Semiclassical constant for the eigenvalue sum in three dimensions · revision history

Revision 2

Accepted · core change · record

Changes against revision 1:

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 · changed · core
Revision 1: 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.
Revision 2: 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.
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

Decisions