MATHPH In the literature: open

Exponent at which the semiclassical constant becomes optimal

In plain words

For large enough exponent $\gamma$ the best constant equals the simple semiclassical value; for small $\gamma$ it is larger. Find the crossover exponent in two and three dimensions.

Precise statement

Define gamma_c(d) = inf{gamma : L_{gamma,d} = L^cl_{gamma,d}}, where $L^{\mathrm{cl}}_{\gamma,d} = \Gamma(\gamma+1)/((4 \pi)^{d/2} \Gamma(\gamma+1+d/2))$; $L_{\gamma,d}/L^{\mathrm{cl}}_{\gamma,d}$ is nonincreasing in $\gamma$ (Aizenman, Lieb 1978). Known: $1 \le \gamma_c(d) \le 3/2$ for all d (lower bound Helffer, Robert 1990; upper bound Laptev, Weidl, Acta Math 2000), $\gamma_c(1) = 3/2$, and $\gamma_c(2)$ >= about 1.165, the exponent at which the one-bound-state and semiclassical constants cross. Determine $\gamma_c(2)$ and $\gamma_c(3)$; the 1976 conjecture $\gamma_c(3) = 1$ follows from a proof of mathph.lieb-thirring-constants.kinetic-3d.

What would settle it

Proofs of $L_{\gamma,d} = L^{\mathrm{cl}}_{\gamma,d}$ for $\gamma \ge \gamma_{*}$ together with explicit potentials beating $L^{\mathrm{cl}}$ for $\gamma < \gamma_{*}$, in $d = 2$ and $d = 3$.

Status in the literature

Frank, Gontier, Lewin (CMP 2021) showed $L_{\gamma,d}$ exceeds the one-bound-state value for $\gamma > \max(0, 2 - d/2)$, so in $d = 2$ for $1 < \gamma < \text{about } 1.165$ the sharp constant is neither the one-bound-state nor the semiclassical value.