Positive Self-adjoint Operator Extensions with Applications to Differential Operators

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 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 ⊂ (A ) * and are such that A ⊂ A β ⊂ (A ) * for an operator A β 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 Ω in R n . In this case the self-adjoint extensions of A are determined by boundary conditions on the boundary ∂Ω of Ω.
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 R n , n ≥ 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.
Denote by A F , a F [·, ·] the Friedrichs extension and associated sesquilin- where (·, ·) is the inner product of H, and D(A F ) is dense as a subspace of Q(A F ) with inner product a F [·, ·] (see [7, Chapter IV] for more on the relation between sesquilinear forms, operators and their Friedrichs extension). By the KVB theory,Â is a positive self-adjoint extension of A if and only if,Â = 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 (1.1) Thus any u ∈ Q(A B ) can be uniquely written as

Positive Extensions and the Von Neumann Theory
The von Neumann theory characterises the self-adjoint extensions of any closed densely defined symmetric operator T . Denoting the deficiency spaces ker(T * ∓ iI) by N ± , we have Let u, v ∈ D(T * ). Then by the von Neumann theory, there exist unique 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 : (see [10,Theorem 3]). The triple (N + ,Λ 0 ,Λ 1 ) is a boundary triple (also known as a space of boundary values) for T . Given a self-adjoint extension T S of T , we now choose Then, from (2.2), ker Λ 1 (T S ) = D(T S ) and we obtain, for all u, v ∈ D(T * )

Theorem 2.2. Let A B be a positive self-adjoint extension of the positive operator
. Then on the one hand, we have since w ∈ N , and on the other hand, On combining (2.11) and (2.12) we get j,k b jk w j ζ k and from (2.10) and the fact that (2.14)

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 second-order differential operator in L 2 (Ω) defined by subject to conditions on the coefficients p ij , q and the domain Ω. The assumptions are the ones made in [1] which weaken the smoothness requirements on the coefficients and the boundary of Ω 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  In the list of assumptions to be made, we shall denote the boundary of Ω by Σ, and H s t is a Bessel potential space (a Sobolev space for s ∈ N), which we write as H s when t = 2; see [1, Section 2] for definitions of H s t (Ω) and H s t (Σ).

Assumptions
1. There exists c 0 > 0 such that for all x ∈ Ω and ξ ∈ R n n i,j=1 2. There exists c > 0 such that 3. The boundary Σ is of class B 3 2 r,2 and the coefficients p and q of A lie in H 1 t (Ω) and L t (Ω), respectively, under the constraints n ≥ 2, 2 < r < ∞, Our third assumption is Assumption 2.18 in [1]. Therefore, we . Moreover, in the notation of [1,6], denote the solution of Then by [1,Theorem 5.4], for all s ∈ [0, 2], is continuous, is a homeomorphism, and We remark that under the more restrictive assumptions that Ω is a bounded domain whose boundary is an (n − 1)-dimensional C ∞ manifold, and the coefficients p jk , q of A in (3.1) lie in C ∞ (Ω) these properties were already shown by Grubb in [8].  .14), (3.11) and furthermore, while the right-hand side of (3.13) is and equivalently, As {ζ k } is an arbitrary sequence in 2 , the 'boundary condition' associated with A B is given by ∀k. (3.14) Corollary 3.4. The boundary condition associated with A B is given by
Since (∇p∇−q)ψ k = 0 and u F ∈ Q(A F ), two of the four terms in (3.16) vanish and, as Γ 1 u F = ν 1 u F , we get that the boundary condition in (3.14) becomes ∀k · j b jk w j = (ν 1 u F , γ 0 ψ k ) Σ .
On combining (2.10) and (3.17) we have For the Krein-von Neumann extension Λ K ψ k = 0, so we again get as the Krein-von Neumann boundary condition.
Example 3.8. We consider extensions of the positive operator A = −Δ + 1 when Ω is the unit disc in R 2 . According to (3.15), v = v F + w lies in the domain of an extension A B if and only if Let − Δψ + ψ = 0 and put ψ(r, θ) = R(r)Θ(θ), where x = (r, θ) are polar co-ordinates. Then since and Θ Θ = −n 2 with constant n and Θ(0) = Θ(2π); thus Θ n (θ) = e inθ , n ∈ 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 ≥ 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 ΔK 0 is not zero in the sense of distributions, excluding K 0 from N . Therefore Consequently For the Friedrichs extension, we formally have b jk = ∞ for all j, k in (3.19). This implies that w must be orthogonal to all the ψ k . As w ∈ N , this gives w = 0.

Aharonov-Bohm Operator
Let Ω = {x: |x| < 1}\{0} ⊂ R 2 , and let A be the closure in L 2 (Ω) of Here, the Aharonov-Bohm magnetic potential The sequence {ϕ k (θ) : k ∈ Z}, where ϕ k (θ) = e −ikθ √ 2π , is an orthonormal basis for L 2 (0, 2π) and hence any u ∈ L 2 (Ω) has the representation u(r, θ) = Σ k u k (r)ϕ k (θ), (4.4) where On substituting in (4.3), we have, with λ k = k + α Since min{|λ k |/r: k ∈ Z, 0 < r < 1} ≥ min{α, 1 − α} > 0, it follows that A is strictly positive and its form domain Q(A F ) is the completion of C ∞ 0 (Ω) with respect to the norm given by the square root of  If {ψ k } is an orthonormal basis of Q(B), then we have with the same notation as in Sect. 3, that u = u F + w ∈ D(A B ) if and only if ∀k : The transformation is a unitary operator from L 2 (0, 1; rdr) onto L 2 (0, 1), and as {e imθ / √ 2π} m∈Z is an orthonormal basis of L 2 (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 (0, 1) by the Sturm-Liouville expression i.e., T (m) is the minimal operator in L 2 (0, 1) generated by τ m . With ν = m + α, the set {r 1/2+ν , r 1/2−ν } is a fundamental system for τ m u = 0. The expression τ m is non-oscillatory. For m = −1, 0, it is in the limit-circle case at 0; for all other values of 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 ∈ 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 self-adjoint extension of A.

Remark 4.2.
We note that it is unlikely that all positive self-extensions of A are obtained in this way. This assertion is based on the situation for A 0 = −Δ + 1 from Example 3.8. As in (4.11), where now, T (m) is the minimal operator generated by At 0, τ m is non-oscillatory and in the limit-circle case for m = 0 and is otherwise limit-point. As above for A, if S (m) is a positive self-adjoint extension of T (m) then is a positive self-adjoint 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 !

The Case when τ m is
(4.18) The limits at 0 of the first and the second terms in (4.17) exist separately. To see this, let g (r) g(r) = (ν 2 − 1/4)r −2 =: q(r).