Dynamical Theory of Scattering, Exact Unidirectional Invisibility, and Truncated $\mathfrak{z}\,e^{-2ik_0x}$ 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 $\mathfrak{z}\,e^{-2ik_0 x}$ 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 $\mathfrak{z}$ 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.