Optimal constant for counting bound states in three dimensions
In plain words
The number of bound states of a 3D potential well is at most a constant times the integral of $\mid V \mid^{3/2}$. The best possible constant is unknown.
Precise statement
For $-\Delta + V$ on $L^2(R^3)$, the Cwikel-Lieb-Rozenblum bound reads $N(V) \le L_{0,3} \operatorname{int} V_-^{(3/2)} dx$, $N(V)$ = 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 $L^1_{0,3}$.
What would settle it
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 in the literature
Known upper bounds exceed the semiclassical value by a factor of several; the 2021 Frank-Gontier-Lewin results do not cover $\gamma = 0$ in $d = 3$.