Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion

Abstract

The concept of eigenfunction expansions for the wave equation is generalized to open systems, in which waves escape to the outside. These non-conservative systems are non-Hermitian in the usual sense. It is shown that the natural framework is an eigenfunction expansion within a two-component formalism that treats the wavefunction and its conjugate momentum together. Provided the system approaches spatial infinity rapidly `without tails', and possesses spatial discontinuities, the expansion in terms of the eigenfunctions (which are now quasinormal modes) is shown to be valid.