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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 \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
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