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

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter >0. The high-frequency (or: semi-classical) parameter is ε>0. We let ε and 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 uε radiates in the outgoing direction, uniformly in ε. In particular, the function uε, when conveniently rescaled at the scale ε 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 ε) 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 ε. © 2004 Elsevier Inc. All rights reserved.