Sobolev and Max Norm Error Estimates for Gaussian Beam Superpositions

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.