Abstract
In this paper we consider extensions of positive operators. We study the connections between the von Neumann theory of extensions and characterisations of positive extensions via decompositions of the domain of the associated form. We apply the results to elliptic second order differential operators and look in particular at examples of the Laplacian on a disc and the Aharonov–Bohm operator.
Introduction
Let A be a closed strictly positive symmetric operator with dense domain D(A) and range R(A) in a Hilbert space H. In [11, 12], Krein proved that there is a one to one correspondence between the set of positive selfadjoint extensions \(A_B\) of A and a set of pairs \(\{N_B, B\}\), where \(N_B\) is a subspace of the kernel N of \(A^{*}\) and B is a positive selfadjoint operator with domain and range in \(N_B\). Krein’s result was subsequently developed further by Visik [15] and Birman [3]; this work of the three authors will be referred to as the KVB theory. An important extension of the KVB theory was made in [8] to a pair of closed densely defined operators \(A, A'\), which form a dual pair in the sense that \(A \subset (A')^{*}\) and are such that \(A \subset A_{\beta } \subset (A')^{*}\) for an operator \(A_{\beta }\) with a bounded inverse. The results in [8] include those of KVB when \(A=A'\). Of particular interest to us in [8] is the application of the abstract theory to the case when A is generated by an elliptic differential expression acting in a bounded smooth domain \(\Omega \) in \(\mathbb {R}^n\). In this case the selfadjoint extensions of A are determined by boundary conditions on the boundary \(\partial \Omega \) of \(\Omega \).
In [5], results in Rellich [13], Kalf [9] and Rosenberger [14] were applied to the KVB theory to determine all the positive selfadjoint extensions of a positive Sturm–Liouville operator with minimal conditions on the coefficients. Our objective in this paper is to investigate what can be achieved by applying similar methods to two problems on bounded domains in \(\mathbb {R}^n, n \ge 2\); in the first A is generated by a second order elliptic expression, and in the second it is the Aharonov–Bohm operator on a punctured disc. Our analysis depends on an abstract result which incorporates the von Neumann theory concerning all the selfadjoint extensions of any symmetric operator.
Denote by \(A_F,~ a_F[\cdot ,\cdot ]\) the Friedrichs extension and associated sesquilinear form of A. Then for all \(u \in D(A_F)\) and \(v \in Q(A_F)\) we have
where \((\cdot ,\cdot )\) is the inner product of H, and \(D(A_F)\) is dense as a subspace of \(Q(A_F)\) with inner product \(a_F[\cdot ,\cdot ]\) (see [7, Chapter IV] for more on the relation between sesquilinear forms, operators and their Friedrichs extension). By the KVB theory, \(\hat{A}\) is a positive selfadjoint extension of A if and only if, \(\hat{A} = A_B\), where B is a positive selfadjoint operator acting in a subspace \( N_B\) of N and \(A_B,~B\) have associated forms \(a_B,~b\), respectively which satisfy
Thus any \(u \in Q(A_B)\) can be uniquely written as \( u=u_F + u_N\), where \( u_F \in Q(A_F),~ u_N \in Q(B) \). There are two distinguished positive selfadjoint extensions of A, namely the Friedrichs (or strong) extension \(A_F\) and the Krein–von Neumann (or weak) extension \(A_K\). These are extremal in the sense that any positive selfadjoint extension \(\hat{A}\) of A satisfies \(A_K \le \hat{A} \le A_F\) in the form sense. In (1.1), the Krein–von Neumann extension \(A_K\) corresponds to \(B=0,~ N_B = N\), and the Friedrichs extension \(A_F\) to \(B= \infty ,~Q(B) = 0,\) that is, B acts trivially on a zero dimensional space.
Positive Extensions and the Von Neumann Theory
The von Neumann theory characterises the selfadjoint extensions of any closed densely defined symmetric operator T. Denoting the deficiency spaces \( \text {ker}(T^{*}\mp iI)\) by \(N_{\pm }\), we have
and \(T_S\) is a selfadjoint extension of T if and only if there is a unitary operator \(U(T_S){:}\,N_+ \rightarrow N_\) such that
Let \(u, v \in D(T^{*})\). Then by the von Neumann theory, there exist unique \(u_0,v_0 \in D(T)\) and \(u_\pm ,v_\pm \in N_\pm \) such that \(u= u_0+u_+ + u_\) and \(v= v_0+v_+ + v_\).
It follows that
Let \(P_+\) and \(P_\) denote the projections from \(D(T^{*})\) to \(N_+\) and \(N_\) with respect to the decomposition (2.1) and let \(U{:}\,N_+\rightarrow N_\) be unitary. Set \(\tilde{\Lambda }_0 =UP_++P_\) and \(\tilde{\Lambda }_1=iUP_++iP_\). Then, for any \(u,v\in D(T^{*})\)
(see [10, Theorem 3]). The triple \((N_+, \tilde{\Lambda }_0,\tilde{\Lambda }_1)\) is a boundary triple (also known as a space of boundary values) for T.
Given a selfadjoint extension \(T_S\) of T, we now choose
Then, from (2.2), \(\ker \Lambda _1(T_S) = \mathcal D(T_S)\) and we obtain, for all \(u,v \in D(T^{*})\)
Let \(T=A\) be positive and B a positive selfadjoint operator on a subspace \(N_B\) of the kernel of \(A^{*}\) with domain D(B). By [2, Theorem 3.1], the domain of the selfadjoint extension \(A_B\) of A corresponding to B is
Remark 2.1
The special case \(B=0, N_B =N\) gives the domain of the Krein–von Neumann extension \(A_K\), namely
the sum being a direct sum since A is strictly positive. It follows that
The Friedrichs extension is characterised by the choice of B as acting trivially on \(N_B=\{0\}\). Following the approach of [2], we can set \( b[u] = \infty \) for \(u \in N{\setminus } Q(B)\). It follows from (1.1) that \(Q(A_B) =Q(A_F )\) if and only if \(Q(B) = \{0\}\). Since \(A_F\) is the only selfadjoint extension of A with domain in \(Q(A_F)\) it follows that its domain is determined by \(b[u] = \infty \) for all \(u \in N {\setminus }\{0\}\).
Theorem 2.2
Let \(A_B\) be a positive selfadjoint extension of the positive operator A associated with the pair \(\{B,N_B\}\). Let \(u \in D(A_B)\), where \(u=u_F + w\), \(u_F = u_0 +A_F^{1}(Bw+v), u_0 \in \mathcal {D}(A), w \in \mathcal {D}(B), v \in N \cap \mathcal {D}(B)^{\bot }\). Then
where \(\Lambda _0(A_B) = U(A_B)P_+ +iP_\) and \(\Lambda _1(A_B) =iU(A_B)P_+ +i P_\).
Proof
Let \(\varphi =\theta +\zeta \in Q(A_B)\) with \(\theta \in Q(A_F)\) and \(\zeta \in Q(B)\). Then on the one hand, we have
since \( w\in N\), and on the other hand,
On combining (2.11) and (2.12) we get
and as \(A^{*}\zeta =0\), Eq. (2.6) yields
Since \(\ker \Lambda _1(A_B) = D(A_B)\), (2.10) follows. \(\square \)
Let \(\{\psi _k\}\) be an orthonormalbasis of Q(B), where B is a positive selfadjoint operator in \(N_B\subset N\), and let \(w =\sum _j w_j \psi _j,~ \zeta =\sum \zeta _k\psi _k\) and \(b_{jk} =b[\psi _j,\psi _k]\). Then \(b[w,\zeta ] = \sum _{j,k} b_{jk}w_j \overline{\zeta _k}\) and from (2.10) and the fact that \(\ker \Lambda _1(A_B) = D(A_B)\), \( u = u_F + w \in D(A_B)\) if and only if
Elliptic Differential Operators of Second Order
In this section we shall apply the above abstract theory to the case when A is the closure of a symmetric secondorder differential operator in \(L^2(\Omega )\) defined by
subject to conditions on the coefficients \(p_{ij},~q\) and the domain \(\Omega \). The assumptions are the ones made in [1] which weaken the smoothness requirements on the coefficients and the boundary of \( \Omega \) made by Grubb [8]. In the following definition of a boundary regularity class, \(B^{s}_{p,q}\) is the Besov space of order s (see [1, Section 2]), and we set \(x=(x',x_n), x'\in \mathbb {R}^{n1}, x_n \in \mathbb {R}\).
Definition 3.1
The boundary \(\partial \Omega \) is said to be of class \(B^{M\frac{1}{2}}_{p,q}\) if for each \(x\in \partial \Omega \) there exist an open neighbourhood U satisfying the following: for a suitable choice of coordinates on \(\mathbb {R}^n\), there is a function \(\gamma \in B^{M\frac{1}{2}}_{p,q}(\mathbb {R}^{n1})\) such that \(U\cap \Omega = U\cap \mathbb {R}^n_\gamma \) and \(U\cap \partial \Omega = U\cap \partial \mathbb {R}^n_{\gamma } \), where \( \mathbb {R}^{n}_{\gamma } = \{x \in \mathbb {R}^n{:}\,x_n > \gamma (x')\}\).
In the list of assumptions to be made, we shall denote the boundary of \(\Omega \) by \(\Sigma \), and \( H^s_t\) is a Bessel potential space (a Sobolev space for \(s \in \mathbb {N}\)), which we write as \(H^s\) when \(t=2\); see [1, Section 2] for definitions of \(H^s_t(\Omega ) \) and \(H^s_t(\Sigma )\).
Assumptions

1.
There exists \(c_0>0\) such that for all \(x\in \Omega \) and \(\xi \in \mathbb {R}^{n}\)
$$\begin{aligned} \sum _{i,j=1}^n p_{ij}(x)\xi _i\xi _j dx \ge c_0 \Vert \xi \Vert ^2. \end{aligned}$$ 
2.
There exists \(c >0\) such that
$$\begin{aligned} \Vert u\Vert ^2_1 = \int _{\Omega } \left( p \nabla u^2 + qu^2 \right) dx \ge c \Vert u\Vert ^2,\ \ \ u \in C_0^{\infty }(\Omega ). \end{aligned}$$The completion of \(C_0^{\infty }(\Omega )\) with respect to the norm \( \Vert \cdot \Vert _1 \) is the form domain \(Q(A_F)\) of A.

3.
The boundary \(\Sigma \) is of class \(B^{\frac{3}{2}}_{r,2}\) and the coefficients p and q of A lie in \(H^1_t(\Omega )\) and \(L_t(\Omega )\), respectively, under the constraints \(n\ge 2\), \(2< r<\infty \), \(2< t \le \infty \), and
$$\begin{aligned} 1\tfrac{n}{t}\ge \tfrac{1}{2}\tfrac{n1}{r}> 0. \end{aligned}$$(3.2)
Remark 3.2
Our third assumption is Assumption 2.18 in [1]. Therefore, we have that for \(v \in Q(A_F)\), \(\gamma _0 v =0\), where \(\gamma _0\) is the trace operator which maps v into its value on \(\Sigma \) (see [1, Theorem 2.11]). Moreover, in the notation of [1, 6], denote the solution of
by
Then by [1, Theorem 5.4], for all \( s \in [0,2]\),
is continuous,
is a homeomorphism, and
We remark that under the more restrictive assumptions that \(\Omega \) is a bounded domain whose boundary is an \((n1)\)dimensional \(C^{\infty }\) manifold, and the coefficients \(p_{jk},~q\) of \(A'\) in (3.1) lie in \(C^{\infty }(\overline{\Omega })\) these properties were already shown by Grubb in [8].
Theorem 3.3
Let the above assumptions hold and let \(A_B\) be a positive selfadjoint extension of A. For \(u \in D(A_B)\), we have \(u=u_F+w\) for some \(u_F \in D(A_F),~w \in Q(B)\), and for all \(\zeta \in Q(B)\)
If \(\{\psi _k\}\) is an orthonormal basis of Q(B) then, with \(b_{jk}\) as in (2.14),
Proof
Let \(a_B[\cdot ,\cdot ], a_F[\cdot ,\cdot ], b[\cdot ,\cdot ]\) denote the forms associated with \( A_B, A_F, B\), respectively. For \( u, \varphi \in Q(A_B) \) we have the decompositions
If \( u \in D(A_B)\), it has the decomposition \(u=A_F^{1}A^{*}u+(uA_F^{1}A^{*}u)\), i.e., \(u_F=A_F^{1}A^{*}u\) and \(w=uA_F^{1}A^{*}u,\) since \(u_F\in D(A^{*})\cap Q(A_F) = D(A_F)\) and \(w\in Q(A_B)\cap N=Q(B)\). Now, let \(\varphi =\varphi _F+\zeta \in Q(A_B)\). Then
and furthermore,
Combining (3.11) and (3.12), we get
Now let \(\{\psi _k\}\) be an orthonormalbasis of Q(B), \(w=\sum w_j\psi _j\) and \(\zeta =\sum \zeta _k\psi _k\). Then
while the righthand side of (3.13) is
Consequently
and equivalently,
As \(\{\zeta _k\}\) is an arbitrary sequence in \( \ell ^2\), the ‘boundary condition’ associated with \(A_B\) is given by
\(\square \)
Corollary 3.4
The boundary condition associated with \(A_B\) is given by
where \(\nu _1 u = \sum _{j,k =1}^n n_j \gamma _0[p_{jk}D_k u]\) and \(\mathbf{n} = (n_1,\ldots ,n_n)\) is the interior unit normal.
Proof
Since \(u,\psi _k\in D(A^{*})\), we can use [1, Theorem 6.1] to write
where \( \Gamma _1\) is the “regularised” Neumann operator given by \(\Gamma _1 u = \nu _1 u  P_{\gamma _0, \nu _1} \gamma _0 u\) and \(P_{\gamma _0, \nu _1} \gamma _0 \) is the DirichlettoNeumann map \(P_{\gamma _0, \nu _1} \gamma _0 = \nu _1 K_{\gamma }^0\). Under the smoothness conditions assumed, the trace maps \(\gamma _0,~ \Gamma _1\), map \( D(A^{*})\) continuously into \( H^{1/2}(\Sigma ),~ H^{1/2}(\Sigma )\), respectively. The terms on the righthand side of (3.16) therefore represent in fact, \(H^{1/2},H^{1/2}\)duality products over the boundary \(\Sigma \), which are extensions of the \(L^2(\Sigma )\) inner products (see [1, Theorem 6.1]).
Since \((\nabla p \nabla q )\psi _k=0\) and \(u_F\in Q(A_F)\), two of the four terms in (3.16) vanish and, as \(\Gamma _1 u_F=\nu _1 u_F\), we get that the boundary condition in (3.14) becomes
\(\square \)
Remark 3.5
The Friedrichs extension is determined by the boundary condition \(\gamma _0 u=0\). Under the additional smoothness assumptions on \(\Omega \) and the coefficients of \(A'\) in (3.1) in [8], the Friedrichs extension has domain \(H_0^1(\Omega ) \cap H^2(\Omega )\).
Remark 3.6
The Krein–von Neumann extension corresponds to \(B=0, Q_B = N_B = N =\text {ker}(A^{*})\) and so
when \( u=u_F+w, u_F \in Q(A_F), w \in N\). Thus in (3.15), \(b_{jk} =0\) for all j, k and \(\nu _1 u_F = \Gamma _1 u_F \). Since \(\nu _1 \) maps \(D(A^{*})\) continuously into \(H^{1/2}(\Sigma )\) and \(\gamma _0\) is a homeomorphism of N onto \(H^{1/2}(\Sigma )\), it follows from (3.15) that the boundary condition satisfied by the Krein–von Neumann extension is
Since \( w = K_{\gamma }^0 \gamma _0 u\) we have
Remark 3.7
On combining (2.10) and (3.17) we have
For the Krein–von Neumann extension \(\Lambda _K\psi _k=0\), so we again get
as the Krein–von Neumann boundary condition.
Example 3.8
We consider extensions of the positive operator \(A=\Delta +1\) when \(\Omega \) is the unit disc in \(\mathbb {R}^2\). According to (3.15), \(v=v_F+w\) lies in the domain of an extension \(A_B\) if and only if
for all k, where \(\{\psi _k\}\) is an orthonormalbasis of the subspace Q(B) in \(N=\ker A^{*}\).
Let \(\,\Delta \psi + \psi =0\) and put \(\psi (r, \theta )=R(r) \Theta (\theta )\), where \(x =(r,\theta )\) are polar coordinates. Then since
we get
and \(\dfrac{\Theta ''}{\Theta }=n^2\) with constant n and \(\Theta (0)=\Theta (2\pi )\); thus \( \Theta _n(\theta ) = e^{in\theta }, n\in \mathbb {Z}\) and we seek the \(L^2(0,1;rdr)\) solutions of
These solutions are given by the modified Bessel functions \(I_n(r)\) and \(K_n(r)\).
For \( n \ge 1\), \(K_n(r)\) does not lie in \(L^2(0,1;rdr)\). The function \(K_0(r)\) has a logarithmic singularity at 0, which means that \(\Delta K_0\) is not zero in the sense of distributions, excluding \(K_0\) from N. Therefore
is a basis for N; note that \(I_{k}=I_k\).
For \(k\in \mathbb {Z}\)
and since \(v_F\in D(A_F)\), we have \(\nu _1 v_F=\frac{\partial v_F}{\partial \nu }\). On expanding \(v_F\) in \(\theta \) in terms of its Fourier series,
we derive
Consequently \(v=v_F+w\in D(A_B)\) if and only if for all \(k\in \mathbb {Z}\)
Remark 3.9

1.
For the Krein–von Neumann extension, \(v=v_F+w\in D(A_K)\) if and only if for all \(k\in \mathbb {Z}\) we have
$$\begin{aligned} 0 = 2\pi \frac{\partial v_{F,k}}{\partial r}(1) I_k(1). \end{aligned}$$As \(I_k(1)\ne 0\) for all \(k\in \mathbb {Z}\), this implies that
$$\begin{aligned} v_F(1,\theta ) = \frac{\partial v_{F}}{\partial r}(1,\theta ) = 0 \end{aligned}$$and hence \(v_F\in D(A)\). As there are no restrictions on w, we get \(D(A_K)= D(A) + N\), as expected. Also, the boundary condition satisfied by any \(u \in D(A_K)\) is \( \Gamma _1 u = 0\), where \( \Gamma _1 = \nu _1  P_{\gamma _0, \nu _1} \gamma _0 \) is the regularised Neumann operator.

2.
For the Friedrichs extension, we formally have \(b_{jk}=\infty \) for all j, k in (3.19). This implies that w must be orthogonal to all the \(\psi _k\). As \(w\in N\), this gives \(w=0\).
Aharonov–Bohm Operator
Let \(\Omega = \{x{:}\,x <1\} {\setminus } \{0\} \subset \mathbb {R}^2\), and let A be the closure in \(L^2(\Omega )\) of \(A'\upharpoonleft _{C_0^{\infty }(\Omega )}\), where
Here, the Aharonov–Bohm magnetic potential
where \(x=(r \cos ~\theta , r\sin ~\theta )\) in polar coordinates and \(e_{\theta } = (\sin ~\theta , \cos ~\theta )\) is the unit vector orthogonal to \(e_r = x/r\). Then
For \(u \in C_0^{\infty }(\Omega )\) we have
The sequence \(\{\varphi _k(\theta ): k \in \mathbb {Z}\}\), where \(\varphi _k(\theta ) = \frac{e^{ik\theta }}{\sqrt{2\pi }}\), is an orthonormal basis for \(L^2(0,2\pi )\) and hence any \(u \in L^2(\Omega )\) has the representation
where
On substituting in (4.3), we have, with \(\lambda _k=k+\alpha \)
Since \( \min \{\lambda _k/r{:}\,k \in \mathbb {Z},\ 0<r<1 \} \ge \min \{\alpha ,1\alpha \} >0\), it follows that A is strictly positive and its form domain \(Q(A_F)\) is the completion of \(C_0^{\infty }(\Omega )\) with respect to the norm given by the square root of
Let B be a positive selfadjoint operator acting in a closed subspace \(N_B\) of \(N= \text {ker}~A^{*}\) which is associated with the selfadjoint extension \(A_B\) of A in the KVB theory, and let \(a_B[\cdot ,\cdot ], a_F[\cdot ,\cdot ], b[\cdot ,\cdot ]\) be the forms of \(A_B, A_F, B\), respectively.
For \(u, \varphi \in Q(A_B)\), we have
and since \(v(R,\theta ) = \vartheta (R,\theta ) = 0\) (which follows from the definition of \(Q(A_F)\)),
Remark 4.1
Since \(v(1, \theta ) =0\) for any \(v \in Q(A_F)\), \(Q(A_F)\) coincides with Brasche and Melgaard’s form domain of \(A_F\) in [4], and so \(A_F\) is determined in their Theorem 4.5.
We now proceed as in the proof of Theorem 2.2. For \(u=u_F+w \in D(A_B)\) and \( \varphi = \vartheta + \zeta \in Q(A_B)\)
and
Consequently
If \( \{\psi _k\}\) is an orthonormal basis of Q(B), then we have with the same notation as in Sect. 3, that \(u = u_F + w \in D(A_B)\) if and only if
The transformation
is a unitary operator from \(L^2(0,1;rdr)\) onto \( L^2(0,1)\), and as \(\{e^{im\theta }/\sqrt{2 \pi }\}_{m \in \mathbb {Z}}\) is an orthonormal basis of \(L^2(\mathbb {S}^1)\) we have
In terms of this decomposition it follows that
where \(T^{(m)}\) is the closure in \(L^2(0,1)\) of the operator defined on \(C_0^{\infty }(0,1)\) by the Sturm–Liouville expression
i.e., \(T^{(m)}\) is the minimal operator in \(L^2(0,1)\) generated by \(\tau ^m\). With \(\nu = m+\alpha \), the set \(\{r^{1/2+\nu }, r^{1/2\nu }\}\) is a fundamental system for \( \tau ^m u=0\). The expression \(\tau ^m\) is nonoscillatory. For \(m= 1,~0\), it is in the limitcircle case at 0; for all other values of m, it is in the limitpoint case at 0. It is regular at 1 for all values of m. Thus \(T^{(m)}\) has deficiency indices (2, 2) for \( m=1,~0\) and (1, 1) otherwise. We shall now apply results from [5] to determine the positive selfadjoint extensions of \(T^{(m)}\) in \(L^2(0,1)\) for all \(m \in \mathbb {Z}\). Note that the singular point here is at the left endpoint of the interval [0, 1], i.e., it is the point 0, unlike the analysis of [5], where it is at the right endpoint. If \(S^{(m)}\) is one such extension, then
is a positive selfadjoint extension of A.
Remark 4.2
We note that it is unlikely that all positive selfextensions of A are obtained in this way. This assertion is based on the situation for \(A_0 = \Delta +1\) from Example 3.8. As in (4.11),
where now, \(T_{(m)}\) is the minimal operator generated by
At 0, \( \tau _m\) is nonoscillatory and in the limitcircle case for \(m=0\) and is otherwise limitpoint. As above for A, if \(S_{(m)}\) is a positive selfadjoint extension of \(T_{(m)}\) then
is a positive selfadjoint extension of \(A_0\). All such extensions have boundary conditions which depend on behaviour at 0, in view of the presence of the extension \(S_{(0)}\) of \(T_{(0)}\) which has deficiency indices (1, 1). However in Remark 3.9 we saw that this is not so for the Krein–von Neumann extension of \(A_0!\)
We shall proceed to determine the extensions \(T^{(m)}\) in (4.11).
The Case when \(\tau ^m\) is Limit Point at 0 (\(m \ne 1,0\))
Theorem 2.1 in [5] establishes a oneone correspondence between the positive selfadjoint extensions of \(T^{(m)}\) in this case and the oneparameter family \(\{T^{(m)}_l\}\), \(0 \le l \le \infty \) of restrictions of \((T^{(m)})^{*}\) to the domains
Here \(\psi \) is a real function in \(L^2(0,1)\) which satisfies \(\tau ^m \psi =0\) and \( \psi (1) =1.\) We therefore have
The Case when \(\tau ^m\) is LimitCircle at 0 (\(m=1,0\)) and \(\mathbf dim ~N_B =1\)
From Theorem 2.2 in [5] and writing \(T^{*}\) instead of \(\left( T^{(m)}\right) ^{*}\) for simplicity, it follows that the positive selfadjoint extensions of the operator \(T^{(m)}\) which correspond to the pair \(\{B,N_B\}\) in the KVB theory with \(\text {dim}N_B =1\) form a oneparameter family \(T_{\beta }\) of restrictions of \(T^{*}\) with domains
where \(\psi \) is a real function in \(N_B\) with \(\psi (1) =1\), g is the nonprincipal solution of \(\tau ^m u=0\) and \(\beta \ge 0\). The nonprincipal solution is \(r^{1/2  \nu },~ \nu = m+\alpha \). The Wronskian W is given by
The limits at 0 of the first and the second terms in (4.17) exist separately. To see this, let
Hence by the Jacobi identity [5, Equation (1.10)], for \(v\in \mathcal {D}(T^{*})\) we get
Thus, since \( g \in L^2(0,1)\),
which implies that
both exist. From [9, Remark 3] (see also [5, (2.9)]), \(\lim _{r \rightarrow 0+}(v/g)(r) \) exists, which confirms our assertion that the separate limits exist.
We shall now determine the boundary conditions satisfied by the selfadjoint extensions of \(T^{(m)}\) in the two cases corresponding to \( \nu = m + \alpha ,\ m = 1, 0,\ \alpha \in (0,1)\).
The Case \(m = 1,\ \nu =1 +\alpha \in (1,0)\)
In this case, the nonprincipal solution is \( g(r) = r^{1/2 +\nu }\) and \(\psi (r) = \gamma \left( C_1 r^{1/2\nu } + C_2 r^{1/2+\nu }\right) \), where \( \gamma = (C_1+C_2)^{1}\) for \(C_1,C_2\) are constants and \(C_1\ne 0\). Thus,
and so using (4.18)
The value at \(r=1\) is
By (4.20) and since \( \nu < 0\), the limits at 0 of both terms in (4.21) exist and
Thus the boundary condition for \(A_B\) in this case is
where \( f_1(r) := \left[ r^{1/2+\nu } v'(r) (1/2+\nu ) r^{1/2 +\nu } v(r)\right] \).
The Case \(m =0,\ \nu = \alpha \in (0,1)\)
This time, the nonprincipal solution is \(g(r) = r^{1/2\nu }\) and, with \(\psi \) as above, we have
giving
Therefore
By (4.25) and since \( \nu >0\), both limits at 0 in (4.26) exist and
Thus the boundary condition in this case is
where \( f_2(r) := \left[ r^{1/2\nu } v'(r) (1/2\nu ) r^{1/2\nu }v(r)\right] \).
The Case when \(\tau ^m\) is LimitCircle at 0 (\(m=1,0\)) and \(\hbox {dim}N_B=2\)
From [5, Theorem 2.2], we have
where \(B:= (b_{jk})_{j,k=1,2}\) is a matrix of parameters which is nonnegative, \(\{\psi _1, \psi _2\}\) is a real orthonormal basis of \(N_B\) and \(c_1,~c_2\) are determined by the values of v / g at 0 and 1:
The main difference from the analysis of the previous section is that we now replace \(\psi \) by an orthonormal basis \((\psi _1,\psi _2)\) obtained from the linearly independent basis elements
On using the Gram–Schmidt procedure, we obtain the orthogonal vectors
and the orthonormal system
The nonprincipal solution is \(g(r) = r^{1/2 \nu }\) and we have
and
Let
Then
and we set
Also
giving
We also have
Setting \(V=\Psi c\), where
is invertible and has inverse
The boundary condition in (4.30) is therefore
For the Krein–von Neumann extension \(\left( T^{(m)}\right) _K\) of the onedimensional operator, the boundary condition is determined by \(\Theta _1 = \Theta _2 = 0\): Hence
and
Following Remark 2.1, the Friedrichs extension \(\left( T^{(m)}\right) _F\) is obtained by \(c_1=c_2=0\), so that the right hand side of (4.30) is finite. Thus, from (4.31), the boundary conditions are given by \((v/g)(0)=(v/g)(1)=0\), i.e.,
References
Abels, H., Grubb, G., Wood, I.G.: Extension theory and Kreĭntype resolvent formulas for nonsmooth boundary value problems. J. Funct. Anal. 266(7), 4037–4100 (2014)
Alonso, A., Simon, B.: The Birman–Kreĭn–Vishik theory of selfadjoint extensions of semibounded operators. J. Oper. Theory 4(2), 251–270 (1980)
Birman, M.Š.: On the theory of selfadjoint extensions of positive definite operators. Mat. Sb. N.S. 38(80), 431–450 (1956)
Brasche, J.F., Melgaard, M.: The Friedrichs extension of the Aharonov–Bohm Hamiltonian on a disc. Integral Equ. Oper. Theory 2(3), 419–436 (2005)
Brown, B.M., Evans, W.D.: Selfadjoint and \(m\) sectorial extensions of Sturm–Liouville operators. Integral Equ. Oper. Theory 85(2), 151–166 (2016)
Brown, B.M., Grubb, G., Wood, I.G.: Mfunctions for closed extensions of adjoint pairs of operators, with applications to elliptic boundary problems. Math. Nachr. 282, 314–347 (2009)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Clarendon Press, Oxford (1989)
Grubb, G.: A characterization of the nonlocal boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup. Pisa 3(22), 425–513 (1968)
Kalf, H.: A characterization of the Friedrichs extension of Sturm–Liouville operators. J. Lond. Math. Soc. (2) 17(3), 511–521 (1978)
Kochubei, A.N.: Extensions of symmetric operators and symmetric binary relations. Math. Notes 17(1), 25–28 (1975)
Kreĭn, M.: The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications. I. Rec. Math. [Mat. Sb.] N.S. 20(62), 431–495 (1947)
Kreĭn, M.: The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications. II. Mat. Sb. N.S. 21(63), 365–404 (1947)
Rellich, F.: Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung. Math. Ann. 122, 343–368 (1951)
Rosenberger, R.: A new characterization of the Friedrichs extension of semibounded Sturm–Liouville operators. J. Lond. Math. Soc. (2) 31(3), 501–510 (1985)
Vishik, M.: On general boundary conditions for elliptic differential operators. Trudy Moskov. Mat. Obsc. (Russian). (English translation In: Amer. Math. Soc. Transl. 24, 107–172), 187–246 (1952)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The Authors are grateful to the anonymous referees for providing useful comments that helped improve the paper.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Brown, B.M., Evans, W.D. & Wood, I.G. Positive Selfadjoint Operator Extensions with Applications to Differential Operators. Integr. Equ. Oper. Theory 91, 41 (2019). https://doi.org/10.1007/s0002001925404
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s0002001925404
Mathematics Subject Classification
 Primary 47A20
 Secondary 35J15
 47A07
 47B25
 47F05
Keywords
 Operator extensions
 Von Neumann theory
 Sesquilinear form
 Elliptic operators
 Aharonov–Bohm operator