Sobolev and Max Norm Error Estimates for Gaussian Beam Superpositions

TL;DR

This paper provides Sobolev and max norm error estimates for Gaussian beam superpositions approximating solutions to hyperbolic PDEs and Schrödinger equations, demonstrating convergence rates unaffected by caustics and valid in any dimension.

Contribution

It derives rigorous Sobolev and max norm error bounds for Gaussian beam superpositions, including near caustics, for the first time in any spatial dimension.

Findings

Convergence rate in Sobolev norms is O(ε^{k/2 - s}) for k-th order beams.

Max norm convergence is spectral away from caustics, with slower rates near caustics.

Results are valid in any number of spatial dimensions, unaffected by caustics.

Abstract

This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schr\"odinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wavelength $\varepsilon$. The estimates are performed for the scalar wave equation and the Schr\"odinger equation. Our result demonstrates that a Gaussian beam superposition with $k$-th order beams converges to the exact solution as $O(\varepsilon^{k/2-s})$ in order $s$ Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is $O(\varepsilon^{\lceil k/2\rceil})$ and away from the essential support of the solution, the convergence is spectral in $\varepsilon$. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate $O(\varepsilon^{(k-n)/2})$ in $n$ spatial dimensions.