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- ItemResolvent expansion for discrete non-Hermitian resonant systems [Invited](Washington, DC : Optica, 2022) Simonson, L.; Özdemir, S.K.; Busch, K.; El-Ganainy, R.The linear response of non-Hermitian resonant systems demonstrates various intriguing features such as the emergence of non-Lorentzian lineshapes. Recently, we have developed a systematic theory to understand the scattering lineshapes in such systems and, in doing so, established the connection with the input/output scattering channels. Here, we follow up on that work by presenting a different, more transparent derivation of the resolvent operator associated with a non-Hermitian system under general conditions and highlight the connection with the structure of the underlying eigenspace decomposition. Finally, we also present a simple solution to the problem of self-orthogonality associated with the left and right Jordan canonical vectors and show how the left basis can be constructed in a systematic fashion. Our work provides a unifying mathematical framework for studying non-Hermitian systems such as those implemented using dielectric cavities, metamaterials, and plasmonic resonators.