Helmholtz quasi-resonances are unstable under most single-signed perturbations of the wave speed

TL;DR

The paper addresses how Helmholtz resonances (quasi-resonances) induced by wave-speed perturbations behave under most single-signed perturbations of the wave speed. It develops a robust semiclassical framework using meromorphic continuation via complex scaling, pole counting with trace-class determinants, and a semiclassical maximum principle to bound resolvents away from poles. The main contributions are (i) a polynomial bound on the number of poles in the perturbed problem, and (ii) a resolvent bound that holds for most perturbation values z at fixed frequencies, showing quasi-resonances are unstable under most such perturbations. These results have implications for uncertainty quantification and perturbation analysis of resonances in penetrable obstacle scattering.

Abstract

We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter $z$. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We show that there exists a polynomial function of frequency such that, for any frequency, for most values of $z$, the norm of the solution operator is bounded by that function. This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows superalgebraically, with these frequencies often called $\textit{quasi-resonances}$. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations.

Figures (2)

• Figure 1.1: The absolute value of the field scattered by the plane wave $\exp(i k(x\cos(\pi/6) + y\sin(\pi/6)))$ hitting the penetrable obstacle $\mathcal{O} = B(0,1)$ with $n_i=100$ (the red line denotes the boundary $\mathcal{O}$.) The frequency is $k=0.992772133752486$, which is (an approximation to) a quasi-resonance for the problem when $n_i=100$. The left plot corresponds to $n_i=100$ and $z=0$ (i.e., the setting of the quasi-resonance) and the right plot corresponds to $n_i=100$ and $z=0.01$ (i.e., a small perturbation of the wave speed). We highlight the different scales on the colour bars.
• Figure 1.2: Same as Figure \ref{['fig:QR1']} except now $k=2.19476917403094$, which is also (an approximation to) a quasi-resonance when $n_i=100$.

Theorems & Definitions (45)

• Theorem 1.1: Main result for smooth $n$
• Theorem 1.2: Main result for discontinuous $n$
• Corollary 1.3: Holomorphy of solution operator in $B(z,Ck^{-M})$ for "most" $z$
• proof
• Lemma 2.1: Limiting absorption principle
• proof : References for the proof
• Lemma 2.2: Meromorphic continuation
• Lemma 2.3
• proof : Proof of Lemma \ref{['lem:meromorphic']}
• proof : Proof of Lemma \ref{['lem:joey']}