Donoghue $m$-functions for singular Sturm--Liouville operators (2107.09832v1)

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 $\mathcal{N}i = \ker \big(\big(\dot A\big)^* - i I{\mathcal{H}}\big)$, $\dim \, (\mathcal{N}i)=k\in \mathbb{N} \cup {\infty}$, 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 [ \tau=\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 $b$.