Eigenvalues and eigenfunctions of the perturbed non-Hermitian SSH Hamiltonian with PT symmetry

In the paper we find near-zero eigenvalues (their physical meaning is the electron energy) and eigenfunctions describing the electronic states of the non-Hermitian SSH Hamiltonian for an infinite chain with PT symmetry, perturbed by a $\delta$-shaped potential. We prove that for a small non-Hermitian parameter there are two (generalized) eigenvalues of multiplicity one, and, in contrast to the Hermitian model, the corresponding (generalized) eigenfunctions depending on the parameters of the system can either increase exponentially (which corresponds to resonant, i.e., decaying states) or decrease exponentially (which corresponds to bound states) as $|n|\to\infty$.