MATHPH In the literature: open

Semiclassical constant for the eigenvalue sum in three dimensions

In plain words

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 statement

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.

What would settle it

A proof of the inequality with constant $1/(15 \pi^{2})$, or an explicit potential exceeding it.

Status in the literature

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.