The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach
François Castella
TL;DR
The paper studies the high-frequency Helmholtz equation with a source term under vanishing absorption, analyzing the simultaneous limits $\varepsilon\to0$ and $\alpha_\varepsilon\to0$ with a non-trapping zero-energy classical flow and a transversality condition on zero-energy trajectories. The authors reformulate using the rescaled field $u^\varepsilon(x)=\varepsilon^{-d/2} w^\varepsilon(x/\varepsilon)$ and express $u^\varepsilon$ as a time-integrated semiclassical propagator, then decompose contributions by time and energy scales. They prove that $u^\varepsilon$ converges, uniformly in $\varepsilon$, to the outgoing solution of the constant-coefficient equation $(\tfrac{1}{2}\Delta_x + n^2(0)) w^{out}=S$, implying that $w^\varepsilon$ radiates in the outgoing direction with a uniform radiation condition. The proof combines a uniform version of the Egorov theorem for large times, uniform dispersive estimates via a wave-packet approach for moderate times under the transversality assumption, and small-time dispersive properties of the variable-coefficient operator. Overall, the work provides an $\varepsilon$-uniform limiting absorption principle and a framework to compare variable-coefficient and constant-coefficient Helmholtz problems, with implications for high-frequency radiation conditions.
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 $\eps$.