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 $\mathcal {N}_i = \ker ((\dot {A})^* - i I_{\mathcal {H}})$, $\dim (\mathcal {N}_i)=k\in \mathbb {N} \cup \{\infty \}$, 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 $M_{A,\mathcal {N}_i}^{Do}(z)=zI_{\mathcal {N}_i} + (z^2+1) P_{\mathcal {N}_i} (A - z I_{\mathcal {H}})^{-1} P_{\mathcal {N}_i} \vert _{\mathcal {N}_i}$, $z\in \mathbb {C}\setminus \mathbb {R},$ 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 $\tau =\frac {1}{r(x)}[-\frac {d}{dx}p(x)\frac {d}{dx} + q(x)]$ for a.e. $x\in (a,b) \subseteq \mathbb {R}$, in $L^2((a,b); rdx)$, and, as our principal aim in this paper, systematically construct the associated Donoghue $m$-functions (respectively, $(2 \times 2)$ matrices) in all cases where $\tau$ is in the limit circle case at least at one interval endpoint $a$ or $b$.