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 ∈ R d ), supplemented with a given high frequency source term supported near the origin x = 0. A small absorption parameter αε > 0 is added, which somehow prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is ε > 0. We let ε and αε go to zero simultaneaously. We study the question whether the indirectly prescribed radiation condition at infinity is satisfied uniformly along the asymptotic process ε → 0, or, in other words, whether the conveniently rescaled solution to the considered equation goes to the outgoing solution to the natural limiting Helmholtz equation.This question has been previously studied by the first autor in [4]. In [4], it is proved that the radiation condition is indeed satisfied uniformly in ε, provided the refraction index satisfies a specific 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 optimality of the above mentionned non-refocusing condition, in the following sense. We exhibit a refraction index which 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 does not satisfy the natural radiation condition at infinity. More precisely, we show that the limiting solution is a 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. (2000): Primary 35QXX, Secondary 35J10, 81Q20 Mathematics subject classification1 Here and below we use the standard notation n 2 (x), a squared term, assuming in doing so that the corresponding term is everywhere non-negative. This is a harmless abuse of notation, since the refraction index n 2 (x) that is eventually chosen in our analysis is negative for certain values of x. The reader may safely skip this fact, since the Helmholtz equation also arises in the spectral analysis of Schrödinger operators, where the refraction index becomes E − V (x) where E is an energy and V (x) is a space-dependent potential, and the term E − V (x) may change sign in that context. 2 The limiting case αε = 0 + can be considered along our analysis, see below.

Resonances for homoclinic trapped sets

“…Note also that g − • g + > 0 for all g ± ∈ H ±∞ tang . The non-trapping potential V 1 (r − a)ψ(θ) has been considered for the Helmholtz equation by Castella and Klak in [22,Section 1.3]. Remark 3.10. i) Since the kernel of T 0 (τ ) is bounded from below by a positive constant, A 0 vanishes at some point if and only if mes S n−1 (H −∞ tang ) = 0.…”