2005/03/16 by François Castella, Castella, François
Mathematics · Physics and Astronomy · #35J10 #35Q40 #81Q20 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:35J10 #msc:35Q40 #msc:81Q20
paper · pdf · doi:10.48550/arxiv.math/0503331
arxiv created 2005/03/16 · arxiv updated 2009/12/01
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 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 \eps.