# For most frequencies, strong trapping has a weak effect in frequency-domain scattering

## TL;DR

This paper demonstrates that, despite strong trapping in frequency-domain scattering, the Helmholtz solution operator exhibits at most polynomial growth for most frequencies, which supports the effectiveness of certain numerical methods at high frequencies.

## Contribution

It proves that the Helmholtz solution operator grows polynomially for most frequencies even with strong trapping, extending the understanding of wave behavior in complex geometries.

## Key findings

- Solution operator growth is polynomial for most frequencies.
- Strong trapping does not cause exponential growth at most frequencies.
- Supports polynomial-growth assumptions in numerical methods.

## Abstract

It is well known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.