Optimizers for the finite-rank fractional Lieb--Thirring inequality

Abstract

The finite-rank fractional Lieb--Thirring inequality provides an estimate on the Riesz sum of the first $N$ negative eigenvalues of the fractional Schrödinger operator $(-\Delta {)}^s-V$ in terms of the $L^{\gamma +\frac{d}{2s}}({\mathbb{R}}^d)$ norm of the potential $V$, where the Riesz exponent $\gamma >\max \left\{0,1-d/(2s)\right\}$. We prove the existence of optimizers for every finite rank by establishing a result on the spectrum in the case of dichotomy. We further characterize the spectral gap and qualitative properties of optimizers, including the Euler--Lagrange equation, regularity, and asymptotic behavior at infinity. In particular, under the condition $\gamma +d/(2s)\ge 3+4s/d$, we show that all normalized maximizing sequences are precompact modulo translations. Moreover, we derive a strict binding inequality for the optimal constants in the finite-rank setting, which implies that the optimal fractional Lieb--Thirring constant, if it exists, cannot be attained by a potential with finitely many negative eigenvalues.

arXiv comment37 pages