## 2 Theoretical Background
### Hollow Waveguide of Arbitrary Cross Section
\begin{array}{l}K_{m,p}^{(+,+)}=\sqrt{mp}\times\kappa_{m,p}^{(+,+)},\ K_{m}^{(+,+)}= \sqrt{m}\times\kappa_{m,0}^{(+,+)},\\ \text{where }\kappa_{m,p}^{(+,+)}=\frac{1}{2}\Big{[}\left(h_{m,p}^{00}+h_{m,-p}^{00} \right)+\left(h_{-m,p}^{00}+h_{-m,-p}^{00}\right)\Big{]};\\ K_{m,p}^{(+,-)}=\sqrt{mp}\times\kappa_{m,p}^{(+,-)},\ K_{m}^{(-,+)}=\sqrt{m} \times\kappa_{m,0}^{(-,+)},\\ \text{where }\kappa_{m,p}^{(+,-)}=\frac{1}{2}\Big{[}\left(h_{m,p}^{00}-h_{m,-p}^{00} \right)+\left(h_{-m,p}^{00}-h_{-m,-p}^{00}\right)\Big{]};\\ K_{m,p}^{(-,+)}=\sqrt{mp}\times\kappa_{m,p}^{(+,+)},\ K_{p}^{(+,+)}=\sqrt{p} \times\kappa_{m,p}^{(+,+)},\\ \text{where }\kappa_{m,p}^{(-,+)}=\frac{1}{2}\Big{[}\left(h_{m,p}^{00}+h_{m,-p}^{00} \right)-\left(h_{-m,p}^{00}+h_{-m,-p}^{00}\right)\Big{]};\\ K_{m,p}^{(-,-)}=\sqrt{mp}\times\kappa_{m,p}^{(-,-)},\ K_{p}^{(+,-)}=\sqrt{p} \times\kappa_{0,p}^{(+,-)},\\ \text{where }\kappa_{m,p}^{(-,-)}=\frac{1}{2}\Big{[}\left(h_{m,p}^{00}-h_{m,-p}^{00} \right)-\left(h_{-m,p}^{00}-h_{-m,-p}^{00}\right)\Big{]},\end{array}  \tag{21}\]  
and where the quantities \(K_{m,p}^{(\pm,\pm)}=K_{m,p}^{(\pm,\pm)}(k)\) depend on the spectral parameter \(k\in\mathcal{R}\). Equations (18) and (19) can be made more compact by using (20) to eliminate the coefficient \(\xi_{0}^{(0)}\) via the rearrangement  
\[\xi_{0}^{(0)}=-\frac{2}{\kappa_{0,0}^{(+,+)}}\sum_{p=1}^{\infty}\Big{(}K_{p}^ {(+,+)}X_{p}-K_{p}^{(+,-)}Y_{p}\Big{)}=0. \tag{22}\]  
After substituting (22) in (18) and (19), we arrive at the final form of the equations to be solved, being a coupled ISLAE where the index \(m\) runs over \(m=1,2,3,\ldots\):  
\[X_{m}+\sum_{p=1}^{\infty}\left(K_{m,p}^{(+,+)}-\frac{K_{m}^{(+,+)}}{\kappa_{0, 0}^{(+,+)}}K_{p}^{(+,+)}\right)X_{p}-\sum_{p=1}^{\infty}\left(K_{m,p}^{(+,-)} -\frac{K_{m}^{(+,+)}}{\kappa_{0,0}^{(+,+)}}K_{p}^{(+,-)}\right)Y_{p}=0, \tag{23}\]  
\[Y_{m}-\sum_{p=1}^{\infty}\left(K_{m,p}^{(-,-)}-\frac{K_{m}^{(-,+)}}{\kappa_{0, 0}^{(+,+)}}K_{p}^{(+,-)}\right)Y_{p}+\sum_{p=1}^{\infty}\left(K_{m,p}^{(-,+)}- \frac{K_{m}^{(-,+)}}{\kappa_{0,0}^{(+,+)}}K_{p}^{(+,+)}\right)X_{p}=0. \tag{24}\]  
This is a Fredholm matrix equation of second kind. More precisely, one can show using the definitions and results of [1,2] that the left-hand sides of ISLAE (23), (24) define an infinite-matrix (summation) OVF \(H(k):\;I_{2}^{2}\to I_{2}^{2},\;k\in\mathcal{R},\;\) where \(I_{2}^{2}\) is a Cartesian product of two copies of spaces \(l_{2}\), and Equations (23) and (24) form a Fredholm infinite-matrix (summation) equation system of the second kind equivalent to the initial homogeneous boundary IE (7). This system, written as \(H(k)=0,\) constitutes, in turn, the problem on CNs for OVF \(H(k).\) Since \(H(k)\) is a Fredholm OVF considered in the Hilbert space of sequences \(I_{2}^{2}\) and its resolvent set is nonempty, its \"spectral\" set \(\sigma_{\rm H}\subset\;\mathcal{R}\) of the sought CNs is discrete, i.e., it consists of isolated points on the complex plane with the only accumulation point at infinity. Based on the equivalence, one can verify that every \(k\in\sigma_{\rm H}\) is a CN of the initial integral OVF \(G(k)\) defined by the initial IE (7) and is a sought eigenvalue of the initial Dirichlet BVP.