in this problem, both functions $u_{\text{in}}(t,
\mathbf{x})$ and $u_{\text{ex}}(t, \mathbf{x})$ vanishing. However, the wave equations (\[2-14\]) and (\[2-15\]) have nonzero solutions with complex frequencies, i.e. quasi-normal modes, which satisfy the matching conditions (\[2-16\]) and (\[2-17\]) at the interface $S$ and radiation conditions (\[2-5\]) at spatial infinity. It is important, that the frequencies of oscillations in internal ($D_{\text{in}}$) and external ($D_{\text{ex}}$) regions are the same. A typical example here is the complex eigenfrequencies of a dielectric sphere. This problem has been investigated by Debye in his PhD thesis concerned with the light pressure on a material particles [@Debye].

Let us consider a sphere of radius $a$, consisting of a material which is characterized by permittivity $\varepsilon_1 $ and permeability $\mu_1$. The sphere is assumed to be placed in an infinite medium with permittivity $\varepsilon_2 $ and permeability $\mu_2$. In the case of spherical symmetry the solutions to Maxwell equations are expressed in terms of two scalar Debye potentials $\psi$ (see, for example, textbooks [@Stratton; @Jackson]): $$\begin{aligned}
\mathbf{E}^{\text{TM}}_{lm}&=&\bm{\nabla} \times
\bm{\nabla}\times(\mathbf{r}\psi^{\text{TM}}_{lm}),\quad
\mathbf{H}^{\text{TM}}_{lm}=-i\,\omega \,\bm{\nabla} \times
(\mathbf{r}\psi^{\text{TM}}_{lm})\quad (\text{E-modes}), \nonumber
\\ \mathbf{E}^{\text{TE}}_{lm}&=&i\,\omega \,\bm{\nabla} \times
(\mathbf{r}\psi^{\text{TE}}_{lm}),\quad
\mathbf{H}^{\text{TE}}_{lm}=\bm{\nabla} \times
\bm{\nabla}\times(\mathbf{r}\psi^{\text{TE}}_{lm})\quad
(\text{H-modes})\,{.} \label{3-1}\end{aligned}$$ These potentials obey the Helmholtz equation and have the indicated angular dependence $$\label{3-2} \left ( \mathbf{\nabla}^2+k^2_i\right
