\leq\alpha_3 < \cdots <\beta_p$$ Since $\Delta(x)$ is a polynomial of degree $p$, there are $p$ solutions of $\Delta(x)=2$ and of $\Delta(x)=-2$, so $2p$ points $\{\alpha_j\}_{j=1}^p \cup 
\{\beta_j\}_{j=1}^p$.

The bands are $[\alpha_1,\beta_1],[\alpha_2,\beta_2],\dots, [\alpha_p,\beta_p]$ and the gaps are $(\beta_1,\alpha_2)$, $(\beta_2,\alpha_3), \dots, (\beta_{p-1}, \alpha_{p+1})$. If some $\beta_j=\alpha_{j+1}$, we say the $j$-th gap is closed. Otherwise we say the gap is open.

If we remove the bands from ${{\mathbb{C}}}$, $\Gamma_\pm(z)$ are single-valued analytic functions and holds. Moreover, $\Gamma_+$ has an analytic continuation to the Riemann surface, ${{\mathcal S}}$, of genus $\ell \leq p-1$ where $\ell$ is the number of open gaps. ${{\mathcal S}}$ is defined by the function $[(z-\alpha_1)(z-\beta_p) \prod_{\text{open gaps}} 
(z-\beta_j) (z-\alpha_{j+1})]^{1/2}$. $\Gamma_-$ is precisely the analytic continuation of $\Gamma_+$ to the second sheet.

The Dirichlet data are partially those $x$’s where $${\label}{2.14} 
T_p(x) \binom{1}{0} = c_x \binom{1}{0}$$ that is, points where the $21$ matrix element of $T_p$ vanishes. It can be seen that the Dirichlet data $x$’s occur, one to each gap, that is, $x_1, \dots, x_{p-1}$ with $\beta_j \leq x_j \leq \alpha_{j+1}$. If $x$ is at an edge of a gap, then $c_j \equiv 
c_{x_j}$ is $\pm 1$. Otherwise ${\lvertc_j\rvert} \neq 1$. If ${\lvertc_j\rvert} >1$, we add the sign $\sigma_j =-1$ 