The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave-packet approach

Abstract

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α > 0 . The high-frequency (or: semi-classical) parameter is ɛ > 0 . We let ɛ and α go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution u ɛ radiates in the outgoing direction, uniformly in ɛ . In particular, the function u ɛ , when conveniently rescaled at the scale ɛ close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in ɛ ) of the limiting absorption principle. Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) for very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above-mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in ɛ .