corresponds to $$\label{Formulation1}
            \hspace{-3ex}   {{\mathcal{B}}}_\tau y_\tau^{k+1/2} := \frac{i}{2} {{\mathcal{B}}}_1 {{\mathcal{A}}}_{0, \tau}^{-1/2} \left(y_{2, \tau}^{k+1/2} - y_{1, \tau}^{k+1/2}\right) + \frac{1}{2}{{\mathcal{B}}}_2 \left(y_{1, \tau}^{k+1/2} + y_{2, \tau}^{k+1/2}\right)$$ in the formulation .

-   A less natural discretization is as follows $$\label{Formulation2-a}
            {{\mathcal{B}}}_1 {{\mathcal{A}}}_0^{-1/2} {{\mathcal{A}}}_{0, \tau}^{1/2}\left(\frac{\varphi_\tau^k + \varphi_\tau^{k+1}}{2}\right) + {{\mathcal{B}}}_2 \left(\frac{\varphi_\tau^{k+1} - \varphi_\tau^k}{\tau} \right),$$ which corresponds to $$\label{Formulation2}
            \hspace{-3ex}       {{\mathcal{B}}}y_\tau^{k+1/2} := \frac{i}{2} {{\mathcal{B}}}_1 {{\mathcal{A}}}_{0}^{-1/2} \left(y_{2, \tau}^{k+1/2} - y_{1, \tau}^{k+1/2}\right) + \frac{1}{2}{{\mathcal{B}}}_2 \left(y_{1, \tau}^{k+1/2} + y_{2, \tau}^{k+1/2}\right)$$ in the formulation .

Note that Theorem \[Thm-Main\], as stated, can only handle the second formulation in which the observation operator does not depend on $\tau >0$, though it corresponds to a time-discrete observation operator of the form for , which seems less natural than .

Whether or not system is observable through for general observation operators ${{\mathcal{B}}}_1, {{\mathcal{B}}}_2$ when the corresponding continuous system is observable is an open problem.

Though, if ${{\mathcal{B}}}_2 =0$, a trick allows us to get the same result as in Theorem \[Thm-Main\] for an observation operator of the form $${{\mathcal{B}}}_1\left(\frac{\varphi_\tau^k + \varphi_\tau^{k+1}}{2}\right).$$ Indeed, first apply Theorem \[Thm-Main\] to the observation . There we obtain, for $\delta >0$, $T$ as in , and initial data in $\mathfrak{C}(\delta/\tau)$ back in the variable $\varphi_\tau$: $$\begin{gathered}
        {\left\Vert\frac{\varphi_\tau^{1}-\varphi_\tau^0}{\tau}\right\Vert}_{{{\mathcal{X}}}_0}^2 
        + {\left\Vert{{\mathcal{A}}}_{0, 