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
.