Uniform real-frequency stability of outgoing finite-cluster approximations

Remove or verify, under natural geometric hypotheses, the uniform real-frequency stability assumption for outgoing finite-cluster Helmholtz problems used to approximate the frequency-dependent capacitance operator in free space, and establish a direct real-frequency convergence theorem under minimal assumptions.

Background

The paper proves convergence of outgoing finite-cluster approximations under a uniform nonresonance assumption that bounds both the outgoing Dirichlet-to-Neumann maps and the inverses of the associated finite-cluster variational operators uniformly in the cluster size.

The authors note that this stability assumption does not follow from the infinite-volume exterior spectral gap and may fail when growing clusters develop resonances near the physical reference frequency. Complex-frequency absorption and Hermitian symmetrization provide an unconditional computational alternative, but the corresponding direct real-frequency theory remains unresolved.

References

Several challenging questions remain open. First, when the reference exterior wavenumber squared belongs to the spectrum of minus the exterior Dirichlet Laplacian, the exterior DtN map is singular, and the leading splitting is expected to be of order $\delta{1/2}$. It would be interesting to extend the results of in this case to the waveguiding problem with a line defect. Second, another remaining analytical question is to remove or verify, under natural geometric hypotheses, the uniform real-frequency stability assumption eq:outgoing-patch-stability used for the outgoing finite-cluster approximation in free space. The complex-frequency stabilization developed in this paper provides an unconditional computational alternative, but a direct real-frequency convergence theorem under minimal assumptions would further connect rigorous bounded-patch construction and practical radiation computations.

— Waveguiding in systems of high contrast resonators: Theory and fast computations  (2608.26906 - Ammari et al., 27 Aug 2026) in Section 5, Concluding remarks