AI 中文总结
该研究刻画了球面n-月牙形上测地弹子球的周期轨道，确定了特定张角月牙形的谱分布，证明了多类月牙形满足波利亚猜想并给出特征值渐近公式与显式界。
AI 中文摘要
我们刻画了$\n$维球面$\n\b{S}^{n}$上月牙形区域上测地弹子球的周期轨道。对于张角为正整数$p$对应的$\nπ/p$形式的情形，我们完全确定了其Dirichlet（狄利克雷）谱和Neumann（诺依曼）谱。随后我们证明，张角小于$π$且不是$π$的有理倍数的月牙形，或是张角为$p>1$对应的$π/p$形式的月牙形，最终都满足Pólya（波利亚）猜想，这与对应测地弹子球是否满足非周期条件无关。对于张角为$π/p$的月牙形，我们进一步基于精确上下界给出了特征值的双项渐近公式，以及借助测地弹子球方法建立的对应双项计数函数。最后，我们用维度给出了$p$的显式界，以确保对应月牙形的所有特征值都满足Pólya猜想。
英文摘要
We characterise the periodic orbits of geodesic billiards on spherical lunes on $\mathbb{S}^{n}$. In the case of angle openings of the form $π/p$ for positive integer $p$ we fully determine their Dirichlet and Neumann spectra. We then show that lunes with an angle opening smaller than $π$ which is not a rational multiple of $π$, or those with an angle opening of the form $π/p$ for $p$ larger than one satisfy Pólya's conjecture eventually, independently of whether the corresponding geodesic billiards satisfy the nonperiodicity condition or not. For lunes with an angle opening $π/p$ we further provide a two-term asymptotic formula for the eigenvalues based on sharp upper and lower bounds, together with a corresponding two-term counting function established using the geoesic billiards approach. Finally,we give an explicit bound on $p$ in terms of the dimension ensuring the corresponding lunes satisfy Pólya's conjecture for all eigenvalues.
Comments56 pages
