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Semiclassical means counting states as if each occupies a fixed volume of position-momentum space.","posed_since":"1976","precise":"For $-\\Delta + V$ on $L^2(R^3)$ (units $\\hbar^2/(2m) = 1$) with negative eigenvalues $E_j$, prove or disprove $\\sum_j \\mid E_j \\mid \\le L^{\\mathrm{cl}}_{1,3} \\int V_-^{5/2}\\, dx$ with $L^{\\mathrm{cl}}_{1,3} = 1/(15 \\pi^2)$. Dual form: for orthonormal $\\phi_1..\\phi_N$ in $L^2(R^3)$, $\\sum_j \\int \\mid \\mathrm{grad}\\, \\phi_j \\mid^2 \\ge (3/5)(6 \\pi^2)^{2/3} \\int \\rho^{5/3}$, $\\rho = \\sum_j \\mid \\phi_j \\mid^2$. Answer: a proof, or a potential with ratio greater than 1.","problem_ref":"2ff4ec14e0a81f529f9b586e6c446dc012ced89d0408e7f04050ea1468dba1da","references":"","settled_by":"A proof of the inequality with constant $1/(15 \\pi^{2})$, or an explicit potential exceeding it.","status_note":"Best bound $L_{1,3}/L^{\\mathrm{cl}}_{1,3} \\le 1.456$ (Frank, Hundertmark, Jex, Nam, JEMS 2021, valid in every dimension); Frank, Gontier, Lewin (CMP 2021) proved that for $\\gamma \\ge 1$ the optimum is not attained by any potential with finitely many eigenvalues.","title":"Semiclassical constant for the eigenvalue sum in three 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