$Y:=e^{-\lambda\tau}e^{iq_{+}}$, and the real valued functions $$\begin{aligned}
S(q_{-}) & =R_{+}R_{-}=C^{2}\cos{(k_{-}+q_{-})}\cos{(k_{-}-q_{-})},\\
A(\omega,q_{-}) & =C\left(\left[R+a^{2}\cos k_{\tau}\right]\cos{k_{-}}\cos{q_{-}}+\omega\sin{k_{\tau}}\sin{k_{-}}\sin{q_{-}}\right),\\
B(\omega,q_{-}) & =C\left(-a^{2}\sin{k_{\tau}}\sin{k_{-}}\sin{q_{-}}+\omega\cos{k_{\tau}}\cos{k_{-}}\cos{q_{-}}\right),\\
D(\omega) & =R^{2}-\omega^{2}+2Ra^{2}\cos{k_{\tau}},\\
E(\omega) & =2\omega(a^{2}+R\cos{k_{\tau}}).\end{aligned}$$ Note that the dependence on $q_{+}$ in the obtained equation (\[eq:psolY\]) is included only in $Y$. Solving the quadratic equation (\[eq:psolY\]) with respect to $Y$ leads to $$Y_{\pm}(\omega,q_{-})=\frac{1}{S(q_{-})}\left(A(\omega,q_{-})+iB(\omega,q_{-})\pm\sqrt{\zeta(\omega,q_{-})}\right),$$ with $$\zeta(\omega,q_{-})=A^2(\omega,q_{-})-B^{2}(\omega,q_{-})-S(q_-)D(\omega)+i\left[2A(\omega,q_{-})B(\omega,q_{-})-S(q_-)E(\omega)\right].$$ Since there are two solutions $Y_{\pm}$, one obtains two functions for the pseudo-continuous spectrum $$\gamma_{\pm}(\omega,q_{-})=-\log|Y_{\pm}|=-\frac{1}{2}\log\left(Y_{\pm}Y_{\pm}^{\ast}\right).$$ The spectrum possesses the following symmetries $$Y_{\pm}(\omega,q_{-}+\pi)=-Y_{\mp}(\omega,q_{-})\label{eq:Ysym1}$$ and $$Y_{\pm}(-\omega,-q_{-})=Y_{\pm}^{\ast}(\omega,q_{-}).\label{eq:Ysym2}$$ The first relation (\[eq:Ysym1\]) implies that it is sufficient to consider only one of the two functions $\gamma_{\pm}(\omega,q_{-})$, since they are related to each other by the shift $q_{-}\to q_{-}+\pi$ as $$\gamma_{+}(\omega,q_{-}+\pi)=\gamma_{-}(\omega,q_{-}).\label{eq: SL 2-torus gamma shift property}$$ This also indicates that the pseudo-continuous spectrum is twofold degenerate in the limit of $M,N\to\infty$. The second property (\[eq:Ysym2\]) implies that the spectrum has the reflection-symmetry in the $(\omega,q_{-})$-plane: $$\gamma_{\pm}(-\omega,-q_{-})=\gamma_{\pm}(\omega,q_{-}).$$ Note that in the special case $k_{-}=0$ the additional symmetry-relations $Y_{\pm}(-\omega,q_{-})=Y_{\pm}^{\ast}(\omega,q_{-})$ and $Y_{\pm}(\omega,-q_{-})=Y_{\pm}(\omega,q_{-})$ hold. Figure \[pic: SL psol pcs\] illustrates the function $\gamma_{\max}(\omega,q_{-})=\max\left\{ \gamma_{+},\gamma_{-}\right\} $ for different values of $k_{-}$ and $k_{\tau}$.

The eigenvalues $\lambda$ are known as *characteristic exponents* or *Floquet-exponents* and are related to the *Floquet-multipliers* via $\mu=e^{\lambda T}=\exp(\frac{2\pi\gamma}{\Omega\tau})\exp(\frac{2\pi i\omega}{\Omega})$. As known from the 