Explicit Robin Green's Functions and Resonance Spectra for the Helmholtz Equation on Balls in All Dimensions
Abstract
This work constructs explicit closed-form Green's functions for the Helmholtz equation with Robin boundary conditions on balls in all dimensions d≥ 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.
Related
- Dispersion-corrected compact finite difference discretizations for the elastic Helmholtz equation in two and three dimensions
- Explicit Robin Green's Functions, Resonance Spectra, and Impedance Recovery on Annuli and Spherical Shells
- Wavenumber-explicit stability of three-dimensional Cartesian UPMLs with piecewise-constant profiles for Helmholtz and Maxwell equations
- On solutions of the modified Helmholtz equation in a planar domain
- Cartesian PML truncation for the Helmholtz equation is exponentially accurate at high frequency
BibTeX
@misc{yang2026explicit,
title = {{Explicit Robin Green's Functions and Resonance Spectra for the Helmholtz Equation on Balls in All Dimensions}},
author = {Ming Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37701},
eprint = {2609.37701},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37701},
}
RIS
TY - UNPB
TI - Explicit Robin Green's Functions and Resonance Spectra for the Helmholtz Equation on Balls in All Dimensions
AU - Ming Yang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37701
UR - https://arxiv.org/abs/2609.37701
AB - This work constructs explicit closed-form Green's functions for the Helmholtz equation with Robin boundary conditions on balls in all dimensions d≥ 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.
ER -