The wave automaton for the time-dependent Schrödinger, classical wave and Klein-Gordon equations

The "wave automaton" model, recently studied by Vanneste et al. [Europhys. Lett. 17 (1992) 715], is a discrete in space and time S-matrix formulation of time-dependent wave propagation, in which the evolution operator is given explicitly. Here, we show how it can be related to various partial differential wave equations and demonstrate that, depending upon the parametrization of the S-matrices, it is equivalent to a finite-difference discretized version of the time-dependent Schrödinger, hyperbolic wave and Klein-Gordon equations. Compared to finite-difference versions of partial differential equations, the wave automaton has the advantage of dealing directly with measurable physical quantities, such as the local fluxes.