Extending completeness of the eigenmodes of an open system beyond its boundary, for Green's function and scattering-matrix calculations

The asymptotic completeness of a set of the eigenmodes of an open system with increasing number of modes enables an accurate calculation of the system response in terms of these modes. Using the exact eigenmodes, such completeness is limited to the interior of the system. Here we show that when the eigenmodes of a target system are obtained by the resonant-state expansion, using the modes of a basis system embedding the target system, the completeness extends beyond the boundary of the target system. We illustrate this by using the Mittag-Leffler series of the Green's function expressed in terms of the eigenmodes, which converges to the correct solution anywhere within the basis system, including the space outside the target system. Importantly, this property allows one to treat perturbations outside the target system and to calculate the scattering cross-section using the boundary conditions for the basis system. Choosing a basis system of spherical geometry, these boundary conditions have simple analytical expressions, allowing for an efficient calculation of the response of the target system, as we demonstrate for a resonator in a form of a finite dielectric cylinder.