Exact propagators on the lattice with applications to diffractive effects

TL;DR

This paper derives exact propagators for the discrete Schrödinger equation, revealing their properties through path summation, and applies them to analyze diffraction and scattering phenomena in discrete and periodic media.

Contribution

It introduces a new exact propagator for the discrete Schrödinger equation and explores its applications to diffraction and scattering in lattice systems.

Findings

Exact propagator derived for discrete Schrödinger equation

Analytic solutions for diffraction in discrete space and continuous time

Connections established with tight-binding arrays and photonic crystals

Abstract

The propagator of the discrete Schr\"odinger equation is computed and its properties are revealed through a Feynman path summation in discrete space. Initial data problems such as diffraction in discrete space and continuous time are studied analytically by the application of the new propagator. In the second part of this paper, the analogy between time propagation and 2D scattering by 1D obstacles is explored. New results are given in the context of diffraction by edges within a periodic medium. A connection with tight-binding arrays and photonic crystals is indicated.