# Positive Self-adjoint Operator Extensions with Applications to Differential Operators

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## 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.

## Keywords

Operator extensions Von Neumann theory Sesquilinear form Elliptic operators Aharonov–Bohm operator## Mathematics Subject Classification

Primary 47A20 Secondary 35J15 47A07 47B25 47F05## 1 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 self-adjoint 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 self-adjoint 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 self-adjoint 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 self-adjoint 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 self-adjoint extensions of any symmetric operator.

*A*. Then for all \(u \in D(A_F)\) and \(v \in Q(A_F)\) we have

*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 self-adjoint extension of

*A*if and only if, \(\hat{A} = A_B\), where

*B*is a positive self-adjoint operator acting in a subspace \( N_B\) of

*N*and \(A_B,~B\) have associated forms \(a_B,~b\), respectively which satisfy

*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 self-adjoint 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.

## 2 Positive Extensions and the Von Neumann Theory

*T*. Denoting the

*deficiency spaces*\( \text {ker}(T^{*}\mp iI)\) by \(N_{\pm }\), we have

*T*if and only if there is a unitary operator \(U(T_S){:}\,N_+ \rightarrow N_-\) such that

*T*.

*T*, we now choose

*B*a positive self-adjoint operator on a subspace \(N_B\) of the kernel of \(A^{*}\) with domain

*D*(

*B*). By [2, Theorem 3.1], the domain of the self-adjoint extension \(A_B\) of

*A*corresponding to

*B*is

### Remark 2.1

*A*is strictly positive. It follows that

*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 self-adjoint 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

*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

### Proof

*Q*(

*B*), where

*B*is a positive self-adjoint 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

## 3 Elliptic Differential Operators of Second Order

*A*is the closure of a symmetric second-order differential operator in \(L^2(\Omega )\) defined by

*s*(see [1, Section 2]), and we set \(x=(x',x_n), x'\in \mathbb {R}^{n-1}, x_n \in \mathbb {R}\).

### 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 \(\gamma \in B^{M-\frac{1}{2}}_{p,q}(\mathbb {R}^{n-1})\) 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 thatThe completion of \(C_0^{\infty }(\Omega )\) with respect to the norm \( \Vert \cdot \Vert _1 \) is the form domain \(Q(A_F)\) of$$\begin{aligned} \Vert u\Vert ^2_1 = \int _{\Omega } \left( p |\nabla u|^2 + q|u|^2 \right) dx \ge c \Vert u\Vert ^2,\ \ \ u \in C_0^{\infty }(\Omega ). \end{aligned}$$
*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$$\begin{aligned} 1-\tfrac{n}{t}\ge \tfrac{1}{2}-\tfrac{n-1}{r}> 0. \end{aligned}$$(3.2)

### Remark 3.2

*v*into its value on \(\Sigma \) (see [1, Theorem 2.11]). Moreover, in the notation of [1, 6], denote the solution of

### Theorem 3.3

*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 \(\zeta \in Q(B)\)

*Q*(

*B*) then, with \(b_{jk}\) as in (2.14),

### Proof

*Q*(

*B*), \(w=\sum w_j\psi _j\) and \(\zeta =\sum \zeta _k\psi _k\). Then

### Corollary 3.4

### Proof

### 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

*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

### Remark 3.7

### Example 3.8

*k*, where \(\{\psi _k\}\) is an orthonormal-basis of the subspace

*Q*(

*B*) in \(N=\ker A^{*}\).

*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

*N*. Therefore

*N*; note that \(I_{-k}=I_k\).

### 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 haveAs \(I_k(1)\ne 0\) for all \(k\in \mathbb {Z}\), this implies that$$\begin{aligned} 0 = 2\pi \frac{\partial v_{F,k}}{\partial r}(1) I_k(1). \end{aligned}$$and hence \(v_F\in D(A)\). As there are no restrictions on$$\begin{aligned} v_F(1,\theta ) = \frac{\partial v_{F}}{\partial r}(1,\theta ) = 0 \end{aligned}$$
*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 \(w=0\).

## 4 Aharonov–Bohm Operator

*A*be the closure in \(L^2(\Omega )\) of \(A'\upharpoonleft _{C_0^{\infty }(\Omega )}\), where

*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

*B*be a positive self-adjoint operator acting in a closed subspace \(N_B\) of \(N= \text {ker}~A^{*}\) which is associated with the self-adjoint 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.

### 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.

*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

*m*, it is in the limit-point 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 self-adjoint 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

*A*.

### Remark 4.2

*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),

*A*, if \(S_{(m)}\) is a positive self-adjoint extension of \(T_{(m)}\) then

We shall proceed to determine the extensions \(T^{(m)}\) in (4.11).

### 4.1 The Case when \(\tau ^m\) is Limit Point at 0 (\(m \ne -1,0\))

### 4.2 The Case when \(\tau ^m\) is Limit-Circle at 0 (\(m=-1,0\)) and \(\mathbf dim ~N_B =1\)

*g*is the non-principal solution of \(\tau ^m u=0\) and \(\beta \ge 0\). The non-principal solution is \(r^{1/2 - |\nu |},~ \nu = m+\alpha \). The Wronskian

*W*is given by

We shall now determine the boundary conditions satisfied by the self-adjoint extensions of \(T^{(m)}\) in the two cases corresponding to \( \nu = m + \alpha ,\ m = -1, 0,\ \alpha \in (0,1)\).

#### 4.2.1 The Case \(m = -1,\ \nu =-1 +\alpha \in (-1,0)\)

#### 4.2.2 The Case \(m =0,\ \nu = \alpha \in (0,1)\)

### 4.3 The Case when \(\tau ^m\) is Limit-Circle at 0 (\(m=-1,0\)) and \(\hbox {dim}N_B=2\)

*v*/

*g*at 0 and 1:

## Notes

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