Transfer matrices as nonunitary S matrices, multimode unidirectional invisibility, and perturbative inverse scattering

We show that in one dimension the transfer matrix M of any scattering potential v coincides with the S matrix of an associated time-dependent non-Hermitian 2 x 2 matrix Hamiltonian H(tau). If v is real valued, H(tau) is pseudo-Hermitian and its exceptional points correspond to the classical turning