The Lieb–Thirring constant phase-alternative conjecture

Let $d\geq1$ and let $\gamma$ satisfy the constraint defining the admissible Lieb–Thirring regime. Let $L_{\gamma,d}$ be the optimal Lieb–Thirring constant, let $L_{\gamma,d}^{(N)}$ be the best constant when the first $N$ negative eigenvalues are retained, and let $V$ be a potential on $\mathbb{R}^d$. Value of the Lieb–Thirring constant. For all $d\geq1$ and admissible $\gamma$, either there exist $N\in\mathbb{N}$ and a potential $V\in L^{\gamma+d/2}(\mathbb{R}^d)$ with exactly $N$ negative eigenvalues that optimize the nonperiodic Lieb–Thirring inequality, so that

$$L_{\gamma,d}=L_{\gamma,d}^{(N)},$$

or

$$L_{\gamma,d}^{(N)}<L_{\gamma,d}\qquad\text{for all }N\geq1,$$

and there exists an optimal periodic potential $V\in L^{\gamma+d/2}_{\rm loc}(\mathbb{R}^d)$ optimizing the periodic Lieb–Thirring inequality. This potential may be constant, in which case $L_{\gamma,d}=L_{\gamma,d}^{\rm sc}$, or nonconstant. The conjecture proposes a dichotomy between a finite bound-state optimizer and a periodic optimizer. Results preceding this formulation rule out finite-bound-state optimality in substantial parameter ranges, but do not settle the proposed alternative in general.