A theorem on boundary functions for quantum shutters

We prove for one-dimensional time-dependent quantum absorbing (and reflecting) slits that for right-moving incident waves, the Laplace transform of the boundary function must have singular points at the complex roots of s±i(iɛ/ℏ)=0. We test our result against the exact case of the Moshinsky absorbing (and reflecting) shutter, and the agreement is perfect. In the same Moshinsky problem, when the approximated Kirchhoff boundary condition is used, the transmitted wave is a superposition of right- and left-moving Moshinsky packets. Neglecting the wrong directed wave components we get the exact solution.