# General superpositions of Gaussian beams and propagation errors

## TL;DR

This paper extends Gaussian beam superpositions to arbitrary phase space sets, deriving propagation error estimates and analyzing their sharpness, thereby broadening the applicability of high-frequency PDE solutions.

## Contribution

It introduces a superposition of Gaussian beams over arbitrary phase space sets and applies recent tools to estimate propagation errors, advancing high-frequency solution methods.

## Key findings

- Propagation error of order $k^{1- rac{N}{2}- rac{d-m}{4}}$ derived
- Error estimate sharpness analyzed through examples
- Method applicable to general high frequency PDE solutions

## Abstract

Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension $m$ in phase space, and show that the tools recently developed in [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp., 82: 919--952, 2013] can be applied to obtain the propagation error of order $k^{1- \frac{N}{2}- \frac{d-m}{4}}$, where $N$ is the order of beams and $d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.