Donoghue-Type $m$-Functions for Schrödinger Operators with Operator-Valued Potentials

Abstract

Given a complex, separable Hilbert space $\mathcal{H}$, we consider self-adjoint $L^2$-realizations of differential expressions $\tau =-(d^2/d{x}^2)I_{\mathcal{H}}+V(x)$, 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 $\Vert V(\cdot ){\Vert }_{\mathcal{B}(\mathcal{H})}$ is locally integrable, and $V(\cdot )=V(\cdot {)}^{\ast }$ 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 ${\mathcal{N}}_z=\ker ({\dot{A}}^{\ast }-z{I}_{\mathcal{H}})$ and corresponding orthogonal projections $P_{{\mathcal{N}}_z}$ onto ${\mathcal{N}}_z$ is given by $$M_{A,{\mathcal{N}}_i}^{Do}(z)=z{I}_{{\mathcal{N}}_i}+(z^2+1)P_{{\mathcal{N}}_i}(A-z{I}_{\mathcal{H}}{)}^{-1}{P}_{{\mathcal{N}}_i}{|}_{{\mathcal{N}}_i},\mathrm{Im}(z)\ne 0.$$ For half-line and full-line Schrödinger operators, the role of $\dot{A}$ is played by a suitably defined minimal Schrödinger 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 Schrödinger operators encode the entire spectral information of the latter.

44 pages. arXiv admin note: substantial text overlap with arXiv:1301.0682, arXiv:1109.1613