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 $\pm \mathrm{\infty }$). 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 $\mathrm{\parallel }V(\cdot ){\mathrm{\parallel }}_{\mathcal{B}(\mathcal{H})}$ 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.