From Robin to Neumann: Green's Functions, Spectrum, and Error Estimates for the Helmholtz Equation

Abstract: We study the singular limit $\alpha\to0^+$ from Robin to Neumann boundary conditions for the Helmholtz equation, on balls in all dimensions $d\ge2$ and, at the structural level, on arbitrary bounded Lipschitz domains. Away from the Neumann spectrum the Green kernels converge linearly in the energy norm, with certified \emph{a priori} and \emph{a posteriori} error bounds whose constants on the ball are computable to any prescribed accuracy; the \emph{a posteriori} bound uses only the computed Robin boundary trace. Near resonance we prove a sharp spectrally weighted two-sided estimate and a global error law $\|G_\alpha-G_N\|_{H^1}\asymp\frac{\alpha}{\delta+\alpha}|G_N|$ in the spectral distance $\delta$, together with a matching two-sided operator-norm degeneration $\alpha/\delta^2$ on general domains, obtained through a Kre\u{\i}n-type identity that also yields an \emph{a posteriori} operator bound. At a Neumann eigenvalue the resolvent develops a simple pole whose residue, the eigenvalue-branch slopes and the degeneration rates are three faces of one object, the boundary-mass operator; the static zero mode is regularized by a rank-one subtraction and exhibits the square-root branch-point behavior $k_{0,1}(\alpha)\sim\sqrt{d\alpha/R}$. Numerical validation with fully computable constants confirms the practical tightness of the certified bounds.