\sum_{n \, \geq \, 0} 
\sum_{\substack{j \, \geq \, 0 \\ \ell \, \geq \, 0}} 
\sum_{\substack{0 \leq m \leq N_j \\ 0 \leq q \leq N_\ell}} 
\overline{ \Big( B E_m^{n,n+j} \Big) (v')}
\langle \chi_m^{n,n + j},\chi_q^{n,n + \ell} \rangle \\
& \times \left( - \frac{i e^{i(j + \ell + 2)\Phi}}{\sqrt{u}
\sqrt{4 - u}} + \frac{i e^{i\vert j - \ell \vert\Phi}}{\sqrt{u} \sqrt{4 - u}} \right)
\Big( A E_q^{n,n + \ell} \Big) (v) ,
\end{split}$$ throughout the double change of variables $$\label{eq4,60}
\frac{z + 2\sqrt{k} - (k + 1)}{\sqrt{k}} = u = 4 \sin^2 \left( \frac{\Phi}{2} \right), 
\quad {\operatorname{Im}}(\Phi) > 0.$$

Let $\varphi \in \ell^2(\mathcal{V})$ and $z \in \rho (-L + k + 1)$ the resolvent set of the operator $-L + k + 1$. Thanks to , we have $$\label{eq4,7}
\begin{split}
(-L + k + 1 - z)^{-1} & = \sum_{n \, \geq \, 0} P_n (-L_n + k + 1 - z)^{-1} P_n \\
& = \sum_{n \, \geq \, 0} P_n U^\ast U (-L_n + k + 1 - z)^{-1} U^\ast U P_n,
\end{split}$$ where $U$ is the unitary operator defined by . Thus, for any vector $\psi \in \ell^2(\mathcal{V})$, we have $$\label{eq4,8}
\begin{split}
& \biggl\langle  A (-L 