Explicit Robin Green's Functions and Resonance Spectra for the Helmholtz Equation on Balls in All Dimensions

Abstract: This work constructs explicit Green's functions for the Helmholtz equation with Robin boundary conditions on balls in all dimensions $d\ge 2$, in the form of modal series whose coefficients are given in closed form and are computable to machine precision. Despite the canonical geometry, computable representations of these kernels have not been available in general dimension: the classical Mie theory treats the \emph{scattering} of an incident field by an impedance sphere, not the impulse response of the cavity, and the lack of such kernels is reflected in the extrapolated-boundary approximation still standard in biomedical optics and duct acoustics. The obstruction is structural: the Robin condition couples field and normal derivative, preventing image constructions, while eigenfunction expansions lead to transcendental spectral problems whose implicit roots preclude computable closed forms. By a decomposition--expansion method, Graf's and the hyperspherical addition theorems expand the free-space fundamental solution, and the regular correction is 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 kernels yield a complete characterisation of the resonance spectrum: each spectral branch increases strictly with the impedance, 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 both Dirichlet and Neumann extremes. These explicit kernels furnish exact benchmarks for wave simulations in impedance cavities, eliminating geometric discretisation error and providing reference data for validating finite-element and boundary-element algorithms in acoustics, electromagnetism, and quantum mechanics.