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 (−Δ)s−V in terms of the Lγ+d/2s(ℝd) norm of the potential V, where the Riesz exponent γ>max{0,1−d/(2s)}. 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 γ+d/(2s)≥ 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.
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BibTeX
@misc{chen2026optimizers,
title = {{Optimizers for the finite-rank fractional Lieb--Thirring inequality}},
author = {Lu Chen and Lianfeng Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21716},
eprint = {2610.21716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21716},
}
RIS
TY - UNPB
TI - Optimizers for the finite-rank fractional Lieb--Thirring inequality
AU - Lu Chen
AU - Lianfeng Yang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21716
UR - https://arxiv.org/abs/2610.21716
AB - 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 (−Δ)^s−V in terms of the L^γ+d/2s(ℝ^d) norm of the potential V, where the Riesz exponent γ>max{0,1−d/(2s)}. 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 γ+d/(2s)≥ 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.
ER -