SUSY Transformations for Quasinormal Modes of Open Systems

TL;DR

This paper extends supersymmetry (SUSY) concepts to open quantum systems with quasinormal modes, revealing systematic eigenstate counting, preservation of wave structure, and implications for system inversion.

Contribution

It generalizes SUSY to include outgoing-wave boundary conditions, enabling systematic eigenstate analysis and insights into system uniqueness based on spectra.

Findings

SUSY can be applied to quasinormal modes with complex eigenvalues

The wave structure of outgoing states is preserved under SUSY transformations

Multiple states at the same frequency are maintained in SUSY frameworks

Abstract

Supersymmetry (SUSY) in quantum mechanics is extended from square-integrable states to those satisfying the outgoing-wave boundary condition, in a Klein-Gordon formulation. This boundary condition allows both the usual normal modes and quasinormal modes with complex eigenvalues. The simple generalization leads to three features: the counting of eigenstates under SUSY becomes more systematic; the linear-space structure of outgoing waves (nontrivially different from the usual Hilbert space of square-integrable states) is preserved by SUSY; and multiple states at the same frequency (not allowed for normal modes) are also preserved. The existence or otherwise of SUSY partners is furthermore relevant to the question of inversion: are open systems uniquely determined by their complex outgoing-wave spectra?