AI 中文总结
该研究确定三维、四维离散Hardy常数的精确值，证明九维、十维时该常数小于连续介质系数，得到适用于$N\ge3$的显式上界，推导三维、四维的正空间余项及高维离散Schrödinger算子的谱结论。
AI 中文摘要
对于$N\ge3$，设$C(N)$为$\mathbb Z^N$上满足$u(0)=0$的最近邻Hardy不等式的最优常数，记$A_N=(N-2)^2/4$。我们证明连续介质系数在所有维度上仍为上界，即$C(N)\le A_N$，并确定精确值$C(3)=A_3=1/4$、$C(4)=A_4=1$。在更高维度中，我们证明$N=9、10$时$C(N)<A_N$，并得到显式上界$C(N)\le 3N-\sqrt{N^2+8N-8}<2N\\ (N\ge3)$，该上界从十一维起小于$A_N$。低维等式由平移径向超解、角凸性及离散基态表示推导得到。我们还证明该幂次平移机制从五维起无法适用于连续介质系数。九维问题通过高斯Rayleigh-Ritz构造结合精确Jacobi theta函数估计处理，更高维界则通过有限维轨道压缩得到。最后，我们推导三维和四维中的正空间余项，以及高维区域中相关离散Schrödinger算子的谱结果。
英文摘要
For $N\ge3$, let $C(N)$ be the optimal constant in the nearest-neighbour Hardy inequality on $\mathbb Z^N$ with $u(0)=0$, and set $A_N=(N-2)^2/4$. We prove that the continuum coefficient remains an upper bound, $C(N)\le A_N$, in every dimension, and determine the exact values $C(3)=A_3=1/4$ and $C(4)=A_4=1$. In higher dimensions we show $C(N)<A_N$ for $N=9,10$ and obtain the explicit bound \[ C(N)\le 3N-\sqrt{N^2+8N-8}<2N\qquad(N\ge3), \] which lies below $A_N$ from dimension eleven onward. The low-dimensional equalities follow from shifted radial supersolutions, angular convexity, and a discrete ground-state representation. We also show that this shifted-power mechanism cannot work at the continuum coefficient from dimension five onward. Dimension nine is treated by a Gaussian Rayleigh--Ritz construction combined with exact Jacobi theta-function estimates, while the higher-dimensional bounds are obtained through finite-dimensional orbit compressions. Finally, we derive positive spatial remainders in dimensions three and four and a spectral consequence for the associated discrete Schrödinger operators in the high-dimensional regime.
Comments34 pages. Submitted for publication
