mod $p$), we need to analyze the structure of $p_n$ in terms of $\Gamma_+, \Gamma_-$. For $z$ not a branch point (or closed gap), $\Gamma_+ \neq 
\Gamma_-$. $\Gamma_+$ is well-defined on ${{\mathbb{C}}}\backslash\text{bands}$ since ${\lvert\Gamma_+\rvert} > {\lvert\Gamma_-\rvert}$. On the bands, ${\lvert\Gamma_+\rvert}={\lvert\Gamma_-\rvert}$ and, indeed, the boundary values on the two sides of a band are distinct. But $\Gamma_+$ is analytic on ${{\mathbb{C}}}\backslash\text{bands}$, so for such $z$, we can define $P_\pm$ by $${\label}{2.23} 
T_p(z) = \Gamma_+ P_+ + \Gamma_- P_-$$ where $P_+, P_-$ are $2\times 2$ rank one projections obeying $${\label}{2.24} 
P_+^2 = P_+ \qquad P_-^2 =P_- \qquad P_+ P_- = P_- P_+ =0$$ and $${\label}{2.25} 
P_+ + P_- = \boldsymbol{1}$$ It follows from and that $$\begin{aligned}
P_+ &= {\frac}{T_p(z) - \Gamma_- (z) \boldsymbol{1}}{\Gamma_+ - \Gamma_-} {\label}{2.26} \\
P_- &= {\frac}{T_p(z) - \Gamma_+ (z) \boldsymbol{1}}{\Gamma_- - \Gamma_+} {\label}{2.27} \end{aligned}$$ which, in particular, shows that $P_+$ is a meromorphic function on ${{\mathcal S}}$ whose second-sheet values are just $P_-$.

Define $$\begin{aligned}
a(z) &= \bigg\langle \binom{0}{1}, P_+(z) \binom{1}{0}\bigg\rangle {\label}{2.28} \\
b(z) &= \bigg\langle \binom{1}{0}, P_+(z) \binom{1}{0}\bigg\rangle {\label}{2.29} \end{aligned}$$ so implies $$\begin{aligned}
\biggl\langle \binom{0}{1}, P_-(z) \binom{1}{0}\bigg\rangle &= -a(z) {\label}{2.30} \\
\bigg\langle \binom{1}{0}, P_-(z) \binom{1}{0}\bigg\rangle &= 1-b(z) {\label}{2.31} \end{aligned}$$

Under most circumstances, $a(z)$ has a pole at 