## 4 Classification of BSCs in the Array of Spheres  
In the previous section we presented the theory for scattering of plane waves by a periodic array of dielectric spheres based on the approach by [PERSON] et al. [24]. If there is no incident wave we have \(\widehat{L}\mathbf{a}=0\) whose solutions are bound modes of the array. There might be two kinds of the bound modes. The first type of modes have wave number \(\beta>k_{0}\) and describe guided waves along the array. These solutions found by [PERSON] et al. exist in some interval of the material parameters of spheres, dielectric constant \(\epsilon\) or radius \(R\), and the Bloch wave number \(\beta\)[24]. The second type of bound modes with \(\beta<k_{0}\) resides above the light cone (BSCs). It is much more difficult to establish the existence of the second type of bound states because a tuning of material parameters is required. However there might exist symmetry protected BSCs which are robust with respect to the material parameters. These BSCs have been already considered in the linear array of infinitely long dielectric rods [31; 33; 35; 37; 40; 41; 42; 43; 44; 45].  
The axial symmetry of the array implies that the matrices \(\mathcal{A}\) and \(\mathcal{B}\) split into the irreducible representations of the azimuthal number \(m\) which therefore classifies the BSCs. Next, the discrete translational symmetry along the \(z\)-axis implies that the respective wave number \(\beta\) specifies the BSC. At last, additional optional symmetries arise due to the inversion symmetry transformation \(\widehat{K}f(x,y,z)=f(x,y,-z)\) for \(\beta=0,\pi\). It follows from Equation (11) that \(s_{2k+1}=0\), and respectively from Equations (5) and (60) we obtain \(\mathcal{A}_{VL}^{mm}=0\) if \(l+\
u\) is odd, and \(\mathcal{B}_{VL}^{mm}=0\) if \(l+\
u\) is even. Moreover for arbitrary \(\beta\): \(\mathcal{B}_{vl}^{00}=0\) (see Appendix A). These relations establish the selection rules for the amplitudes \(a_{l}^{m},\;b_{l}^{m}\) which determine allowed BSC modes listed in the Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{m}\) & \(\mathbf{\beta}\) & Type I of BSC & Type II of BSC \\ \hline \(\
eq 0\) & 0 & \((a_{2k}^{m},\;b_{2k+1}^{m})\) & \((a_{2k+1}^{m},\;b_{2k}^{m})\) \\ \hline \(0\) & \(\
eq 0\) & \((a_{l}^{0},\;0),E_{z}=0\) & \((0,\;b_{l}^{0}),H_{z}=0\) \\ \hline \(0\) & 0 & \((a_{2k}^{0},\;0),E_{z}=0\) & \((0,\;b_{2k}^{0}),H_{z}=0\) \\ \hline \(0\) & 0 & \((0,\;b_{2k+1}^{0}),H_{z}=0\) & \((a_{2k+1}^{0},\;0),E_{z}=0\) \\ \hline \end{tabular}
\end{table}
Table 1: Classification of the bound states in the radiation continuum (BSCs).  
The cartesian components of the vector spherical functions transform under the inversion of \(z\) as follows  
\[M^{m}_{l,x,y}(\pi-\theta)=-(-1)^{l-m}M^{m}_{l,x,y}(\theta),\ \ M^{m}_{l,z}(\pi- \theta)=(-1)^{l-m}M^{m}_{l,z}(\theta),\] \[N^{m}_{l,x,y}(\pi-\theta)=(-1)^{l-m}N^{m}_{l,x,y}(\theta),\ \ N^{m}_{l,z}(\pi- \theta)=-(-1)^{l-m}N^{m}_{l,z}(\theta). \tag{22}\]  
For \(\beta=0\) we have  
\[\sum_{j}M^{m}_{l,x,y}(\mathbf{r}-\mathbf{R}_{j})=-(-1)^{l-m}\sum _{j}M^{m}_{l,x,y}(\hat{K}\mathbf{r}-\mathbf{R}_{j}),\] \[\sum_{j}M^{m}_{l,z}(\mathbf{r}-\mathbf{R}_{j})=(-1)^{l-m}\sum_{j} M^{m}_{l,z}(\hat{K}\mathbf{r}-\mathbf{R}_{j}).\] \[\sum_{j}N^{m}_{l,x,y}(\mathbf{r}-\mathbf{R}_{j})=-(-1)^{l-m}\sum _{j}N^{m}_{l,x,y}(\hat{K}\mathbf{r}-\mathbf{R}_{j}),\] \[\sum_{j}N^{m}_{l,z}(\mathbf{r}-\mathbf{R}_{j})=(-1)^{l-m}\sum_{j} N^{m}_{l,z}(\hat{K}\mathbf{r}-\mathbf{R}_{j}). \tag{23}\]  
Then from these equations and Equation (2) one can obtain the following symmetric properties for the cartesian components of EM fields collected in Table 2.  
Tables 1 and 2 will be useful for the symmetry classification of the bound modes in the next sections.