'Transmutation techniques and observability for time-discrete approximation schemes of conservative systems[^3]'
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Introduction
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Setting
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Assume that ${{\mathcal{A}}}$ is a skew-adjoint unbounded operator defined in a Hilbert space ${{\mathcal{X}}}$ with dense domain $\mathcal{D}({{\mathcal{A}}}) \subset {{\mathcal{X}}}$ and compact resolvent. We consider the equation $$\label{FD-Abstract}
        y' = {{\mathcal{A}}}y, \quad t \in {\mathbb{R}}, \qquad y(0) = y^0 \in {{\mathcal{X}}}.$$

This equation is well-posed for $t \in {\mathbb{R}}$ and, ${{\mathcal{A}}}$ being skew-symmetric, solutions $y(t)$ of have constant norms ${\left\Verty(t)\right\Vert}_{{\mathcal{X}}}$. Often ${\left\Verty(t)\right\Vert}_{{\mathcal{X}}}^2/2$ is referred to as the energy of the system.

Several classical systems fit this abstract setting, as for instance the wave equation, Schrödinger’s and Maxwell’s equations, among many others.

Our primary goal is to establish a link between the solutions $y$ of and the solutions of time-discrete versions of . Since the solutions $y$ of have constant energy, we will reduce our analysis to time-discretization schemes which preserve the ${{\mathcal{X}}}$-norm ${\left\Vert\cdot\right\Vert}_{{\mathcal{X}}}$ of the solutions.

To state our results precisely, we need to introduce filtered spaces of solutions. For, we write down the spectral decomposition of the operator ${{\mathcal{A}}}$, which is given by a sequence of purely imaginary eigenvalues $(i \mu_{j})$, corresponding to an orthonormal basis of ${{\mathcal{X}}}$ constituted by the eigenvectors $\Phi_j$. 