Donoghue 𝑚-functions for Singular Sturm–Liouville operators

Let A ˙<!-- ˙ --> Ȧ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space H H with equal deficiency indices and denote by N i = ker ⁡<!-- ⁡ --> ( ( A ˙<!-- ˙ --> ) ∗<!-- ∗ --> −<!-- − --> i I H ) N_i = ((Ȧ)^* - i IH) , dim ⁡<!-- ⁡ --> ( N i ) = k ∈<!-- ∈ --> N ∪<!-- ∪ --> { ∞<!-- ∞ --> } (N_i)=k∈ N ∪ \{∞ \} , the associated deficiency subspace of A ˙<!-- ˙ --> Ȧ . If A A denotes a self-adjoint extension of A ˙<!-- ˙ --> Ȧ in H H , the Donoghue m m -operator M A , N i D o ( ⋅<!-- ⋅ --> ) MA,N_iDo (\,· \,) in N i N_i associated with the pair ( A , N i ) (A,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 MA,N_iDo(z)=zIN_i + (z^2+1) PN_i (A - z IH)⁻¹ PN_i N_i , z ∈<!-- ∈ --> C ∖<!-- ∖ --> R , z∈ C R, with I N i IN_i the identity operator in N i N_i , and P N i PN_i the orthogonal projection in H H onto N i 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<!-- τ --> = 1 r ( x ) [ −<!-- − --> d d x p ( x ) d d x + q ( x ) ] τ =1/r(x)[-d/dxp(x)d/dx + q(x)] for a.e. x ∈<!-- ∈ --> ( a , b ) ⊆<!-- ⊆ --> R x∈ (a,b) ⊆ 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 × 2) matrices) in all cases where τ<!-- τ --> τ is in the limit circle case at least at one interval endpoint a a or b b .