SUSY transformations for quasinormal modes of open systems

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 ω. 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?