Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity

The scalar analog of electromagnetism in one dimension is discussed with reference to a cavity formed by a distribution of dielectric material with dielectric constant ε̃(x,ω). Provided that for large ω, ε̃(x,ω) or its spatial derivative of any order has a discontinuity in x at the edge of the cavity (e.g., the interface between two materials), the discrete quasinormal modes (QNM’s) of the system are shown to form a complete set inside the cavity. In terms of these, time-independent perturbation of the dielectric constant can be formulated. Both the completeness relation and the perturbative series are verified explicitly, for example, in the context of which the QNM’s are also analyzed and shown to belong to two classes: those that are intrinsically absorptive (which decouple when the absorption is switched off) and those that are intrinsically leaky in character.