高维离散Hardy–Rellich常数的三项渐近行为
Three-term asymptotics for discrete Hardy--Rellich constants in high dimension
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中文总结 AI 辅助
该研究确定了高维离散Hardy–Rellich常数的三项渐近展开式，给出显式系数，解决了随维度增长的退化性问题，并验证了$\ell=1$时的具体结果。
中文摘要 AI 辅助
精确Hardy–Rellich常数是具有临界反幂势算子的谱阈值，记$C_\ell(N)$为$\Z^N$上$\ell$阶离散Hardy–Rellich不等式的最优常数，近期独立工作确定了对任意固定$\ell$的高维主导行为$C_\ell(N)\sim2^\ell N^\ell$。本文确定了接下来的两个阶次，证明$C_\ell(N)=2^\ell N^\ell+\gamma_\ell N^{\ell-1} +\delta_\ell N^{\ell-2}+O_\ell(N^{\ell-3})$，其中$\gamma_\ell$和$\delta_\ell$为显式系数。核心难点是随维度增长的退化性：共轭后主导算子在原点的$2N$个最近邻上是标量，本文利用符号置换对称性和四个格轨道上的有效算子解决该问题，通过加权环面估计和Feshbach–Schur约化验证全算子内的有限维展开，剩余界和Temple不等式给出所述余项，特别地$C_1(N)=2N-4-20/(3N)+O(N^{-2})$。
英文摘要
Sharp Hardy--Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let $C_\ell(N)$ denote the optimal constant in the $\ell$th-order discrete Hardy--Rellich inequality on $\Z^N$. Recent independent work established the leading high-dimensional behavior $C_\ell(N)\sim2^\ell N^\ell$ for every fixed $\ell$. We determine the next two orders and prove \[ C_\ell(N)=2^\ell N^\ell+γ_\ell N^{\ell-1} +δ_\ell N^{\ell-2}+O_\ell(N^{\ell-3}), \] with explicit coefficients $γ_\ell$ and $δ_\ell$. 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, $C_1(N)=2N-4-20/(3N)+O(N^{-2})$.
