## Appendix A Resolvent Theory for Classical Operators  
The Fredholm integral equation of the second type in general form may be written as [56]  
\[E(\mathbf{x})-\
u\int\limits_{5}g\Big{(}\mathbf{x},\mathbf{x}^{\prime}\Big{)}E \Big{(}\mathbf{x}^{\prime}\Big{)}d\mathbf{x}^{\prime}=E_{0}(\mathbf{x}), \tag{10}\]  
where \(E(\mathbf{x})\) is an unknown function, \(\
u\) is a given parameter, and \(E_{0}(\mathbf{x})\) (right-hand term) and \(g\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}\) (kernel) are a priori given functions. The eigenfunctions \(u_{n}(\mathbf{x})\) of the integral Fredholm operator are defined by the homogeneous version of Equation (10):  
\[u_{n}(\mathbf{x})=\
u_{n}\int\limits_{5}g\Big{(}\mathbf{x},\mathbf{x}^{\prime }\Big{)}u_{n}\Big{(}\mathbf{x}^{\prime}\Big{)}d\mathbf{x}^{\prime}. \tag{11}\]  
For the scattering problems, the kernel of the integral equation is the Green function of the Helmholtz equation. For the two-dimensional (2D) case \(g\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}=-\frac{i}{4}H_{0}^{(1)}\big{(}k |\mathbf{x}-\mathbf{x}^{\prime}|\big{)}\) where \(H_{0}^{(1)}(x)\) is the Hankel function of the zero order and the first type, as one can see, the integral operator kernel is symmetric \(g\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}=g\big{(}\mathbf{x}^{\prime}, \mathbf{x}\big{)}\) but non-Hermitian (\(g\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}\
eq g^{*}(\mathbf{x}^{\prime}, \mathbf{x})\)). Regarding the validity of the resolvent approach for this class of integral equations, in principle, the eigen-functions for non-Hermitian operators may not generate the complete basis system, and may not exist at all. However, the opera tor \(\overline{\mathbf{A}}(\cdot)=\int_{S}g\big{(}\mathbf{x},\mathbf{x}^{{}^{\prime}} \big{)}(\cdot)d\mathbf{x}^{{}^{\prime}}\) has some fundamental properties due to the special form of its kernel. First, the kernel has a weak singularity (integrable in the ordinary sense). Second, we can split the kernel \(g\big{(}\mathbf{x},\mathbf{x}^{{}^{\prime}}\big{)}\) into the real and imaginary parts and write \(\overline{\mathbf{A}}(\cdot)=\overline{\mathbf{A}}_{R}(\cdot)+i\overline{ \mathbf{A}}_{I}(\cdot)\), where subscripts \(R\), \(I\) mean real and imaginary components, respectively. The singular component of operator \(\overline{\mathbf{A}}(\cdot)\) is its real (Hermitian) part, while its imaginary part is regular. Thus, unbounded inverse \(\left(\overset{\raisebox{-0.86 pt}{\scalebox{0.8}{$\frown$}}}{\mathbf{I}}-v \overline{\mathbf{A}}\right)_{R}^{-1}(\cdot)\) exists. Combining these two facts means that operator \(\overline{\mathbf{A}}(\cdot)\) is a weak perturbation of Hermitian operator \(\overline{\mathbf{A}}_{R}(\cdot)\)[57]. For weak perturbations of Hermitian operators, one can use the fundamental mathematical theory of pseudodifferential operators in Sobolev spaces [57] (in particular, the theorems on basis properties). For real \(k\) and \(u(\mathbf{x})\in\mathbf{L}^{2}(S)\), we have \(\mathrm{Im}\int_{S}\int_{S}g\big{(}\mathbf{x},\mathbf{x}^{{}^{\prime}}\big{)} u(\mathbf{x})u^{*}(\mathbf{x}^{{}^{\prime}})d\mathbf{x}d\mathbf{x}^{{}^{ \prime}}<\infty\) (\(\mathbf{L}^{2}(S)\) (which is a space of Lebesgue squared integrable functions). In such cases, the operator \(\overline{\mathbf{A}}(\cdot)\) is a bounded operator acting from \(\mathbf{L}^{2}(S)\) to Sobolev space \(\mathbf{H}^{2}(S)\). Such an operator is a completely continuous (compact) one. This means that the eigenvalues \(v_{n}\) exist as non-zero, and lie in the upper half-plane (\(\mathrm{Im}(v_{n})>0\)), and the resolvent set is non-empty [57]. It is easy to prove the orthogonality of eigenfunctions in the following form  
\[\int_{S}u_{n}(\mathbf{x})\cdot u_{m}(\mathbf{x})d\mathbf{x}=\delta_{mn} \tag{10}\]  
using the standard procedure. To that end, we should take Equation (10) with index \(m\), multiply it by \(u_{n}(\mathbf{x})\), multiply (10) by \(u_{m}(\mathbf{x})\), subtract one from another, and integrate over \(S\).