Donoghue-Type $m$-Functions for Schr\"odinger Operators with Operator-Valued Potentials

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.