Donoghue $m$-functions for singular Sturm--Liouville operators

Abstract

Let $\dot{A}$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by ${\mathcal{N}}_i=\ker \left(\right(\dot{A}{)}^{\ast }-i{I}_{\mathcal{H}})$, $\dim ({\mathcal{N}}_i)=k\in \mathbb{N}\cup \{\infty \}$, the associated deficiency subspace of $\dot{A}$ . If $A$ denotes a self-adjoint extension of $\dot{A}$ in $\mathcal{H}$, the Donoghue $m$-operator $M_{A,{\mathcal{N}}_i}^{Do}(\cdot )$ in ${\mathcal{N}}_i$ associated with the pair $(A,{\mathcal{N}}_i)$ is given by \[ M_{A,\mathcal{N}_i}^{Do}(z)=zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A - z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i} \big\vert_{\mathcal{N}_i}\,, \quad z\in \mathbb{C} \backslash \mathbb{R}, \] with $I_{{\mathcal{N}}_i}$ the identity operator in ${\mathcal{N}}_i$, and $P_{{\mathcal{N}}_i}$ the orthogonal projection in $\mathcal{H}$ onto ${\mathcal{N}}_i$. Assuming the standard local integrability hypotheses on the coefficients $p,q,r$, we study all self-adjoint realizations corresponding to the differential expression \[ τ=\frac{1}{r(x)}\left[-\frac{d}{dx}p(x)\frac{d}{dx} + q(x)\right] \, \text{ for a.e. $x\in (a,b)\subseteq \mathbb{R}$,} \] in $L^2((a,b);rdx)$, and, as the principal aim of this paper, systematically construct the associated Donoghue $m$-functions (resp., $2\times 2$ matrices) in all cases where $\tau$ is in the limit circle case at least at one interval endpoint $a$ or $b$.

35 pages. arXiv admin note: text overlap with arXiv:1910.13117, arXiv:2102.00685