Exceptional Points and Global Resonant Branches of Complex Robin--Helmholtz Balls

Abstract: We study the complex resonance wavenumbers of impedance-loaded balls governed by the Helmholtz equation with complex Robin boundary conditions. Separation of variables reduces the non-self-adjoint problem to an explicit Bessel determinant, which is treated throughout as a single global geometric object --- its zeros are the full resonance spectrum, their coalescence points are the exceptional points, and their global gluing is the branch structure --- rather than as a certificate from which spectral bounds are compressed. Read through the Riccati variable $zJ_\mu'/J_\mu$, the classical Lommel first integral locates all exceptional points in advance, proves that they are simple square-root branch points, and rules out higher-order degeneracies. In the physical dimensions $d=2,3$, no passive exceptional point exists in any angular sector: the degeneracies and their $\sqrt{\delta}$ tolerance law are confined to the gain half-plane. On the negative real axis the lowest branch becomes the purely imaginary ground state of the attractive Robin problem inside a non-oscillatory band. The same explicit spectral relation also yields sharp inverse-spectral consequences: closed-form single-mode placement by one Bessel-quotient evaluation, exact passive feasibility bands separated by gain-requiring gaps, a computable stiffness margin, and an effectively two-parameter quality-factor budget. The analysis is rigorous where analytic and certified or explicitly flagged where validated numerics are used, and it provides a design framework for robust passive impedance cavities while identifying gain-loaded exceptional points as a controllable resource.