## 7 Emergence of the BSC in Scattering  
Scattering of plane waves by periodic two-dimensional arrays of dielectric spheres originates since pioneering papers by [PERSON] et al. [3; 4] (see also Ref. [5]). Scattering by aggregates of finite number of spheres was considered in the framework of multi sphere Mie scattering [6; 9; 11], however to our knowledge the scattering by the one-dimensional infinite array of dielectric spheres was not considered yet. The following subsections aim to present results of numerical calculation of differential and total cross-sections of the infinite array with focus to follow resonant traces of the BSCs in the cross-sections similar to the scattering by array of dielectric rods [37; 44; 45]. For the present case of the array of dielectric spheres we revealed different types of the BSCs. We show that excitation of corresponding quasi-BSCs needs in different ways of injection of EM waves. The BSC has zero coupling with the continuum, i.e., the BSC has infinite quality factor [76]. Also the BSC is unique by that the solution of the scattering problem becomes ambiguous [25] and can be written as superposition of particular scattering state and the BSC [77; 78]. \[\Psi=\Psi_{S}+\alpha\Psi_{BSC} \tag{41}\]  
where \(\alpha\) is arbitrary coefficient. This equation is well known mathematical consequence of the linear Lippmann-Schwinger Equation (1) when the inverse of the matrix \(\widehat{L}\) does not exist, i.e., when \(Det(\widehat{L})=0\). Then the BSC is the eigenmode of the matrix \(\widehat{L}\)  
\[\widehat{L}\Psi_{BSC}=\lambda_{c}\Psi_{BSC} \tag{42}\]  
with real eigenvalue \(\lambda_{c}\). At the first sight the BSCs are not interesting because they are invisible for probing waves incident from the continuum. However the BSC point is isolated in the parametric space and experimentalist can approach to this point only approximately. Other words, in reality we have only the quasi-BSCs whose quality factor is restricted by set-up imperfectness. Moreover for the case of arrays of dielectric rods or nanoparticles we have always finite number of them to give rise to a leakage of trapped modes into the diffraction continua. In this case the parameter \(\alpha\) in Equation (47) is defined by the point in the parametric space. When the point is sufficiently close to the BSC point \(\alpha\) can achieve enormous value however as dependent on the way of approaching [78]. That observation is important for applications [26; 52; 78; 79]. According to above equation the BSC is a null eigenvector of matrix \(\widehat{L}\) with zero eigenvalue. As soon as one deviates from the BSC point in the parametric space the BSC emerges in the form of a collapsing Fano resonance. That phenomenon was observed in scattering of EM waves by arrays of rods [30; 33; 37; 40; 44; 45; 52; 62]. The Fano resonance for the present system can be interpreted as interference of two optical paths, one through the spheres and another between the spheres. In what follows we highlight these features of the BSCs using the biorthogonal basis of eigenvectors of the non-Hermitian matrix \(\widehat{L}\)[37; 80]. \[\widehat{L}\mathbf{X}_{f}=L_{f}\mathbf{X}_{f},\ \ \widehat{L}^{+}\mathbf{Y}_{f}=L _{f}^{\dagger}\mathbf{Y}_{f},\ \ \mathbf{Y}_{f}^{+}\mathbf{X}_{f^{\prime}}=\delta_{ff^{ \prime}}. \tag{43}\]  
It immediately follows that  
\[\widehat{L}^{-1}=\sum_{f}\mathbf{X}_{f}\frac{1}{L_{f}}\mathbf{Y}_{f}^{+}. \tag{44}\]  
Because of the axial symmetry matrix \(\widehat{L}\) has OAM preserving block structure  
\[L_{ll^{\prime}}^{(m)}=\left( \begin{array}{cc}Z_{TE,I}^{-1}\delta_{ll^{\prime }}-\mathcal{A}_{ll^{\prime}}^{mm}&-\mathcal{B}_{ll^{\prime}}^{mm}\\ -\mathcal{B}_{ll^{\prime}}^{mm}&Z_{TM,I}^{-1}\delta_{ll^{\prime}}-\mathcal{A} _{ll^{\prime}}^{mm}\end{array} \right), \tag{45}\]  
where each block correspond to a specific value \(m\). In the nearest vicinity of the BSC point one of the complex eigenvalues \(L_{c}\) is close to zero.