Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion

Abstract

We consider the high frequency Helmholtz equation with a variable refraction index $n^2(x)$ ($x \in \R^d$), supplemented with a given high frequency source term supported near the origin $x=0$. A small absorption parameter $α_{\varepsilon}>0$ is added, which somehow prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is $\varepsilon>0$. We let $\eps$ and $\a_\eps$ go to zero {\em simultaneaously}. We study the question whether the indirectly prescribed radiation condition at infinity is satisfied {\em uniformly} along the asymptotic process $\eps \to 0$, or, in other words, whether the conveniently rescaled solution to the considered equation goes to the {\em outgoing} solution to the natural limiting Helmholtz equation. This question has been previously studied by the first autor. It is proved that the radiation condition is indeed satisfied uniformly in $\eps$, provided the refraction index satisfies a specific {\em non-refocusing condition}, a condition that is first pointed out in this reference. The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin $x=0$ at time $t=0$, should not refocus at some later time $t>0$ near the origin again. In the present text we show the {\em optimality} of the above mentionned non-refocusing condition, in the following sense. We exhibit a refraction index which {\em does} refocus the rays of geometric optics sent from the origin near the origin again, and, on the other hand, we completely compute the asymptotic behaviour of the solution to the associated Helmholtz equation: we show that the limiting solution {\em does not} satisfy the natural radiation condition at infinity. More precisely, we show that the limiting solution is a {\em perturbation} of the outgoing solution to the natural limiting Helmholtz equation, and that the perturbing term explicitly involves the contribution of the rays radiated from the origin which go back to the origin. This term is also conveniently modulated by a phase factor, which turns out to be the action along the above rays of the hamiltonian associated with the semiclassical Helmholtz equation.