aa r X i v : . [ m a t h - ph ] J a n On the domains of Bessel operators
Jan DerezińskiDepartment of Mathematical Methods in Physics,Faculty of Physics, University of Warsaw,Pasteura 5,02-093 Warszawa, Poland,email: [email protected] GeorgescuLaboratoire AGM, UMR 8088 CNRS,CY Cergy Paris Université,F-95000 Cergy, France,email: [email protected] 5, 2021
Abstract
We consider the Schrödinger operator on the halfline with the potential ( m − ) x , often called theBessel operator. We assume that m is complex. We study the domains of various closed homogeneousrealizations of the Bessel operator. In particular, we prove that the domain of its minimal realizationfor | Re( m ) | < and of its unique closed realization for Re( m ) > coincide with the minimal secondorder Sobolev space. On the other hand, if Re( m ) = 1 the minimal second order Sobolev space is asubspace of infinite codimension of the domain of the unique closed Bessel operator. The propertiesof Bessel operators are compared with the properties of the corresponding bilinear forms. Keywords: Schrödinger operatros, Bessel operators, unbounded operators, Sobolev spaces.MSC 2020: 47E99, 81Q80
The Schrödinger operator on the half-line given by the expression L α := − d d x + (cid:16) α − (cid:17) x (1.1)is often called the Bessel operator . The name is justified by the fact that its eigenfunctions and manyother 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 ofclosed realizations of (1.1) for complex α is also interesting. Let us recall the basic elements of this theory,following [6, 5]. 1or 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 α = (1 + i ξ ) , ξ ∈ R , (1.2)(or, if we write α = α R + i α I , by α R + p α + α = 2 ). To the right of this parabola, that is, for | Re √ α | ≥ , we have L min α = L max α . For | Re √ α | < , that is to the left of (1.2), D ( L min α ) has codimension inside D ( L max α ) . The operators D ( L min α ) and D ( L max α ) are homogeneous of degree − .Let us note that in the region | Re √ α | < the operators L min α and L max α are not the most importantrealizations 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-adjointones corresponding to real α and real boundary conditions). All of this is described in [6].Among these realizations for α = 0 only two, and for α = 0 only one, are homogeneous of degree − .All of them are covered by the holomorphic family of closed operators H m , introduced in [5] and definedfor Re( m ) > − as the restriction of L max m to functions that behave as x + m near zero. Note that L min m = H m = L max m , Re( m ) ≥ (1.3) L min m ( H m ( L max m , | Re( m ) | < . (1.4) Our new results give descriptions of the domains of various realizations of L α for various α ∈ C . Firstof all, we prove that for | Re √ α | < the domain of L min α does not depend on α and coincides with the minimal 2nd order Sobolev space H ( R + ) := { f ∈ H ( R + ) | f (0) = f ′ (0) = 0 } , (1.5)where H ( R + ) := { f ∈ L ( R + ) | f ′′ ∈ L ( R + ) } (1.6)is the (full) 2nd order Sobolev space . We also show that { α | | Re √ α | < } ∋ α L min α (1.7)is a holomorphic family of closed operators.We find the constancy of the domain of the minimal operator quite surprising and interesting. Itcontrasts with the fact that D ( L max α ) for | Re √ α | < depends on α . Similarly, D ( H m ) for | Re( m ) | < depends on m .The holomorphic family L min α for | Re √ α | < consists of operators whose spectrum covers the wholecomplex plane. Therefore, the usual approach to holomorphic families of closed operators based on thestudy of the resolvent is not available.We also study H m for Re( m ) ≥ (which by (1.3) coincides with L min m and L max m ). We prove thatfor Re( m ) > its domain also coincides with H ( 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 ( 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 theclosure in L ( R + ) of the operator − ∂ x + α − x with domain C ∞ c ( R + ) .1) If α < then L min α is Hermitian (symmetric) but not self-adjoint and its domain is H ( R + ) .2) If α = 1 then L min α is self-adjoint and H ( R + ) is a dense subspace of infinite codimension of its domain.3) If α > then L min α is self-adjoint with domain H ( R + ) .As a side remark, let us mention two open problems about Bessel operators.2 pen Problem 1.1
1. Can the holomorphic family H m be extended beyond Re( m ) > − ? (Probably not).2. Can the holomorphic family L min α (hence also L max α ) be extended beyond | Re √ α | < ? (Probablynot). Question 1 has already been mentioned in [5]. We hope that both questions can be answered by methodsof [9].
With every operator T on a Hilbert space H one can associate the sesquilinear form ( f | T g ) , f, g ∈ D ( T ) . (1.8)One can try to extend (1.8) to a larger domain. If T is self-adjoint, there is a natural extension to theso-called the form domain of T , Q ( T ) := D ( p | T | ) . Interpreting T as a bounded map from Q ( T ) to itsanti-dual, we obtain the sesquilinear form ( f | T g ) , f, g ∈ Q ( T ) , (1.9)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 complexconjugation. 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 − ∂ x + (cid:16) m − (cid:17) x = (cid:16) ∂ x + + mx (cid:17)(cid:16) − ∂ x + + mx (cid:17) (1.10) = (cid:16) ∂ x + − mx (cid:17)(cid:16) − ∂ x + − mx (cid:17) . (1.11)In Theorems 8.2 and 8.3 for each Re( m ) > − we interpret 1.10 and (1.11) as factorizations of the Besseloperator H m into two closed 1st order operators. They define a natural bilinear forms, which we call Bessel forms . For each
Re( m ) > − 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 ( R + ) , the major role is now played by the minimal 1st order Sobolev space H ( R + ) := { f ∈ H ( R + ) | f (0) = 0 } , (1.12)subspace of the (full) 1st order Sobolev space H ( R + ) := { f ∈ L ( R + ) | f ′ ∈ L ( R + ) } . (1.13)Note that H ( R + ) is the domain of Bessel forms for Re( m ) > .The analysis of Bessel forms and their factorizations shows a variety of behaviors depending on theparameter m . In particular, there is a kind of a phase transition at Re( m ) = 0 . Curiously, in the analysisof the domain of Bessel operators the phase transition occurs elsewhere: at Re( m ) = 1 .3 .4 Comparison with literature The fact that D ( L min α ) does not depend on α for real α ∈ [0 , 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 − | < . Theproof is based on the bound k Q k = of the operator Q on L ( R + ) given by the integral kernel Q ( x, y ) = 1 x ( x − y ) θ ( x − y ) , (1.14)where θ is the Heaviside function. Our proof is quite similar. Instead of (1.14) we consider for | Re( m ) | < the operator Q m with the kernel Q m ( x, y ) = 12 mx ( x + m y − m − x − m y + m ) θ ( x − y ) . (1.15)Note that Q coincides with (1.14). We prove that the norm of Q m is the inverse of the distance of m to the parabola (1.2). A simple generalization of the Kato-Rellich Theorem to closed operators impliesthen our result about D ( L min α ) In the paper [5] on page 567 it is written “If m = 1 / then D ( L min m ) = H .” (In that paper L min m wasdenoted 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 > − , is well known–it isessentially equivalent to the famous Hardy inequality . We will not discuss the literature on this subject,except that we want to mention a recent interesting paper [10] about a refinement of Hardy’s inequality.This paper contains many references about the Hardy inequality and factorizations of the Bessel operatorsin the self-adjoint case.Results about Bessel forms and their factorizations for complex parameters are borrowed to a largeextent from [5]. We include them in this paper, because they provide an interesting complement to theanalysis of domains of Bessel operators.
The main topic of this preliminary section are closed homogeneous realizations of L α . We recall theirdefinitions following [5, 6].We will denote by R + the open positive half-line, that is ]0 , ∞ [ . We will use L ( R + ) as our basicHilbert space. We define L max α to be the operator given by the expression L α with the domain D ( L max α ) = { f ∈ L ( R + ) | L α f ∈ L ( R + ) } . 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 . It is easy to see that thespace of solutions of the differential equation L α f = 0 (2.1)is spanned for α = 0 by x + m , x − m , and for α = 0 by x , x log x . Note that both solutions are squareintegrable near iff | Re( m ) | < . This is used in [5] to show that we have D ( L max α ) = D ( L min α ) + C x + m ξ + C x − m ξ, | Re √ α | < , α = 0; (2.2) D ( L max0 ) = D ( L min0 ) + C x ξ + C x log( x ) ξ, α = 0; (2.3) D ( L max α ) = D ( L min α ) , | Re √ α | ≥ . (2.4)4bove (and throughout the paper) ξ is any C ∞ c [0 , ∞ [ function such that ξ = 1 near .For Re( m ) > − one can also introduce another family of closed realizations of Bessel operators: theoperators H m defined as the restrictions of L max m to D ( H m ) := D ( L min m ) + C x + m ξ. (2.5)We will use various basic concepts and facts about 1-dimensional Schrödinger operators with complexpotentials. We will use [8] as the main reference, but clearly most of them belong to the well-knownfolklore. In particular, we will use two kinds of Green’s operators. Let us recall this concept, following[8]. Let L ( R + ) be the set of integrable functions of compact support in R + . We will say that an operator G : L ( R + ) → AC ( R + ) is a Green’s operator of L α if for any g ∈ L ( R + ) L α Gg = g. (2.6) Let us introduce the operator G → α defined by the kernel G → α ( x, y ) := 12 m (cid:0) x + m y − m − x − m y + m (cid:1) θ ( x − y ) , α = 0; (3.1) G → ( x, y ) := x y log (cid:16) xy (cid:17) θ ( x − y ) , α = 0 . (3.2)Note that G → α is a Green’s operator in the sense of (2.6). Besides, supp G → α g ⊂ supp g + R + , (3.3)which is why it is sometimes called the forward Green’s operator .Unfortunately, the operator G → α is unbounded on L ( R + ) . To make it bounded, for any a > we cancompress it to the finite interval [0 , a ] , by introducing the operator G a → α with the kernel G a → α ( x, y ) := 1l [0 ,a ] ( x ) G → α ( x, y )1l [0 ,a ] ( y ) . (3.4)It is also convenient to consider the operator L α restricted to [0 , a ] . One of its closed realizations, isdefined by the zero boundary condition at and no boundary conditions at a (see [8] Def. 4.14). It willbe denoted L aα, . By Prop. 7.3 of [8] we have G a → α = ( L aα, ) − , and hence D ( L aα, ) = G a → α L [0 , a ] . (3.5)Now we can describe the domain of L min α with the help of the forward Green’s operator. Proposition 3.1
Suppose that f ∈ D ( L max α ) . Then the following statements are equivalent:1. f ∈ D ( L min α ) .2. For some a > and g a ∈ L [0 , a ] we have f (cid:12)(cid:12)(cid:12) [0 ,a ] = G → α g a (cid:12)(cid:12)(cid:12) [0 ,a ] .3. For all a > there exists g a ∈ L [0 , a ] such that f (cid:12)(cid:12)(cid:12) [0 ,a ] = G → α g a (cid:12)(cid:12)(cid:12) [0 ,a ] . roof. The boundary space ([8] Def. 5.2) of L α is trivial at ∞ (see [8] Prop. 5.15). Therefore, for any a > we have f ∈ D ( L min α ) ⇔ f (cid:12)(cid:12)(cid:12) [0 ,a ] ∈ D ( L aα, ) . (3.6)Hence it is enough to apply (3.5). ✷ Define the operator Q α := x G → α . Its integral kernel is Q α ( x, y ) = 12 m ( x − + m y − m − x − − m y + m ) θ ( x − y ) , α = 0; (3.7) Q ( x, y ) := x − y log (cid:16) xy (cid:17) θ ( x − y ) , α = 0 . (3.8) Proposition 3.2
Assume that | Re √ α | < . Then the operator Q α is bounded on L ( R + ) , and k Q α k = 1dist (cid:0) α, (1 + i R ) (cid:1) (3.9) Proof.
Introduce the unitary operator U : L ( R + ) → L ( R ) given by ( U f )( t ) := e t f (e t ) . (3.10)Note that if an operator K has the kernel K ( x, y ) , then U KU − , has the kernel e t K (e t , e s )e s . Therefore, U Q α U − has the kernel m (e − ( t − s )(1 − m ) − e − ( t − s )(1+ m ) ) θ ( t − s ) , α = 0; (3.11) e − ( t − s ) ( t − s ) θ ( t − s ) , α = 0 . (3.12)Thus, it is the convolution by the function t → m (e − t (1 − m ) − e − t (1+ m ) ) θ ( t ) , α = 0; (3.13) t → e − t tθ ( t ) , α = 0 . (3.14)Assume now that | Re √ α | < . Then the function (3.13) is integrable and we can apply the Fouriertransformation. After this transformation the operator U Q α U − becomes the multiplication wrt theFourier transform of (3.13) or (3.14), that is ξ ξ ) − m . (3.15)Thus the norm of U Q α U − , and hence also of Q α , is the supremum of the absolute value of (3.15). ✷ Remark 3.3
The operator Q α belongs to the class of operators analyzed in [16] on p. 271, which goesback to Hardy-Littlewood-Polya [11] p. 229. Proposition (3.2) for α = is especially important and simple. This case was noted in cf. [5, p. 566]and [1, Lemma 2.2]. It can be written as g ( x ) := x − Z x ( x − y ) f ( y )d y ⇒ k g k ≤ k f k . (3.16)The proof of the following proposition uses only the simple estimate (3.16).6 roposition 3.4 D ( L max α ) ∩ D ( L max β ) = H ( R + ) if α = β .Proof. We have f ∈ D ( L max α ) if and only if f ∈ L ( R + ) and − f ′′ + ( α − / x − f ∈ L ( R + ) hence ifwe also have f ∈ D ( L max β ) then ( α − β ) x − f ∈ L ( R + ) and since α = β we get x − f ∈ L ( R + ) hence f ′′ ∈ L ( R + ) . Recall that f, f ′′ ∈ L ( R + ) implies f ∈ H ( R + ) and k f ′ k L ( R + ) ≤ k f k L ( R + ) k f ′′ k L ( R + ) . Itfollows that f is absolutely continuous and f ( x ) = a + R x f ′ ( y )d y for some constant a and f ′ is absolutelycontinuous and f ′ ( x ) = b + R x f ′′ ( y )d y for some constant b , thus f ( x ) = a + bx + Z x Z y f ′′ ( z )d z d y = a + bx + x g ( x ) , g ( x ) := x − Z x ( x − y ) f ′′ ( y )d y. Then, by (3.16) k g k L ( R + ) ≤ k f ′′ k L ( R + ) . (3.17)Thus x − f ( x ) = ax − + bx − + g ( x ) where g ∈ L ( R + ) , so R | x − f ( x ) | d x < ∞ if and only if a = b = 0 ,so that f ( x ) = R x ( x − y ) f ′′ ( y )d y and f ′ ( x ) = R x f ′′ ( y )d y , hence f ∈ H ( R + ) .Reciprocally, if f ∈ H ( R + ) then x − f ∈ L ( R + ) with k x − f k L ( R + ) ≤ k f ′′ k L ( R + ) by (3.16), hence f ∈ D ( L α ) for all α . ✷ | Re( m ) | < Below we state the first main result of our paper (which is an extension of a result of [1]).
Theorem 4.1 If | Re √ α | < , then D ( L min α ) = H ( R + ) . Moreover, (cid:8) α ∈ C | | Re √ α | < (cid:9) ∋ α L min α (4.1) is a holomorphic family of closed operators. The proof of this theorem is based on the following lemma.
Lemma 4.2
Let | Re √ α | < and f ∈ D ( L min α ) . Then k x − f k ≤ (cid:0) α, (1 + i R ) (cid:1) k L min α f k . (4.2) Proof.
Let a > . Set g := L min α f , f a := f (cid:12)(cid:12)(cid:12) [0 ,a ] , g a := g (cid:12)(cid:12)(cid:12) [0 ,a ] . Let G a → α be as in (3.4). As in the proof ofProp. 3.1, f a = G a → α g a . (4.3)So k x − f k = lim a →∞ k x − f a k (4.4) = lim a →∞ k x − G a → α g a k = k Q α g k ≤ (cid:0) α, (1 + i R ) (cid:1) k g k . ✷ (4.5) Proof of Theorem 4.1.
We can cover the region on the lhs of (4.1) by disks touching the boundary of thisregion, that is, (1.2). Inside each disk we apply Thm A.1 and Lemma 4.2. We obtain in particular, thatif | Re √ α i | < , i = 1 , , then D ( L min α ) = D ( L min α ) . But clearly D ( L min ) = H ( R + ) . ✷ heorem 4.3 We have D ( L max α ) = H + C x + m ξ + C x − m ξ, | Re √ α | < , α = 0; (4.6) D ( L max α ) = H + C x ξ + C x log( x ) ξ, α = 0 . (4.7) Besides, D ( L max α ) = D ( L max α ) , α = α , | Re √ α i | < , i = 1 , . (4.8) Furthermore, (cid:8) α ∈ C | | Re √ α | < (cid:9) ∋ α L max α (4.9) is a holomorphic family of closed operators.Proof. Using D ( L min α ) = H , (2.2) and (2.3) can be now rewritten as (4.6) and (4.7).Clearly, x + m ξ and x log( x ) ξ do not belong to H ( R + ) (because their second derivatives are notsquare integrable). Therefore, D ( L max α ) = H ( R + ) . This together with Proposition 3.4 implies (4.8).We have ( L min α ) ∗ = L max α . Therefore, to obtain the holomorphy we can use Proposition A.2. ✷ The most important holomorphic family of Bessel operators is { m ∈ C | Re( m ) > − } ∋ m H m . (4.10)Its holomorphy has been proven in [5]. Using arguments similar to those in the proof of Theorem 4.3 weobtain a closer description of this family in the region | Re( m ) | < . Theorem 4.4
We have D ( H m ) = H + C x + m ξ, | Re( m ) | < . (4.11) Besides, if m = m and | Re( m i ) | < , i = 1 , , then D ( H m ) = D ( H m ) . Let us introduce the operator G m with the kernel G m ( x, y ) := 12 m (cid:16) x + m y − m θ ( y − x ) + x − m y + m θ ( x − y ) (cid:17) . (5.1)It is one of Green’s operators of L m in the sense of (2.6), Following [8], we will call it a two-sided Green’soperator .The operator G m is not bounded on L ( R + ) for any m ∈ C . However, it is useful in the L analysis,at least for Re( m ) > − : Proposition 5.1
Let
Re( m ) > − and a > .1. If g ∈ L [0 , a ] , then f ( x ) = G m g ( x ) = Z ∞ G m ( x, y ) g ( y )d y (5.2) is well defined, belongs to ∈ AC ]0 , ∞ [ and L α f = g .2. Conversely, if f ∈ AC ]0 , ∞ [ , L α f = g ∈ L [0 , a ] , then there exist c + , c − such that f ( x ) = c + x + m + c − x − m + G m g ( x ) . (5.3)8et us introduce the operator Z m := x G m with the kernel Z m ( x, y ) = 12 m (cid:16) x − + m y − m θ ( y − x ) + x − − m y + m θ ( x − y ) (cid:17) . (5.4) Proposition 5.2
Let
Re( m ) > . Then Z m is bounded and k Z m k = 1dist (cid:0) m , (1 + i R ) (cid:1) (5.5) Proof. If U is given by (3.10), then U Z m U − has the kernel m (cid:16) e − ( m − s − t ) θ ( s − t ) + e − ( m +1)( s − t ) θ ( t − s ) (cid:17) . (5.6)If Re( m ) > , after the Fourier transformation, it becomes the multiplication by the function ξ m (cid:16) m − − i ξ ) + 11 + m + i ξ ) (cid:17) = 1 m − (1 + i ξ ) , (5.7)whose supremum is the right hand side of (5.5). ✷ Re( m ) > For
Re( m ) ≥ there is a unique closed Bessel operator. We will see in the following theorem that itsdomain is again the minimal 2nd order Sobolev space, except at the boundary Re( m ) = 1 , cf. Section 7. Theorem 6.1
Let
Re( m ) > . Then D ( H m ) = H ( R + ) .Proof. We know that H ( R + ) ⊂ D ( L max m ) for any m . But for Re( m ) > we have L max m = H m . Thisproves the inclusion H ( R + ) ⊂ D ( H m ) .Let us prove the converse inclusion. Let f ∈ D ( H m ) . It is enough to assume that f ∈ L [0 , . Let g := H m f . Then g ∈ L [0 , . By Prop. 5.1, we can write f ( x ) = c + x + m + c − x − m + x + m m Z x y − m g ( y )d y + x − m m Z x y + m g ( y )d y. (6.1)For x > we have f ( x ) = c + x + m + x − m (cid:18) c − + 12 m Z y + m g ( y )d y (cid:19) , (6.2)hence c + = 0 . We have, for x → , (cid:12)(cid:12)(cid:12)(cid:12) x + m Z x y − m g ( y )d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ x Z | g ( x ) | d y → (6.3) (cid:12)(cid:12)(cid:12)(cid:12) x − m Z x y + m g ( y ) (cid:12)(cid:12)(cid:12)(cid:12) d y ≤ x Z x | g ( y ) | d y → . (6.4) x − m is not square integrable near zero. Hence c − = 0 . Thus f ( x ) = x + m m Z x y − m g ( y )d y + x − m m Z x y + m g ( y )d y. (6.5)9y (6.3) and (6.4), lim x → f ( x ) = 0 . Now f ′ ( x ) = ( + m ) x − + m m Z x y − m g ( y )d y + ( − m ) x − − m m Z x y + m g ( y )d y, (6.6) (cid:12)(cid:12)(cid:12)(cid:12) x − − m Z x y + m g ( y )d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z x | g ( y ) | d y → , (6.7) (cid:12)(cid:12)(cid:12)(cid:12) x − + m Z x y − m g ( y )d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ǫ | g ( y ) | d y + x − +Re( m ) Z ǫ y − Re( m ) | g ( y ) | d y. (6.8)For any ǫ > , the second term on the right of (6.8) goes to zero. The first, by making ǫ small, can bemade arbitrarily small. Therefore (6.8) goes to zero. Thus lim x → f ′ (0) = 0 .Finally f ′′ ( x ) + g ( x ) = ( m − ) x − + m m Z x y − m g ( y )d y + ( m − ) x − − m m Z x y + m g ( y )d y (6.9) = (cid:16) m − (cid:17) Z m g ( x ) . (6.10)By Proposition 5.2 Z m is bounded. Hence f ′′ ∈ L ( R + ) . Therefore, f ∈ H ( R + ) . ✷ Re( m ) = 1 In this section we treat the most complicated situation concerning the domain of H m , namely the case Re( m ) = 1 . By (1.3) we then have H m = L min m = L max m . We prove the following theorem. Theorem 7.1
Let
Re( m ) = 1 .1. H ( R + ) is a dense subspace of D ( H m ) of infinite codimension.2. If ξ is a C [0 , ∞ [ function equal near zero, then x + m ξ ∈ D ( H m ) but x + m ξ
6∈ H ( R + ) .3. If Re( m ′ ) = 1 and m = m ′ , then D ( H m ) ∩ D ( H m ′ ) = H ( R + ) . By (1.3), it is clear that H ( R + ) ⊂ D ( H m ) and x + m ξ ∈ D ( H m ) . The density of H ( R + ) in D ( H m ) is a consequence of H m = L min m . The last assertion of the theorem is a special case of Proposition 3.4.In the rest of this section we construct an infinite dimensional vector subspace of F of D ( H m ) such that F ∩ (cid:0) H ( R + ) + C x + m ξ (cid:1) = 0 , which will finish the proof of the theorem.Let us study the behaviour at zero of the functions in D ( H m ) . For functions in the subspace H ( R + )+ C x + m ξ this is easy, cf. the next lemma, but this is not so trivial for the other functions. Lemma 7.2 If f = f + cx + m ξ ∈ H ( R + ) + C x + m ξ then c = lim x → x − − m f ( x ) . (7.1) Proof. If f ∈ H ( R + ) then f ( x ) = x R ( x − y ) f ′′ ( y )d y . Therefore, √ | f ( x ) | ≤ x k f ′′ k L [0 ,x ] and since Re( m + ) = we get lim x → x − m − f ( x ) = 0 , which implies (7.1). ✷ Let a > . Let G am be the operator G m compressed to the interval [0 , a ] . Its kernel is G am ( x, y ) = 1l [0 ,a ] ( x ) G m ( x, y )1l [0 ,a ] ( y ) . (7.2)We will write L a, max α for the maximal realization of operator L α on L [0 , a ] .10 emma 7.3 Let
Re( m ) > − . Then G am is a bounded operator on L [0 , a ] . If g ∈ L [0 , a ] , then G am g ∈ D ( L a, max m ) and L a, max m G am g = g . Consequently, G am is injective.Proof. We check that (7.2) belongs to L (cid:0) [0 , a ] × [0 , a ] (cid:1) . This proves that G am is Hilbert Schmidt, hencebounded. G am is a right inverse of L a, max m , because G m is a right inverse of L m (see Proposition 5.1). ✷ Lemma 7.4
Let
Re( m ) = 1 . Let g ∈ L [0 , a ] and f = G am g . Then lim x → (cid:18) mx − − m f ( x ) − Z ax y − m g ( y )d y (cid:19) = 0 . (7.3) Therefore, if lim x → Z ax y − m g ( y )d y (7.4) does not exist, then f = Rg
6∈ H ( R + ) + C x + m ξ .Proof. We have mx − − m f ( x ) = Z ax y − − m g ( y )d y + x − m Z x y + m g ( y )d y. (7.5)Since Re( m ) = 1 the absolute value of the second term on the right hand side is less than x − Z x ( y/x ) | g ( y ) | d y ≤ x − Z x | g ( y ) | d y ≤ k g k L [0 ,x ] This proves (7.3).If f = Rg ∈ H ( R + ) + C x + m ξ , then by (7.3) and (7.1) there exists (7.4). This proves the secondstatement of the lemma. ✷ Lemma 7.5
Let
Re( m ) = 1 . There exists an infinite dimensional subspace F ⊂ D ( H m ) such that F ∩ (cid:0) H ( R + ) + C x + m ξ (cid:1) = { } . (7.6) Proof.
For each α ∈ ] , let g α ∈ C (]0 , , for < x < given by g α ( x ) = x − + m (cid:0) ln(1 /x ) (cid:1) − α . Then for x < we have | g α ( x ) | = x − (cid:0) ln(1 /x ) (cid:1) − α = (2 α − − dd x (cid:0) ln(1 /x ) (cid:1) − α . Hence Z | g α ( x ) | d x = (2 α − − (ln 2) − α , and g ∈ L ( I ) . Moreover, if x < then x − m g α ( x ) = x − (cid:0) ln(1 /x ) (cid:1) − α = ( α − − dd x (cid:0) ln(1 /x ) (cid:1) − α . Hence Z x y − g α ( y )d y = ( α − − (ln 2) − α + (1 − α ) − (cid:0) ln(1 /x ) (cid:1) − α → ∞ as x → ∞ . G be the vector subspace of L ( I ) generated by the functions g α with < α < . Note that each finiteset { g α | α ∈ A } with A ⊂ ] , finite is linearly independent. Indeed, if P α ∈ A c α g α = 0 and β = min A and α = β then g α ( x ) g β ( x ) = (cid:0) ln(1 /x ) (cid:1) β − α → as x → so we get c β = 0 , etc. Moreover, for each not zero g = P α ∈ A c α g α ∈ G (with c α = 0 ) we have lim x → (cid:12)(cid:12)(cid:12)R x g ( y )d y (cid:12)(cid:12)(cid:12) = ∞ . Indeed, we may assume c β = 1 and then, Z x y − g ( y )d y =(1 − β ) − (cid:0) ln(1 /x ) (cid:1) − β + X α ∈ A c α ( α − − (ln 2) − α + X α = β c α (1 − α ) − (cid:0) ln(1 /x ) (cid:1) − α , and the first term on the right hand side tends to + ∞ more rapidly than all the other, hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x y − g ( y )d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ − β ) (cid:0) ln(1 /x ) (cid:1) − β if x is small enough.Finally, it suffices to define F as a set of functions in f ∈ C ( R + ) which are zero for x > and suchthat f (cid:12)(cid:12) [0 , = G am g for some g ∈ G (recall that G am is injective). ✷ As noted in the introduction, in this section we will avoid complex conjugation. Thus in the place of theusual sesquilinear scalar product ( f | g ) := Z ∞ f ( x ) g ( x )d x, (8.7)we will prefer to use the bilinear product h f | g i := Z ∞ f ( x ) g ( x )d x, (8.8)Clearly, (8.8) is well defined for f, g ∈ L ( R + ) . Instead of the usual adjoint T ∗ we will use the transpose T , defined with respect to (8.8), see [8].An important role will be played by the 1st order operators given by the formal expression A ρ := ∂ x − ρx . (8.9)A detailed analysis of (8.9) has been done in [5], where the notation was slightly different: A ρ := − i( ∂ x − ρx ) . 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 [5]. For the proof of theinfinite codimensionality assertion in 6 see the proof of Lemma 3.9 there (where γ is arbitrary > ). Theorem 8.1 A min ρ ⊂ A max ρ .2. A min ρ = − A max − ρ , A max ρ = − A min − ρ .3. A min ρ and A max ρ are homogeneous of degree − . . A min ρ = A max ρ iff | Re( ρ ) | ≥ . If this is the case, we will often denote them simply by A ρ
5. If
Re( ρ ) = , then D ( A min ρ ) = H .6. If Re( ρ ) = , then H + C x ρ ξ is a dense subspace of D ( A ρ ) of infinite codimension.7. If | Re( ρ ) | < , then D ( A max ρ ) = H + C x ρ ξ = H .8. If Re( ρ ) , Re( ρ ′ ) ∈ ] − , ] and ρ = ρ ′ then D ( A max ρ ) = D ( A max ρ ′ ) . Now let us describe possible factorizations of H m into operators of the form A min ρ and A max ρ . On theformal level they correspond to one of the factorizations (1.10) and (1.11). Theorem 8.2
1. For
Re( m ) > − we have h f | H m g i = h A max + m f | A max + m g i , f ∈ D ( A max + m ) , g ∈ D ( A max + m ) ∩ D ( H m ) . (8.10) Moreover, D ( H m ) = n f ∈ D ( A max + m ) | A max + m f ∈ D ( A min − − m ) o , (8.11) H m f = A min − − m A max + m f, f ∈ D ( H m ) . (8.12)
2. For
Re( m ) > we have h f | H m g i = h A min − m f | A min − m g i , f ∈ D ( A min − m ) , g ∈ D ( A min − m ) ∩ D ( H m ) . (8.13) Moreover, D ( H m ) = n f ∈ D ( A min − m ) | A min − m f ∈ D ( A max − + m ) o , (8.14) H m f = A max − + m A min − m f, f ∈ D ( H m ) . (8.15)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 [5], especially pages 571–574 and 577. Theorem 8.3
The following bilinear forms are extensions of h f | H m g i = h H m f | g i , f, g ∈ D ( H m ) , (8.16) to larger domains:1. For ≤ Re( m ) , h A + m f | A + m g i = h A − m f | A − m g i , f, g ∈ H . (8.17)
2. For < Re( m ) < , h A + m f | A + m g i = −h A min − m f | A min − m g i , f, g ∈ H . (8.18)
3. For
Re( m ) = 0 , h A + m f | A + m g i , f, g ∈ D ( A + m ) ⊃ H + C x + m ξ, (8.19) h A − m f | A − m g i , f, g ∈ D ( A − m ) ⊃ H + C x − m ξ. (8.20)13 . For − < Re( m ) < , h A max + m f | A max + m g i , f, g ∈ H + C x + m ξ. (8.21)Note that for Re( m ) > both factorizations yield the same quadratic form. This is no longer truefor Re( m ) = 0 , m = 0 , when there are two distinct quadratic forms with distinct domain correspondingto H m . Finally, for − < m < , 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 withThm 4.22 of [5]. Theorem 8.4
For real − < m the operators H m are positive and self-adjoint. The correspondingsesquilinear form can be factorized as follows:1. For ≤ m , ( p H m f | p H m g ) = ( A + m f | A + m g ) = ( A − m f | A − m g ) , f, g ∈ Q ( H m ) = H . (8.22) H m is essentially self-adjoint on C ∞ c ( R + ) .2. For < m < , ( p H m f | p H m g ) = ( A + m f | A + m g ) = ( A min − m f | A min − m g ) , f, g ∈ Q ( H m ) = H . (8.23) H m is the Friedrichs extension of L m restricted to C ∞ c ( R + ) .3. For m = 0 , ( p H f | p H g ) = ( A f | A g ) , f, g ∈ Q ( H ) = D ( A ) ) H + C x ξ. (8.24) H is both the Friedrichs and Krein extension of L restricted to C ∞ c ( R + ) .4. For − < m < , ( p H m f | p H m g ) = ( A max + m f | A max + m g ) , f, g ∈ Q ( H m ) = H + C x + m ξ. (8.25) H m is the Krein extension of L m restricted to C ∞ c ( R + ) . A Holomorphic families of closed operators and the Kato-RellichTheorem
In this appendix we describe a few general concepts and facts from the operator theory, which we use inour paper.The definition (or actually a number of equivalent definitions) of a holomorphic family of boundedoperators is quite obvious and does not need to be recalled. In the case of unbounded operators thesituation 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 whosevalues are closed operators on H . We say that this is a holomorphic family of closed operators if for each z ∈ Θ there exists a neighborhood Θ of z , a Banach space K and a holomorphic family of boundedoperators Θ ∋ z A ( z ) ∈ B ( K , H ) such that Ran A ( z ) = D ( H ( z )) and Θ ∋ z H ( z ) A ( z ) ∈ B ( K , H ) is a holomorphic family of bounded operators.The following theorem is essentially a version of the well-known Kato-Rellich Theorem generalizedfrom self-adjoint to closed operators: 14 heorem A.1 Suppose that A is a closed operator on a Hilbert space H . Let B be an operator D ( A ) →H such that k Bf k ≤ c k Af k , f ∈ D ( A ) . (A.26) Then for | z | < c the operator A + zB is closed on D ( A ) and (cid:8) z ∈ C | | z | < c − (cid:9) ∋ z A + zB (A.27) is a holomorphic family of closed operators.Proof. We easily check that the norms p k f k + k Af k and p k f k + k ( A + zB ) f k are equivalent for | z | < c . By restriction to the closure of D ( A ) we can assume that A is densely defined, so that we candefine A ∗ . The operator ( A ∗ A + 1l) − is unitary from H to D ( A ) . Clearly, it is bounded in the sense of H . Now C ∋ z ( A + zB )( A ∗ A + 1l) − (A.28)is obviously a polynomial of degree 1 with values in bounded operators (hence obviously a holomorphicfamily). ✷ Let us also quote the following fact proven by Bruk [4], see also [9]:
Proposition A.2 If z A ( z ) is a holomorphic family of closed operators, then so is z A ( z ) ∗ . Acknowledgements
Jan Dereziński is grateful to F. Gesztesy for drawing attention to the references [1, 2, 3] andfor useful comments. His work was supported by National Science Center (Poland) under the grantUMO-2019/35/B/ST1/01651.