Fig. 5: Structure of the eigenspace and its fingerprint on the linear response.
From: Linear response theory of open systems with exceptional points
a The underlying vector space associated with the non-Hermitian Hamiltonian \(\hat{{{{{{{{\mathcal{H}}}}}}}}}\) can be divided into two subspaces: \({{{{{{{{\mathcal{D}}}}}}}}}_{\phi }=\{\left|{\psi }_{n}^{\,r}\right\rangle :n=1,2,\ldots ,N-M\}\) and \({{{{{{{{\mathcal{D}}}}}}}}}_{{{{{{\rm{EP}}}}}}}=\{\left|{J}_{m}^{\,r}\right\rangle :m=1,2,\ldots ,M\}\). The former is spanned by the non-degenerate right eigenvectors of \(\hat{{{{{{{{\mathcal{H}}}}}}}}}\), while the latter is spanned by the right generalized eigenvectors. Note that in this classification, the degenerate (or exceptional) vector belongs to \({{{{{{{{\mathcal{D}}}}}}}}}_{{{{{{\rm{EP}}}}}}}\). The dual spaces \({{{{{{{{\mathcal{D}}}}}}}}}_{\phi }^{\,d}\) and \({{{{{{{{\mathcal{D}}}}}}}}}_{{{{{{\rm{EP}}}}}}}^{\,d}\) are defined in a similar fashion for the left ordinary and generalized eigenvectors. b Pictorial representation of the linear response associated with a non-Hermitian system having an EP. Typical Lorentzian response arises due to coupling between an input and an output channel that belong to the same modal class. On the other hand, super-Lorentzian responses emerge when the input signal matches a particular generalized eigenmodes of certain order while the output signal matches a lower order generalized eigenvector (including the exceptional vector) according to Eq. (6). The arrows, together with the legends, illustrate a few possible responses explicitly. The symbol Lm indicates a super-Lorentzian response of order m, i.e., a Lorentzian raised to the power m.
