Explicit Robin Green's Functions, Resonance Spectra, and Impedance Recovery on Annuli and Spherical Shells

Abstract: This paper constructs explicit modal Green's functions, with
closed-form modal coefficients, for the
Helmholtz equation on annular and spherical-shell domains with two
independent Robin impedances on the inner and outer boundaries. Graf's
and the hyperspherical addition theorems reduce each angular mode to an
explicit $2\times2$ linear system, and the resonance spectrum is governed
by a characteristic determinant bilinear in the two impedances. This
bilinearity has a direct inverse-problem consequence: two resonant
frequencies of one non-radial angular mode generate at most two candidate
impedance pairs via an explicit quadratic equation, a third resonance
selects the physical pair, and an exact reflection-symmetry obstruction
identifies where the radial spherical mode cannot recover the
impedances. Spectrally, we prove the branches positive, simple and
strictly increasing in both impedances; derive first-order asymptotics at
the four corners of the impedance plane, with coefficients given by
boundary masses and normal derivatives of the limiting eigenfunctions,
and an explicit mixed second-order coefficient at the Dirichlet--Dirichlet corner, which for the radial mode evaluates in closed form to $2\pi/(R_2-R_1)^3$; establish a low-frequency resonance-free band, with a
second-order threshold expansion explicit in dimension three and a
rational approximation accurate over the whole impedance range; and prove
the universal high-frequency spacing law with a shell-curvature
correction. Jacobian-based sensitivity and conditioning criteria are
included. The kernels and spectra are computable to machine precision,
all asymptotic regimes are confirmed numerically, and the kernels provide
reference solutions for finite-element and boundary-element validation.