Spatiotemporal deformations of reflectionless potentials

Reflectionless potentials for classical or matter waves represent an important class of scatteringless systems encountered in different areas of physics. Here we mathematically demonstrate that there is a family of non-Hermitian potentials that, in contrast to their Hermitian counterparts, remain reflectionless even when deformed in space or time. These are the profiles that satisfy the spatial Kramers-Kronig relations. We start by considering scattering of matter waves for the Schrödinger equation with an external field, where a moving potential is observed in the Kramers-Henneberger reference frame. We then generalize this result to the case of electromagnetic waves, by considering a slab of reflectionless material that both is scaled and has its center displaced as an arbitrary function of position. We analytically and numerically demonstrate that the backscattering from these profiles remains zero, even for extreme deformations. Our results indicate the supremacy of non-Hermitian Kramers-Kronig potentials over reflectionless Hermitian potentials in keeping their reflectionless property under deformation and could find applications to, e.g., reflectionless optical coatings of highly deformed surfaces based on perfect absorption.