\biggr\rangle \overline{\biggl\langle \psi,A E_q^{n,n+\ell} 
\biggr\rangle} \\
& \times \int_{\mathbb{T}} \big( -2\sqrt{k} \cos (\theta) + k + 1 - z \big)^{-1} 
\biggl\langle U E_m^{n,n+j} (\theta),U E_q^{n,n+\ell} (\theta) 
\biggr\rangle \, d\theta \\
& = \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}} 
\biggl\langle \biggl\langle \varphi,B E_m^{n,n+j} \biggr\rangle 
A E_q^{n,n+\ell},\psi \biggr\rangle \\
& \times \int_{\mathbb{T}} \big( -2\sqrt{k} \cos (\theta) + k + 1 - z \big)^{-1} 
\biggl\langle U E_m^{n,n+j} (\theta),U E_q^{n,n+\ell} (\theta) 
\biggr\rangle \, d\theta.
\end{split}$$ According to the construction of the unitary operator $U$ and , for $p = m$, $q$ and $p' = j$, $\ell$ respectively, we have $$U E_p^{n,n+p'} (\theta) = \frac{1}{\sqrt{\pi}} \chi_p^{n,n+p'} 
\sin \big( (p' + 1) \theta \big).$$ Putting this together with , we obtain $$\label{eq4,12}
\begin{split}
& \biggl\langle A (-L + k + 1 - z)^{-1} B^\ast 
\varphi,\psi \biggr\rangle \\
& = \frac{1}{\pi} \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}} 
\biggl\langle \biggl\langle \varphi,B E_m^{n,n+j} \biggr\rangle 
\langle \chi_m^{n,n+j},\chi_q^{n,n+\ell} \rangle A E_q^{n,n+\ell},\psi \biggr\rangle \\
& \times \int_{\mathbb{T}} \big( -2\sqrt{k} \cos 