Kramers-Kronig potentials for the discrete Schrödinger equation

In a seminal work, Horsley and collaborators [S. A. R. Horsley, Nat. Photon. 9, 436 (2015)1749-488510.1038/nphoton.2015.106] showed that, in the framework of non-Hermitian extensions of the Schrödinger and Helmholtz equations, a localized complex scattering potential with spatial distributions of the real and imaginary parts related to one another by the spatial Kramers-Kronig relations are reflectionless and even invisible under certain conditions. Here we consider the scattering properties of Kramers-Kronig potentials for the discrete version of the Schrödinger equation, which generally describes wave transport on a lattice. Contrary to the continuous Schrödinger equation, on a lattice a stationary Kramers-Kronig potential is reflective. However, it is shown that a slow drift can make the potential invisible under certain conditions.