Exact solutions of non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions

We study one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore to analytically solve the systems under some specific boundary conditions. Although introduction of terms prevents us from finding analytical solutions for arbitrary parameters, we identify existence exact when parameters fulfill constraint relations, which give Our results show that wave functions take simple forms are independent range, while eigenvalue spectra display rich model-dependent structures. Particularly, find a special point coined as pseudo-periodic condition, eigenvalues same periodical system certain conditions, whereas eigenstates skin effect.