SUSY Transformations for Quasinormal and Total-Transmission Modes of Open Systems

TL;DR

This paper explores supersymmetric transformations in open systems' quasinormal modes, revealing how these transformations can generate isospectral partners, affect mode spectra, and impact the inverse spectral problem.

Contribution

It introduces SUSY transformations for quasinormal and total-transmission modes, illustrating their effects on spectra and mode properties in open systems.

Findings

Multiple isospectral SUSY partners can exist, each missing a different eigenstate.

SUSY transformations preserve a bilinear orthonormality, analogous to inner products.

Doubled quasinormal and total-transmission modes can result from SUSY transformations.

Abstract

Quasinormal modes are the counterparts in open systems of normal modes in conservative systems; defined by outgoing-wave boundary conditions, they have complex eigenvalues. The conditions are studied for a system to have a supersymmetric(SUSY) partner with the same complex quasinormal-mode spectrum (or the same except for one eigenvalue). The discussion naturally includes total-transmission modes as well(incoming at one extreme and outgoing at the other). Several types of SUSY transformations emerge, and each is illustrated with examples, including the transformation among different Poschl-Teller potentials and the well-known identity in spectrum between the two parity sectors of linearized gravitational waves propagating on a Schwarzschild background. In contrast to the case of normal modes, there may be multiple essentially isospectral partners, each missing a different state. The SUSY transformation preserves orthonormality under a bilinear map which is the analog of the usual inner product for conservative systems. SUSY transformations can lead to doubled quasinormal and total-transmission modes; this phenomenon is analysed and illustrated. The existence or otherwise of SUSY partners is also relevant to the question of inversion: are open wave systems uniquely determined by their complex spectra?