Dynamical theory of scattering, exact unidirectional invisibility, and truncated z e(-2ik0x) potential

The dynamical formulation of time-independent scattering theory that is developed in (2014 Ann. Phys., NY 341 70-85) offers simple formulas for the reflection and transmission amplitudes of finite-range potentials in terms of the solution of an initial-value differential equation. We prove a theorem that simplifies the application of this result and use it to give a complete characterization of the invisible configurations of the truncated z e(-2ik0x) potential to a closed interval, [0, L], with k(0) being a positive integer multiple of pi/L. This reveals a large class of exact unidirectionally and bidirectionally invisible configurations of this potential. The former arise for particular values of z that are given by certain zeros of Bessel functions. The latter occur when the wavenumber k is an integer multiple of pi/L but not of k(0). We discuss the optical realizations of these configurations and explore spectral singularities of this potential.