  Given a complex, separable Hilbert space $\mathcal{H}$, we consider
self-adjoint $L^2$-realizations of differential expressions $\tau = -
(d^2/dx^2) I_{\mathcal{H}} + V(x)$, on half-lines and on the real line
(assuming the limit-point property of $\tau$ at $\pm \infty$). Here $V$ denotes
a bounded operator-valued potential $V(\cdot) \in \mathcal{B}(\mathcal{H})$
such that $V(\cdot)$ is weakly measurable, the operator norm
$\|V(\cdot)\|_{\mathcal{B}(\mathcal{H})}$ is locally integrable, and $V(\cdot)
= V(\cdot)^*$ a.e.
  In a nutshell, a Donoghue-type $m$-function $M_{A,\mathcal{N}_i}^{Do}(\cdot)$
associated with self-adjoint extensions $A$ of a closed, symmetric operator
$\dot A$ in $\mathcal{H}$ with deficiency spaces $\mathcal{N}_z = \ker
\big({\dot A}^* - z I_{\mathcal{H}}\big)$ and corresponding orthogonal
projections $P_{\mathcal{N}_z}$ onto $\mathcal{N}_z$ is given by $$
M_{A,\mathcal{N}_i}^{Do}(z) = zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A
- z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i}\big\vert_{\mathcal{N}_i} \,, \quad
{\rm Im}(z)\neq 0. $$ For half-line and full-line Schr\"odinger operators, the
role of $\dot A$ is played by a suitably defined minimal Schr\"odinger operator
which will be shown to be completely non-self-adjoint. The latter property is
used to prove that the corresponding operator-valued measures in the
Herglotz--Nevanlinna representations of the Donoghue-type $m$-functions
corresponding to self-adjoint half-line and full-line Schr\"odinger operators
encode the entire spectral information of the latter.
