Critical Optimizers for the Finite-Rank Fractional Lieb--Thirring Inequality
Abstract
We study optimal potentials for the finite-rank fractional Lieb--Thirring problem at the eigenvalue-counting endpoint, for 0<s<1 and d>2s. Using the zero-energy Birman--Schwinger formulation, we prove a profile decomposition of normalized optimizing sequences with exact splitting of rank and potential mass, and strong approximation in Ld/(2s). Strict subadditivity yields existence and compactness modulo translations and dilations. At rank two, the optimal constant equals the rank-one constant, the supremum is not attained, and every normalized optimizing sequence splits into two Sobolev potentials. We also prove strict increase of the minimal potential masses and a uniform spectral gap for optimizing sequences. For each optimizer, we derive an Euler--Lagrange system and establish boundedness, local Hölder regularity and V(x)=|x|−4s(c∞+o(1)) as |x|→∞, with c∞≥0. The conformal orbit of every optimizer contains a representative with a strictly positive asymptotic coefficient.
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BibTeX
@misc{li2026critical,
title = {{Critical Optimizers for the Finite-Rank Fractional Lieb--Thirring Inequality}},
author = {Zhuocheng Li and Xiaoyu Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41148},
eprint = {2610.41148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41148},
}
RIS
TY - UNPB
TI - Critical Optimizers for the Finite-Rank Fractional Lieb--Thirring Inequality
AU - Zhuocheng Li
AU - Xiaoyu Zeng
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41148
UR - https://arxiv.org/abs/2610.41148
AB - We study optimal potentials for the finite-rank fractional Lieb--Thirring problem at the eigenvalue-counting endpoint, for 0<s<1 and d>2s. Using the zero-energy Birman--Schwinger formulation, we prove a profile decomposition of normalized optimizing sequences with exact splitting of rank and potential mass, and strong approximation in L^d/(2s). Strict subadditivity yields existence and compactness modulo translations and dilations. At rank two, the optimal constant equals the rank-one constant, the supremum is not attained, and every normalized optimizing sequence splits into two Sobolev potentials. We also prove strict increase of the minimal potential masses and a uniform spectral gap for optimizing sequences. For each optimizer, we derive an Euler--Lagrange system and establish boundedness, local Hölder regularity and V(x)=|x|^−4s(c_∞+o(1)) as |x|→∞, with c_∞≥0. The conformal orbit of every optimizer contains a representative with a strictly positive asymptotic coefficient.
ER -