On the domains of Bessel operators

We consider the Schr\"odinger operator on the halfline with the potential $(m^2-\frac14)\frac1{x^2}$, often called the Bessel operator. We assume that $m$ is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for $|\Re(m)|<1$ and of its unique closed realization for $\Re(m)>1$ coincide with the minimal second order Sobolev space. On the other hand, if $\Re(m)=1$ the minimal second order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator. The properties of Bessel operators are compared with the properties of the corresponding bilinear forms.


Overview of closed realizations of the Bessel operator
The Schrödinger operator on the half-line given by the expression is often called the Bessel operator. The name is justified by the fact that its eigenfunctions and many other related objects can be expressed in terms of Bessel-type functions. There exists a large literature devoted to self-adjoint realizations of (1.1) for real α. The theory of closed realizations of (1.1) for complex α is also interesting. Let us recall the basic elements of this theory, following [7,6].
For any complex α there exist two most obvious realizations of L α : the minimal L min α , and the maximal L max α . The complex plane is divided into two regions by the parabola defined by (or, if we write α = α R + iα I , by α R + α 2 R + α 2 I = 2). To the right of this parabola, that is, for |Re √ α| ≥ 1, we have L min α = L max α . For |Re √ α| < 1, that is to the left of (1.2), D(L min α ) has codimension 2 inside D(L max α ). The operators D(L min α ) and D(L max α ) are homogeneous of degree −2. Let us note that in the region |Re √ α| < 1 the operators L min α and L max α are not the most important realizations of L α . Much more useful are closed realizations of L α situated between L min α and L max α , defined by boundary conditions near zero. (Among these realizations, the best known are self-adjoint ones corresponding to real α and real boundary conditions). All of this is described in [7].
Among these realizations for α = 0 only two, and for α = 0 only one, are homogeneous of degree −2.

Main results
Our new results give descriptions of the domains of various realizations of L α for α ∈ C. First of all, we prove that for |Re √ α| < 1 the domain of L min α does not depend on α and coincides with the minimal 2nd order Sobolev space where is the (full) 2nd order Sobolev space. We also show that is a holomorphic family of closed operators. We find the constancy of the domain of the minimal operator quite surprising and interesting. It contrasts with the fact that D(L max α ) for |Re √ α| < 1 depends on α. Similarly, D(H m ) for |Re(m)| < 1 depends on m.
The holomorphic family L min α for |Re √ α| < 1 consists of operators whose spectrum covers the whole complex plane. Therefore, the usual approach to holomorphic families of closed operators based on the study of the resolvent is not available.
We also study H m for Re(m) ≥ 1 (which by (1.3) coincides with L min m 2 and L max m 2 ). We prove that for Re(m) > 1 its domain also coincides with H 2 0 (R + ). The most unusual situation occurs in the case Re(m) = 1. In this case we show that the domain of H m is always larger than H 2 0 (R + ) and depends on m.
Specifying to real α, the main result of our paper can be summarized as follows: Let L min α be the closure in is Hermitian (symmetric) but not self-adjoint and its domain is H 2 0 (R + ). 2. Can the holomorphic family L min α (hence also L max α ) be extended beyond |Re √ α| < 1? (Probably not).
Question 1 has already been mentioned in [6]. We hope that both questions can be answered by methods of [10].

Bilinear Bessel forms
With every operator T on a Hilbert space H one can associate the sesquilinear form One can try to extend (1.8) to a larger domain. If T is self-adjoint, there is a natural extension to the so-called form domain of T , Q(T ) := D( |T |). Interpreting T as a bounded map from Q(T ) to its anti-dual, we obtain the sesquilinear form which extends (1.8).
We would like to have a similar construction for Bessel operators, including non-self-adjoint ones. Before we proceed we should realize that identities involving non-self-adjoint operators do not like complex conjugation. Therefore, instead of sesquilinear forms it is more natural to use bilinear forms.
Our analysis of bilinear Bessel forms is based on the pair of formal factorizations of the Bessel operator (1.11) In Theorems 8.2 and 8.3 for each Re(m) > −1 we interpret (1.10) and (1.11) as factorizations of the Bessel operator H m into two closed 1st order operators. They define natural bilinear forms, which we call Bessel forms. For each Re(m) > −1 the corresponding Bessel form is unique, except for Re(m) = 0, m = 0, when the two factorizations yield two distinct Bessel forms. Instead of H 2 0 (R + ), the major role is now played by the minimal 1st order Sobolev space subspace of the (full) 1st order Sobolev space

Comparison with literature
The fact that D(L min α ) does not depend on α for real α ∈ [0, 1[ was first proven in [1], see also [2,3]. Actually, the arguments of [1] are enough to extend the result to complex α such that |α − 1 4 | < 3 4 . The proof is based on the bound Q = 3 4 of the operator Q on L 2 (R + ) given by the integral kernel where θ is the Heaviside function. Our proof is quite similar. Instead of (1.14) we consider for |Re(m)| < 1 the operator Q m 2 with the kernel (1.15) Note that Q 1 4 coincides with (1.14). We prove that the norm of Q m 2 is the inverse of the distance of m 2 to the parabola (1.2). A simple generalization of the Kato-Rellich Theorem to closed operators implies then our result about D(L min α ). In the paper [6] on page 567 it is written "If m = 1/2 then D(L min m ) = H 2 0 ." (In that paper L min m 2 was denoted L min m ). This sentence was not formulated as a proposition, and no proof was provided. Anyway, in view of the results of [2] and of this paper, this sentence was wrong.
The analysis of Bessel forms in the self-adjoint case, that is for real m > −1, is well known-it is essentially equivalent to the famous Hardy inequality. This subject is discussed e.g. in the monograph [4] and in a recent interesting paper [11] about a refinement of the 1-dimensional Hardy's inequality. The latter paper contains in particular many references about factorizations of Bessel operators in the self-adjoint case.
Results about Bessel forms and their factorizations for complex parameters are borrowed to a large extent from [6]. We include them in this paper, because they provide an interesting complement to the analysis of domains of Bessel operators.

Basic closed realizations of the Bessel operator
The main topic of this preliminary section are closed homogeneous realizations of L α . We recall their definitions following [6,7].
We will denote by R + the open positive half-line, that is ]0, ∞[. We will use L 2 (R + ) as our basic Hilbert space. We define L max α to be the operator given by the expression L α with the domain We also set L min α to be the closure of the restriction of L max α to C ∞ c (R + ). We will often write m for one of the square roots of α, that is, α = m 2 . It is easy to see that the space of solutions of the differential equation Note that both solutions are square integrable near 0 iff |Re(m)| < 1. This is used in [6] to show that we have Above (and throughout the paper) ξ is any C ∞ c [0, ∞[ function such that ξ = 1 near 0. Following [6], for Re(m) > −1 we also introduce another family of closed realizations of Bessel operators: the operators H m defined as the restrictions of L max We will use various basic concepts and facts about 1-dimensional Schrödinger operators with complex potentials. We will use [9] as the main reference, but clearly most of them belong to the well-known folklore. In particular, we will use two kinds of Green's operators. Let us recall this concept, following [9]. Let L 1 c (R + ) be the set of integrable functions of compact support in R + . We will say that an operator G :

The forward Green's operator
Let us introduce the operator G → α defined by the kernel Note that G → α is a Green's operator in the sense of (2.6). Besides, which is why it is sometimes called the forward Green's operator. Unfortunately, the operator G → α is unbounded on L 2 (R + ). To make it bounded, for any a > 0 we can compress it to the finite interval [0, a], by introducing the operator G a→ α with the kernel It is also convenient to consider the operator L α restricted to [0, a]. One of its closed realizations, is defined by the zero boundary condition at 0 and no boundary conditions at a (see [9] Def. 4.14). It will be denoted L a α,0 . By Prop. 7.3 of [9] we have G a→ α = (L a α,0 ) −1 , and hence Now we can describe the domain of L min α with the help of the forward Green's operator.
. Then the following statements are equivalent: Proof. The boundary space ( [9] Def. 5.2) of L α is trivial at ∞ (see [9] Prop. 5.15). Therefore, for any Hence it is enough to apply (3.5). ✷ Define the operator Q α := 1 Then the operator Q α is bounded on L 2 (R + ), and Proof. Introduce the unitary operator U : (3.10) Note that if an operator K has the kernel K(x, y), then U KU −1 , has the kernel e t 2 K(e t , e s )e s 2 . Therefore, for any α the operator U Q α U −1 has the kernel Thus, it is the convolution by the function Assume now that |Re √ α| < 1. Then the function (3.13) is integrable and we can apply the Fourier transformation defined by (F u)(ω) = (2π) −1/2 e −iωt u(t) dt. After this transformation the operator U Q α U −1 becomes the multiplication wrt the Fourier transform of (3.13) or (3.14), that is Thus the norm of U Q α U −1 , and hence also of Q α , is the supremum of the absolute value of (3.15). ✷ where we identify f = u ′′ and g = u x 2 . The proof of the following proposition uses only the simple estimate (3.16).
. It follows that f is absolutely continuous and f (x) = a + Then, by (3.16) Below we state the first main result of our paper (which is an extension of a result of [1]).
is a holomorphic family of closed operators.
The proof of this theorem is based on the following lemma. Then . Let G a→ α be as in (3.4). As in the proof of So Proof of Theorem 4.1. We can cover the region on the lhs of (4.1) by disks touching the boundary of this region, that is, (1.2). Inside each disk we apply Thm A.1 and Lemma 4.2. We obtain in particular, that if |Re √ α i | < 1, i = 1, 2, then D(L min α1 ) = D(L min α2 ). But clearly D(L min Besides, is a holomorphic family of closed operators. (4.10) Its holomorphy has been proven in [6]. Using arguments similar to those in the proof of Theorem 4.3 we obtain a closer description of this family in the region |Re(m)| < 1.   Recall that θ is the Heaviside function. (5.1) is one of Green's operators of L m 2 in the sense of (2.6), Following [9], we will call it the two-sided Green's operator. The operator G m is not bounded on L 2 (R + ) for any m ∈ C. However, at least for Re(m) > −1, it is useful in the L 2 setting. 1. If g ∈ L 2 [0, a], then

Conversely
Proof. Note first that Re(m) > 1 implies x 1 2 ±m is locally in L 2 . Using this, the proof of the first part of the proposition is a straightforward computation done, in a more general setting, in [9], see §2.7 and Definition 2.10 there. For the second part, note that L α (f − G m g) = 0 by the first part of the proposition, and that the two functions x 1 2 ±m give a basis of the nullspace of L α . ✷ Let us introduce the operator Z m := 1 Then Z m is bounded and Proof. If U is given by (3.10), then U Z m U −1 has the kernel whose supremum is the right hand side of (5.5). ✷

Domain of Bessel operators for Re(m) > 1
For Re(m) ≥ 1 there is a unique closed Bessel operator. We will see in the following theorem that its domain is again the minimal 2nd order Sobolev space, except at the boundary Re(m) = 1, cf. Section 7. For any ǫ > 0, the second term on the right of (6.8) goes to zero. The first, by making ǫ small, can be made arbitrarily small. Therefore (6.8) goes to zero. Thus lim    D(H m ). For functions in the subspace H 2 0 (R + )+ Cx 1 2 +m ξ this is easy, cf. the next lemma, but this is not so trivial for the other functions. Proof.
and since We will write L a,max α for the maximal realization of operator L α on L 2 [0, a]. Proof. We have x 0 y 1 2 +m g(y)dy. (7.5) Since Re(m) = 1 the absolute value of the second term on the right hand side is less than This proves (7.3). If f = G a m g ∈ H 2 0 (R + ) + Cx Proof. For each τ ∈] 1 2 , 1[ let g τ ∈ C 2 (]0, 1]), for 0 < x < 1 2 given by and arbitrary on [ 1 2 , 1]. Then for x < 1 2 we have Let G be the vector subspace of L 2 [0, 1] generated by the functions g τ with 1 2 < τ < 1. Note that each finite set {g τ | τ ∈ A} with A ⊂] and the first term on the right hand side tends to +∞ more rapidly than all the other, hence 1]. Let us define V as the space of functions on R + of the form f = ϕG m g with g ∈ G. By Lemma 7.3, G m 1l [0,1] (x) is injective. Hence V is infinite dimensional and it satisfies (7.6) by Lemma 7.4. ✷

Bilinear forms associated with Bessel operators
As noted in the introduction, in this section we will avoid complex conjugation. Thus in the place of the usual sesquilinear scalar product we will prefer to use the bilinear product Clearly, (8.8) is well defined for f, g ∈ L 2 (R + ). Instead of the usual adjoint T * we will use the transpose T # , defined with respect to (8.8), see [9]. An important role will be played by the 1st order operators given by the formal expression A detailed analysis of (8.9) has been done in [6], where the notation was slightly different: Let us recall the main points of that analysis. In the usual way we define two closed realizations of A ρ : the minimal and the maximal one, denoted A min ρ , resp. A max ρ . The following theorem was (mostly) proven in Section 3 of [6]. For the proof of the infinite codimensionality assertion in 6 see the proof of Lemma 3.9 there (where γ is arbitrary > 1 2 ).

A min
ρ and A max ρ are homogeneous of degree −1.

A min
If this is the case, we will often denote them simply by A ρ 5. If Re(ρ) = 1 2 , then D(A min ρ ) = H 1 0 .
) The factorizations described in Theorem 8.2 can be used to define bilinear forms corresponding to H m . For details of the proof, we refer again to [6], especially pages 571-574 and 577. to larger domains: 1. For 1 ≤ Re(m), 2. For 0 < Re(m) < 1, Note that for Re(m) > 0 both factorizations yield the same quadratic form. This is no longer true for Re(m) = 0, m = 0, when there are two distinct quadratic forms with distinct domain corresponding to H m . Finally, for −1 < m < 0, and also for m = 0, we have a unique factorization.
Let us finally specialize Theorem 8.3 to real m. The following theorem is essentially identical with Thm 4.22 of [6].

A Holomorphic families of closed operators and the Kato-Rellich Theorem
In this appendix we describe a few general concepts and facts from the operator theory, which we use in our paper. The definition (or actually a number of equivalent definitions) of a holomorphic family of bounded operators is quite obvious and does not need to be recalled. In the case of unbounded operators the situation is more subtle.
Suppose that Θ is an open subset of C, H is a Banach space, and Θ ∋ z → H(z) is a function whose values are closed operators on H. We say that this is a holomorphic family of closed operators if for each z 0 ∈ Θ there exists a neighborhood Θ 0 of z 0 , a Banach space K and a holomorphic family of bounded operators Θ 0 ∋ z → A(z) ∈ B(K, H) such that RanA(z) = D(H(z)) and is a holomorphic family of bounded operators.
The following theorem is essentially a version of the well-known Kato-Rellich Theorem generalized from self-adjoint to closed operators: is obviously a polynomial of degree 1 with values in bounded operators (hence obviously a holomorphic family). ✷ Let us also quote the following fact proven by Bruk [5], see also [10]: is a holomorphic family of closed operators, then so is z → A(z) * .