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 [Ann. Phys. (NY) 341, 77-85 (2014)] 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 k0 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 k0. We discuss the optical realizations of these configurations and explore spectral singularities of this potential..