A note on the validity of the Schr\"odinger approximation for the Helmholtz equation

TL;DR

This paper rigorously justifies the Schr"odinger approximation for the Helmholtz equation describing electromagnetic waves, providing bounds that confirm the approximation's validity despite the ill-posed nature of the original system.

Contribution

It offers a mathematical validation of the Schr"odinger approximation for the Helmholtz equation, addressing the challenge of ill-posedness in the evolution along the propagation axis.

Findings

Established bounds between the approximation and true solutions.

Proved the Schr"odinger approximation's validity despite ill-posedness.

Enhanced understanding of wave evolution modeling.

Abstract

Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation $\Delta u+\omega ^{2}u=0 $ for $ (x,y,z) \in \mathbb{R}^3 $. For the evolution of such waves along the $z$-axis a Schr\"odinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as an evolutionary system along the $z$-axis.