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Title: Semiclassical constant for the eigenvalue sum in three dimensions
Plain: Prove that in three dimensions the total binding energy of all bound states is never larger than the simple semiclassical estimate. Semiclassical means counting states as if each occupies a fixed volume of position-momentum space.
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 |grad phi_j|^2 >= (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.
Settled by: A proof of the inequality with constant $1/(15 \pi^{2})$, or an explicit potential exceeding it.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
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.
Posed since: 1976