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 $\a>0$. The high-frequency (or: semi-classical) parameter is $\eps>0$. We let $\eps$ and $\a$ 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 $u^\eps$ radiates in the outgoing direction, {\bf uniformly} in $\eps$. In particular, the function $u^\eps$, when conveniently rescaled at the scale $\eps$ close to the origin, is shown to converge towards the {\bf outgoing} solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in $\eps$) of the limiting absorption principle. Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schr\"odinger 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 Schr\"odinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in $\eps$.