{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"c87cbdb04e162ac3fa04569ba6047d83cf6b998fcc6eca3fc487371f00019f36","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"5a22024856d2acea1b267b80cb8ed83bcbfea6b8383da89ec4d18046822d41f6","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":"1","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 $\\operatorname{sum}_j \\left|E_j\\right|^\\gamma \\le L_{\\gamma,1} \\operatorname{int} V_-^{(\\gamma + 1/2)} dx$, 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":null,"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 < 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