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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 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