Resonances in Quantum Scattering Theory and Electromagnetics

It is shown that the resonant states (RSs) in the quantum scattering theory and electromagnetics have a common nature: in one-dimensional setting, they are associated with complex eigenvalues of the nonselfadjoint Sturm–Liouville problems on the line that give rise to singularities of the analytical continuation to the complex domain of the solution, scattering matrices, and finally the scattered fields. For a homogeneous dielectric slab situated in free space, in a rectangular waveguide with perfectly conducting (PEC) walls, or between two parallel PEC planes, the dispersion equations (DEs) for eigenvalues coincide with those obtained in the quantum scattering problems involving rectangular barriers or single or multiple rectangular quantum wells. The existence of infinite sets of complex DE zeros is proved which explains the known oscillatory behaviour of the transmission coefficients and provides fundamental knowledge about the background of resonance phenomena.

I. Introduction

The occurence of resonances studied in quantum physics, electromagnetics, acoustics, and other fields of science and technology manifests itself as oscillations (local extrema) of various characteristics of the scattered fields. Note e.g. an indicative behaviour of the radar cross-sections considered with respect to frequency or other parameter of the irradiated obstacle [1], [2]. Both the nature and mathematical background of the phenomenon and their explanations may be different depending on the area of research or application. In view of this, there are several notions connected with the analysis of resonances. Among them, the particular quantities commonly referred to as resonant states (RSs) have been reported in quantum mechanics since the early studies [3], [4].