Exceptional points and defective resonances in an acoustic scattering system with sound-hard obstacles

TL;DR

This work demonstrates that non-Hermitian degeneracies and exceptional points can arise in acoustic scattering from sound-hard obstacles, by recasting resonances as eigenvalues of a holomorphic operator-valued function $A(k)$ and exploiting boundary-integral formulations. A perturbation framework for holomorphic Fredholm operator-valued functions yields fractional-power (Puiseux) sensitivity of defective resonances, enabling numerical identification via Nyström discretization and Sakurai--Sugiura. Numerical experiments in a 2D disk-array geometry provide evidence of a defective resonance, including near-degenerate eigenvalues, a short Jordan chain, and a clear square-root splitting under perturbation, together with an exceptional-point encircling demonstration. These results suggest that exceptional-point phenomena can occur in simple scattering systems without high-contrast materials, with implications for sensing and dissipation in non-Hermitian physics and potential connections to periodic-band structures.

Abstract

This paper is concerned with non-Hermitian degeneracy and exceptional points associated with resonances in an acoustic scattering problem with sound-hard obstacles. The aim is to find non-Hermitian degenerate (defective) resonances using numerical methods. To this end, we characterize resonances of the scattering problem as eigenvalues of a holomorphic integral operator-valued function. This allows us to define defective resonances and associated exceptional points based on the geometric and algebraic multiplicities. Based on the theory on holomorphic Fredholm operator-valued functions, we show fractional-order sensitivity of defective resonances with respect to operator perturbation. This property is particularly important in physics and associated with intriguing phenomena, e.g., enhanced sensing and dissipation. A defective resonance is sought based on the perturbation analysis and Nyström discretization of the boundary integral equation. Numerical evidence of the existence of a defective resonance is provided. The numerical results combined with theoretical analysis provide a new insight into novel concepts in non-Hermitian physics.

Figures (8)

• Figure 1: Discrete mechanical systems. The point masses, springs, and dampers are characterized by positive constants $M$, $G$, and $\gamma$, respectively.
• Figure 2: Schematic illustration of eigenvalues of $A+\varepsilon B$ as a function of $\varepsilon$, where $A$ has an eigenvalue $k_0$ with geometric multiplicity $m$ and algebraic multiplicity $p$. (a) Two paths (solid and dashed lines) in the neighborhood $V$. (b) and (c) Trajectory of eigenvalues in $\Lambda_0$ when $\varepsilon$ continuously changes along the paths. (b) $m=p=1$. (c) $m=1$ and $p=2$.
• Figure 3: $20\times 2$ disks aligned on the two-dimensional Cartesian grid. The top and bottom disks have the radii $R_1$ and $R_2$, respectively.
• Figure 4: (a) and (b) The real part of resonant states $u^{(1)}$ and $u^{(2)}$ corresponding to the two resonances $k^{(1)}$ and $k^{(2)}$, respectively. The values are scaled such that $u^{(1)}(9.5,0)=u^{(2)}(9.5,0)=1$.
• Figure 5: Distance between the two resonances $k^{(1)}(\varepsilon)$ and $k^{(2)}(\varepsilon)$ as a function of $\varepsilon$.

Theorems & Definitions (3)

• Theorem 3.1
• proof
• Corollary 3.2