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Uniform Resolvent Estimates for Subwavelength Resonators: The Minnaert Bubble Case

Long Li, Mourad Sini

TL;DR

This work analyzes subwavelength resonators formed by Minnaert bubbles, establishing uniform resolvent estimates and a robust link between two prevalent resonance definitions. Using a scaled boundary-volume Lippmann-Schwinger framework and a spectral decomposition of the Neumann-Poincaré type operator, the authors derive space-and-frequency uniform asymptotics for the scattered field and resolvent as the bubble size $e\to 0$, revealing two Minnaert resonances in the lower half-plane that converge to $\pm\omega_M$. They show the scattered field is dominated by a point-scatterer term located at the bubble, with uniform control across space, and relate the scattering resonances to poles of the natural wave operator, thereby unifying resonance concepts within a single Hamiltonian framework. The results have implications for acoustic metamaterials, cloaking, and wave control by small, high-contrast inclusions, providing rigorous tools to predict resonance-induced field amplification and their lifetimes. Key methods include boundary- and volume-integral operator analysis, the Lippmann-Schwinger equation, and a novel operator-inversion technique based on a spectral projection of $K_0^*$.

Abstract

Subwavelength resonators are small scaled objects that exhibit contrasting medium properties (eigher in intensity or sign) while compared to the ones of a uniform background. Such contrasts allow them to resonate at specific frequencies. There are two ways to mathematically define these resonances. First, as the frequencies for which the related system of integral equations is not injective. Second, as the frequencies for which the related resolvent operator of the natural Hamiltonian, given by the wave-operator, has a pole. In this work, we consider, as the subwavelength resonator, the Minneart bubble. We show that these two mentioned definitions are equivalent. Most importantly, 1. we derive the related resolvent estimates which are uniform in terms of the size/contrast of the resonators. As a by product, we show that the resolvent operators have no scattering resonances in the upper half complex plane while they exhibit two scattering resonances in the lower half plane which converge to the real axis, as the size of the bubble tends to zero. As these resonances are poles of the natural Hamiltonian, given by the wave-operator, and have the Minnaert frequency as their dominating real part, this justifies calling them Minnaert resonances. 2. we derive the asymptotic estimates of the generated scattered fields which are uniform in terms of the incident frequency and which are valid everywhere in space (i.e. inside or outside the bubble). The dominating parts, for both the resolvent operator and the scattered fields, are given by the ones of the point-scatterer supported at the location of the bubble. In particular, these dominant parts are non trivial (not the same as those of the background medium) if and only if the used incident frequency identifies with the Minnaert one.

Uniform Resolvent Estimates for Subwavelength Resonators: The Minnaert Bubble Case

TL;DR

This work analyzes subwavelength resonators formed by Minnaert bubbles, establishing uniform resolvent estimates and a robust link between two prevalent resonance definitions. Using a scaled boundary-volume Lippmann-Schwinger framework and a spectral decomposition of the Neumann-Poincaré type operator, the authors derive space-and-frequency uniform asymptotics for the scattered field and resolvent as the bubble size , revealing two Minnaert resonances in the lower half-plane that converge to . They show the scattered field is dominated by a point-scatterer term located at the bubble, with uniform control across space, and relate the scattering resonances to poles of the natural wave operator, thereby unifying resonance concepts within a single Hamiltonian framework. The results have implications for acoustic metamaterials, cloaking, and wave control by small, high-contrast inclusions, providing rigorous tools to predict resonance-induced field amplification and their lifetimes. Key methods include boundary- and volume-integral operator analysis, the Lippmann-Schwinger equation, and a novel operator-inversion technique based on a spectral projection of .

Abstract

Subwavelength resonators are small scaled objects that exhibit contrasting medium properties (eigher in intensity or sign) while compared to the ones of a uniform background. Such contrasts allow them to resonate at specific frequencies. There are two ways to mathematically define these resonances. First, as the frequencies for which the related system of integral equations is not injective. Second, as the frequencies for which the related resolvent operator of the natural Hamiltonian, given by the wave-operator, has a pole. In this work, we consider, as the subwavelength resonator, the Minneart bubble. We show that these two mentioned definitions are equivalent. Most importantly, 1. we derive the related resolvent estimates which are uniform in terms of the size/contrast of the resonators. As a by product, we show that the resolvent operators have no scattering resonances in the upper half complex plane while they exhibit two scattering resonances in the lower half plane which converge to the real axis, as the size of the bubble tends to zero. As these resonances are poles of the natural Hamiltonian, given by the wave-operator, and have the Minnaert frequency as their dominating real part, this justifies calling them Minnaert resonances. 2. we derive the asymptotic estimates of the generated scattered fields which are uniform in terms of the incident frequency and which are valid everywhere in space (i.e. inside or outside the bubble). The dominating parts, for both the resolvent operator and the scattered fields, are given by the ones of the point-scatterer supported at the location of the bubble. In particular, these dominant parts are non trivial (not the same as those of the background medium) if and only if the used incident frequency identifies with the Minnaert one.
Paper Structure (13 sections, 20 theorems, 207 equations, 1 figure)

This paper contains 13 sections, 20 theorems, 207 equations, 1 figure.

Table of Contents

  1. Introduction and statement of the main results
  2. The Mathematical model
  3. The Minnaert frequency and the acoustic fields
  4. The Minnaert frequency and the resolvent of the acoustic propagator
  5. The associated scaled Hamiltonian
  6. The resolvent of the original acoustic propagator
  7. Comparison with related works
  8. Asymptotic estimates of auxiliary operators
  9. Global asymptotics of the acoustic field in both space and frequency
  10. Resolvent's asymptotics of the scaled Hamiltonian
  11. Minnaert resonance as a pole of the scaled Hamiltonian
  12. Equivalence of the two definitions
  13. Asymptotics of Minnaert resonances

Key Result

Theorem 1.1

Let $I\subset \mathbb R_+$ be a bounded interval containing $\omega_M$ given by eq:45. Assume that $\alpha > 1/2$ and $\varepsilon>0$. We have with holding uniformly with respect to all $\omega \in I$. Here, $d_{I,\max}:= \max_{z\in I} |z|$, $d_{I,\min}:=\min_{z\in I} |z|$ and $C_{d_{I,\max},d_{I,\min}}$ is a constant independent of $\varepsilon$ and $\omega$. In addition, the weighted space $L_

Figures (1)

  • Figure 1.1: Geometric setting of $\Omega$ and $\Omega_\varepsilon$

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 31 more