In this paper, we develop a numerical method for the computation of (quasi-)resonances in spherical symmetric heterogeneous Helmholtz problems with piecewise smooth refractive index. Our focus lies in resonances very close to the real axis, which characterize the so-called whispering gallery modes. Our method involves a modal equation incorporating fundamental solutions to decoupled problems, extending the known modal equation to the case of piecewise smooth coefficients. We first establish the well-posedeness of the fundamental system, then we formulate the problem of resonances as a nonlinear eigenvalue problem, whose determinant will be the modal equation in the piecewise smooth case. In combination with the numerical approximation of the fundamental solutions using a spectral method, we derive a Newton method to solve the nonlinear modal equation with a proper scaling. We show the local convergence of the algorithm in the piecewise constant case by proving the simplicity of the roots. We confirm our approach through a series of numerical experiments in the piecewise constant and variable case.

翻译：本文针对具有分段光滑折射率的球对称非均匀亥姆霍兹问题，发展了一种计算（准）共振模式的数值方法。我们重点关注紧邻实轴的共振模式，这类模式表征了所谓的回音壁模式。该方法通过引入解耦问题基本解构建模态方程，将已知的模态方程推广至分段光滑系数情形。我们首先建立基本解系统的适定性，随后将共振问题表述为非线性特征值问题，其行列式即为分段光滑情况下的模态方程。结合谱方法对基本解进行数值逼近，我们推导出采用适当缩放比例的牛顿法来求解该非线性模态方程。通过证明根的单重性，我们在分段常数情况下验证了算法的局部收敛性。通过一系列分段常数及变系数情形的数值实验，我们验证了该方法的有效性。