77a169db64308d0","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"B1MkNUGRatSAHOXhUgOMqCF-QBwRcpLkfOpI7WJ1uF7iBIsDGGzOZhGhllWzxLoZSETiE2WmC0upleDxehv5DQ"},"schema":"pubphys.envelope/1"},"record_hash":"e8a936689185e2a498c500728465d7d6e220c459635079634ee3c872eda23a56","leaf_index":3442}