发表机构
School of Mathematics, Southeast University（东南大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过分解-展开方法，为所有维度球上Robin边界条件的亥姆霍兹方程构造了显式闭式格林函数，完整刻画了共振谱，并提供了精确基准解以验证数值算法。
AI 中文摘要
本工作构造了在所有维度$d\ge 2$的球上具有Robin边界条件的亥姆霍兹方程的显式闭式格林函数。尽管几何形状是经典的，但由于Robin条件耦合了场及其法向导数，阻止了镜像法的应用并妨碍了标准特征函数展开，此类核函数此前一直无法获得。通过一种分解-展开方法，自由空间基本解在二维中利用Graf加法定理、在更高维度中利用超球面加法定理进行展开；然后通过逐模式匹配Robin边界数据，代数地确定正则修正项。所得级数在紧致内部子集上绝对收敛，并可计算至机器精度。这些显式核函数给出了共振谱的完整刻画：每个谱分支随阻抗参数严格递增，在Neumann和Dirichlet特征值之间插值。我们建立了具有显式截止估计的低频谱间隙，以及具有二阶修正的普适高频模式间距，该修正将Robin条件与Dirichlet和Neumann极端情况区分开来。闭式表达式为阻抗匹配腔中的波模拟提供了精确的基准解，消除了几何离散化误差，并为声学、电磁学和量子力学中有限元和边界元算法的验证提供了参考数据。
英文摘要
This work constructs explicit closed-form Green's functions for the Helmholtz equation with Robin boundary conditions on balls in all dimensions $d\ge 2$. Despite the canonical geometry, such kernels have remained unavailable because the Robin condition couples the field and its normal derivative, preventing the method of images and obstructing standard eigenfunction expansions. By a decomposition--expansion method, the free-space fundamental solution is expanded via Graf's addition theorem in two dimensions and the hyperspherical addition theorem in higher dimensions; the regular correction is then determined algebraically by matching Robin boundary data mode by mode. The resulting series converge absolutely on compact interior subsets and are computable to machine precision. These explicit kernels yield a complete characterisation of the resonance spectra: each spectral branch increases strictly with the impedance parameter, interpolating between Neumann and Dirichlet eigenvalues. We establish a low-frequency spectral gap with an explicit cutoff estimate, and a universal high-frequency mode spacing with a second-order correction that distinguishes the Robin condition from the Dirichlet and Neumann extremes. The closed-form expressions furnish exact benchmark solutions for wave simulations in impedance-matched cavities, eliminating geometric discretisation error and providing reference data for the validation of finite-element and boundary-element algorithms in acoustics, electromagnetism and quantum mechanics.
