Two-component eigenfunction expansion for open systems described by the wave equation II: linear space structure

For a broad class of open systems described by the wave equation, the eigenfunctions (which are quasinormal modes) provide a complete basis for simultaneously expanding outgoing wavefunctions . In this paper, the linear space structure associated with this expansion is developed. Under a modified inner product, the time-evolution operator is self-adjoint, even though energy is not conserved for the system alone. Thus, the eigenfunctions are mutually orthogonal. Consequently, the usual tools of eigenfunction expansions can be transcribed to these open systems. As an example, the time-independent perturbation theory is developed in straightforward analogy with quantum mechanics, giving the shift in both the real part and the imaginary part of the eigenvalues .