A numerical study on exceptional points in two-dimensional acoustic system with two sound-hard obstacles

This paper is concerned with non-Hermitian degeneracy and exceptional points in a simple two-dimensional scattering system. We study resonant frequencies associated with multiple scattering between a sound-hard disk and an ellipse situated in the ambient air. To this end, a boundary integral equation is used to characterize the system’s resonant frequencies and is discretized by the Nyström method. This allows us to compute the scattering resonances with the help of a nonlinear eigenvalue solver. The system is parameterized by two geometric parameters: position and rotation angle of the ellipse. We define exceptional points of resonant frequencies and seek them numerically based on hierarchical subdivision of the two-dimensional parameter space. The proposed method, a combination of the boundary integral equation and hierarchical algorithm, does not rely on the sensitivity of resonant frequencies with respect to the system parameters. To illustrate the concepts of non-Hermitian degeneracy and exceptional points, we first study a simple mechanical model and show that the degeneracy is associated with perturbation theory, i.e., how eigenvalues of the system depend on the parameters. Subsequently, we formulate the scattering resonance problem and describe the proposed method for pinpointing exceptional points. Numerical results confirm that the proposed method provides strong numerical evidence of the existence of exceptional points.