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

Understand

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.