Donoghue 𝑚-functions for Singular Sturm–Liouville operators

Let A ˙ \dot {A} be a densely defined, closed, symmetric operator in the complex, separable Hilbert space H \mathcal {H} with equal deficiency indices and denote by N i = ker ⁡ ( ( A ˙ ) ∗ − i I H ) \mathcal {N}_i = \ker ((\dot {A})^* - i I_{\mathcal {H}}) , dim ⁡ ( N i ) = k ∈ N ∪ { ∞ } \dim (\mathcal {N}_i)=k\in \mathbb {N} \cup \{\infty \} , the associated deficiency subspace of A ˙ \dot {A} . If A A denotes a self-adjoint extension of A ˙ \dot {A} in H \mathcal {H} , the Donoghue m m -operator M A , N i D o ( ⋅ ) M_{A,\mathcal {N}_i}^{Do} (\,\cdot \,) in N i \mathcal {N}_i associated with the pair ( A , N i ) (A,\mathcal {N}_i) is given by M A , N i D o ( z ) = z I N i + ( z 2 + 1 ) P N i ( A − z I H ) − 1 P N i | N i 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 ∈ C ∖ R , z\in \mathbb {C}\setminus \mathbb {R}, with I N i I_{\mathcal {N}_i} the identity operator in N i \mathcal {N}_i , and P N i P_{\mathcal {N}_i} the orthogonal projection in H \mathcal {H} onto N i \mathcal {N}_i .

Assuming the standard local integrability hypotheses on the coefficients p , q , r p, q,r , we study all self-adjoint realizations corresponding to the differential expression τ = 1 r ( x ) [ − d d x p ( x ) d d x + q ( x ) ] \tau =\frac {1}{r(x)}[-\frac {d}{dx}p(x)\frac {d}{dx} + q(x)] for a.e. x ∈ ( a , b ) ⊆ R x\in (a,b) \subseteq \mathbb {R} , in L 2 ( ( a , b ) ; r d x ) L^2((a,b); rdx) , and, as our principal aim in this paper, systematically construct the associated Donoghue m m -functions (respectively, ( 2 × 2 ) (2 \times 2) matrices) in all cases where τ \tau is in the limit circle case at least at one interval endpoint a a or b b .