/2}{1-i \mu_j \tau /2}\right)^k \Phi_j = \exp( i \mu_{j,\tau} k \tau) \Phi_j, 
        \\
        \hbox{ where $\mu_{j,\tau}$ is defined by }
        \exp( i \mu_{j,\tau} \tau ) = \frac{1 + i \mu_j \tau /2}{1-i \mu_j \tau /2},
    \end{gathered}$$ yielding $$ \mu_{j,\tau}  \tau = 2 \arctan( \mu_j \tau/2).$$

Hence the midpoint scheme fits the assumptions of Theorem \[Thm-Main\] by setting $f(\alpha) = 2 \arctan(\alpha/2)$ and thus $R = \infty$.

The fourth order Gauss Method
-----------------------------

Let us present the so-called fourth order Gauss method for discretizing , which enters the frame of Runge-Kutta methods, see for instance [@Hairer-book].

It reads as follows: $$\label{Gauss}
        \left\{\begin{array}{ll}
            {\displaystyle}\kappa_i^k={{\mathcal{A}}}\left(y_\tau^k+\tau\sum_{j=1}^2 \alpha_{ij}\kappa_j^k\right), \qquad i=1,2,
            \\
            \begin{array}{ll}
                {\displaystyle}y_\tau^{k+1}=y_\tau^k+\frac{\tau}{2}(\kappa_1^k+\kappa_2^k), 
                \\
                {\displaystyle}y_\tau^0 = y^0 \in {{\mathcal{X}}}\ \hbox{given},
            \end{array}  
            \qquad
            (\alpha_{ij}) =  
                \left(\begin{array}{cc}
                \frac{1}{4}& \frac{1}{4}-\frac{\sqrt 3}{6}\\
                \frac{1}{4}+\frac{\sqrt 3}{6}&\frac{1}{4}
            \end{array}\right).
          \end{array}\right.$$ An easy computation shows that, for any $k \in {\mathbb{Z}}$, $$\frac{1}{2}\left(\kappa_1^k + \kappa_2^k \right)= \left(Id - \frac{\tau {{\mathcal{A}}}}{2} + \frac{\tau^2 {{\mathcal{A}}}^2}{12}   \right)^{-1} {{\mathcal{A}}}y_\tau^k,$$ which allows to rewrite as $$y_\tau^{k+1} = \left(Id - \frac{\tau {{\mathcal{A}}}}{2} + \frac{\tau^2 {{\mathcal{A}}}^2}{12}   \right)^{-1} \left(Id + \frac{\tau {{\mathcal{A}}}}{2} + \frac{\tau^2 {{\mathcal{A}}}^2}{12}   \right) y_\tau^k, \quad k \in {\mathbb{Z}}.$$

The spectral decomposition of the semi-discrete scheme can easily be performed. If $y^0 = \Phi_j$, we obtain $$y_\tau^k = \exp\left(i \frac{f(\mu_j\tau)}{\tau} k 