Three-term asymptotics for discrete Hardy--Rellich constants in high dimension
Abstract
Sharp Hardy--Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let Cℓ(N) denote the optimal constant in the ℓth-order discrete Hardy--Rellich inequality on ZN. Recent independent work established the leading high-dimensional behavior Cℓ(N)∼2ℓ Nℓ for every fixed ℓ. We determine the next two orders and prove Cℓ(N)=2ℓ Nℓ+γℓ Nℓ−1 +δℓ Nℓ−2+Oℓ(Nℓ−3), with explicit coefficients γℓ and δℓ. The central difficulty is a degeneracy that grows with the dimension: after conjugation, the leading operator is scalar on the 2N nearest neighbours of the origin. We resolve this cluster using signed-permutation symmetry and an effective operator on four lattice orbits. Weighted torus estimates and Feshbach--Schur reduction justify the finite-dimensional expansion inside the full operator, while a residual bound and Temple's inequality give the stated remainder. In particular, C1(N)=2N−4−20/(3N)+O(N−2).
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BibTeX
@misc{lizama2026threeterm,
title = {{Three-term asymptotics for discrete Hardy--Rellich constants in high dimension}},
author = {Carlos Lizama},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30888},
eprint = {2608.30888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30888},
}
RIS
TY - UNPB
TI - Three-term asymptotics for discrete Hardy--Rellich constants in high dimension
AU - Carlos Lizama
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30888
UR - https://arxiv.org/abs/2608.30888
AB - Sharp Hardy--Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let C_ℓ(N) denote the optimal constant in the ℓth-order discrete Hardy--Rellich inequality on Z^N. Recent independent work established the leading high-dimensional behavior C_ℓ(N)∼2^ℓ N^ℓ for every fixed ℓ. We determine the next two orders and prove C_ℓ(N)=2^ℓ N^ℓ+γ_ℓ N^ℓ−1 +δ_ℓ N^ℓ−2+O_ℓ(N^ℓ−3), with explicit coefficients γ_ℓ and δ_ℓ. The central difficulty is a degeneracy that grows with the dimension: after conjugation, the leading operator is scalar on the 2N nearest neighbours of the origin. We resolve this cluster using signed-permutation symmetry and an effective operator on four lattice orbits. Weighted torus estimates and Feshbach--Schur reduction justify the finite-dimensional expansion inside the full operator, while a residual bound and Temple's inequality give the stated remainder. In particular, C_1(N)=2N−4−20/(3N)+O(N^−2).
ER -