Waves in Open Systems: Eigenfunction Expansions

TL;DR

This paper reviews recent advances in the mathematical treatment of quasinormal modes in open systems, highlighting their completeness and applications in optics and gravitational wave physics.

Contribution

It introduces a unifying mathematical framework for quasinormal modes, enabling exact descriptions of open system dynamics and extending tools from conservative systems.

Findings

QNM sets can be complete for outgoing wavefunctions

Mathematical tools from hermitian systems can be adapted

Applications include optics in microspheres and black hole gravitational waves

Abstract

An open system is not conservative because energy can escape to the outside. An open system by itself is thus not conservative. As a result, the time-evolution operator is not hermitian in the usual sense and the eigenfunctions (factorized solutions in space and time) are no longer normal modes but quasinormal modes (QNMs) whose frequencies $\omega$ are complex. QNM analysis has been a powerful tool for investigating open systems. Previous studies have been mostly system specific, and use a few QNMs to provide approximate descriptions. Here we review recent developments which aim at a unifying treatment. The formulation leads to a mathematical structure in close analogy to that in conservative, hermitian systems. Many of the mathematical tools for the latter can hence be transcribed. Emphasis is placed on those cases in which the QNMs form a complete set for outgoing wavefunctions, so that in principle all the QNMs together give an exact description of the dynamics. Applications to optics in microspheres and to gravitational waves from black holes are reviewed, and directions for further development are outlined.