)\psi_{lm}=0,\quad
k_i^2=\varepsilon_i\,\mu_i\,\frac{\omega^2}{c^2}, \quad i=1,2\quad
(r\neq a); \quad \psi_{lm}(\mathbf{r})=f_l(r)Y_{lm}(\Omega)\,{.}$$

Equations (\[3-2\]) should be supplemented by the boundary conditions at the origin, at the sphere surface, and at infinity. In order for the fields to be finite at $r=0$ the Debye potentials should be regular here. At the spatial infinity we impose the radiation conditions with the goal to find the spectrum of eigenfunctions with complex frequencies (quasi-normal modes in the problem at hand). At the sphere surface the standard matching conditions for electric and magnetic fields should be satisfied [@Stratton].

In view of all this the Helmholtz equation (\[3-2\]) becomes now the spectral problem for the Laplace operator multiplied by the discontinuous factor $-1/(\varepsilon
(r)\,\mu(r))$ $$\label{3-2a}-\, \frac{1}{\varepsilon (r)\,\mu(r)}\Delta
\,\psi_{\omega l m}(r)=\frac{\omega^2}{c^2}\,\psi_{\omega l m}(r),
\quad r\neq a\,{,}$$ where $$\varepsilon (r)\,\mu(r)=
\begin{cases}
\varepsilon_1\,\mu_1\,{,}&r<a \,{,}\\
\varepsilon_2\,\mu_2\,{,}&r>a \,{.}
\end{cases}$$ In this problem the spectral parameter is $\omega^2/c^2$.

In order to obey the boundary conditions at the origin and at spatial infinity formulated above, the solution to the spectral problem (\[3-2a\]) should have the form $$\label{3-2b} f_{\omega l}(r)=C_1\,j_l(k_1r)\,{,} \quad r<a,\quad
f_{\omega l}(r)=C_2\,h^{(1)}_l(k_2r)\,{,} \quad r>a\,{,}$$ where $j_l(z)$ is the spherical Bessel function and $h_l^{(1)}(z)$ is the spherical Hankel function of the first kind [@AS], the latter obeys 