Optimal constants and optimizers for the fractional Lieb–Thirring inequality

Determine the exact values of the optimal constants in the fractional Lieb–Thirring inequalities and establish whether the corresponding optimal constants admit optimizers, beyond the finite-rank setting treated in the paper.

Background

The paper studies the fractional Lieb–Thirring inequality for the fractional Schrödinger operator $(-\Delta)^s-V$ and proves existence of optimizers for every finite-rank constant $L^{(N)}_{\gamma,d,s}$. It also proves that, in the specified parameter range, an optimizer for the full optimal constant $L_{\gamma,d,s}$ cannot have only finitely many negative eigenvalues.

The exact values of the full optimal constants and the existence of optimizers for the unrestricted fractional Lieb–Thirring inequality are identified by the authors as broader unresolved questions. The finite-rank results provide partial information but do not determine these full constants or establish their attainment.