Momentum approach to the 1/r^2 potential as a toy model of the Wilsonian renormalization
aa r X i v : . [ m a t h - ph ] D ec Momentum approach to the 1 /r potentialas a toy model of the Wilsonian renormalization Jan Derezi´nski, Oskar Grocholski,Faculty of Physics, University of Warsaw,Warsaw, PolandDecember 23, 2020
Abstract
The Bessel operator, that is, the Schr¨odinger operator on the halfline with a potential proportional to 1 /x ,is analyzed in the momentum representation. Many features of this analysis are parallel to the approach `a laK. Wilson to Quantum Field Theory: one needs to impose a cutoff, add counterterms, study the renormalizationgroup flow with its fixed points and limit cycles. Keywords: Schr¨odinger operators, Bessel operators, momentum representation, renormalization groupMSC2020: 47E99, 81Q10, 81Q80
Our paper is devoted to one of the most curious operators in mathematical physics, L α := − d d x + (cid:16) α − (cid:17) x , (1)where α ∈ R . Let us briefly list the most important properties of L α . It is always Hermitian (symmetric) on C ∞ c ( R + ). However, it is essentially self-adjoint only for α ≥
1. For α < α > α ≤ α = 0 self-adjoint realizations can be described by specifying boundary conditions at zero as appropriatemixtures of x + m and x − m , where m := √ α . For α = 0, one needs to take mixtures of x and x ln x and thereis an curious “phase transition”: m is real for α ≥ α <
0. Moreover, all self-adjoint realizationsof L α are bounded from below for α ≥ α < L α can be computed explicitly. For instance, eigenfunction expansions related to L α can be given in terms of Bessel functions, as described in the classic book by Titchmarsh [18]. This allows us todescribe explicitely all self-adjoint realizations of L α , as described by many authors, e.g. [1, 2, 4–10, 16].It is useful to note that the d -dimensional Schr¨odinger operator with the inverse square potential L α,d := − ∆ d + gr (2)can be reduced to the 1-dimensional one (1). In fact, if we restrict (2) to spherical harmonics of degree l andmake a simple transformation, then the radial operator coincides with L α , where α depends on d and l . In ourpaper, for simplicity we always stick to the dimension 1, except for a short resume of the reduction of the 3- to the1-dimensional case.One of the most striking properties of L α is its homogeneity of degree −
2. In other words, if U τ denotes thescaling transformation (see (15)), then U τ L α U − τ = e − τ L α . (3)This suggests to introduce the transformation R τ ( B ) := e τ U τ BU − τ , (4)1cting on, say, self-adjoint operators. R ∋ τ R τ is a representation of the group R and preserves the set of self-adjoint realizations of L α . Some of these realizations are “fixed points” of R τ . This suggests an analogy betweenthe action of R τ on self-adjoint extensions of L α and the renormalization group acting on models of Quantum FieldTheory (QFT).In Quantum Field Theory it is usually more convenient to work in the momentum rather than position rep-resentation. This is related to the fact that experimentalists have at their disposal a finite range of energies. Inorder to compute various useful quantities one typically needs to impose a cutoff at momentum Λ and to addappropriate Λ-dependent counterterms. The desired quantities are obtained in the limit Λ → ∞ , provided thatvarious parameters are appropriately adjusted.This procedure was greatly clarified by the Nobel prize winner K. Wilson, who stressed the role of scalingtransformations in the process of renormalization. He also pointed out that it is not so important whether QFThas a well defined limit for Λ → ∞ , what matters is a weak dependence of low energy quantities on the high energycutoff.A number of authors [11–15], mostly with a high energy physics background, noticed that L α has a greatpedagogical potential to illustrate the concept of renormalization group and Wilson’s ideas. Typically, these authorslook at the operator (2) in dimension 3 and the s-wave sector in the momentum representation. Its naive formulationis ill defined due to diverging integrals at large momenta. Besides, it does not select a self-adjoint realization. Thedivergence for high momenta can be cured by introducing a cutoff. It is also indispensable to add an appropriatecounterterm. In the limit Λ → ∞ one obtains well defined cutoff independent quantities, which are associated withone of self-adjoint realizations. Along the way one encounters various objects analogous to renormalization groupflows, fixed points and renormalization group equations known from QFT.Note in parenthesis that the operator L α is in some sense “too good” to ilustrate all ideas of the Wilsonianrenomalization. In fact, one can completely remove the cutoff and obtain a “renormalized theory”, somethingthat may be impossible in QFT, as Wilson suggested. This “renormalized theory” in the case of L α is one of itsself-adjoint realizations.A recent exposition of the “Wilsonian approach” to the inverse square potential can be found in the paper [11],which is the main inspiration for our work. [11] starts from the 3-dimensional operator (2) in the s-wave sectorand develops its renormalization theory as a pedagogical illustration of Wilsonian ideas. The exposition containedin [11] stays all the time on the momentum side, and the reader may find it difficult to connect it with the standardtheory of self-adjoint realizations of L α , more transparent to describe in the position representation.Our paper is devoted to an explanation of the mathematics that underlies the momentum approach to inversesquare potentials. We will see how to make the momentum approach rigorous, including all its tools such as cutoffsand counterterms, and how it relates to the position approach.In Sec. 2 we briefly recall the theory of self-adoint realizations of L α on L ( R + ) in the position representation.We will follow the terminology and conventions of [1, 2].In Sec. 3 we recapitulate the content of [11]. On purpose, we stick to the original terminology and line ofreasoning.Sec. 4 is the main part of our work. Here we describe rigorously the momentum approach to L α . As it shouldbe, it is completely equivalent to the position approach. Most readers will find it probably more complicatedthan the position approach, but it has a special “high energy physics flavor”, which some of the readers may findappealing. We explain how to construct the minimal, maximal and self-adjoint realizations of the Bessel operatorin the momentum representation. The main difficulty is the choice of appropriate counterterms. These countertemsare not needed for the minimal realization. For the maximal realization they are quite simple and sometimes notnecessary. For self-adjoint realizations they are quite elaborate and satisfy a differential equation that resemblesthe renormalization group equations from high energy QFT.When discussing the momentum representation of the Bessel operator we do not use the whole line R . Wealways consider the half-line R + , which is consistent with the many-dimensional analysis, where the radius is alwayspositive. This leads to the following problem: the usual Fourier transformation is adapted to the line, whereas onthe half-line we have two natural cousins of the Fourier transformation: the sine and the cosine transformation. Theformer diagonalizes the Dirichlet Laplacian, and the latter the Neumann Laplacian. We discuss both approaches,because from the 1-dimensional point of view both are equally natural. Actually, from the 3-dimensional point ofview only the approach based on the sine transformation is relevant, because the 3-dimensional Laplacian in thes-wave sector reduces to the Dirichlet Laplacian on the half-line.2 Position approach
Let us recall basic concepts of the theory of self-adjoint extensions of Hermitian operators (often called symmetricoperators). A comprehensive treatment of this topic can be found in [17].Let A and B be operators with domains D ( A ) and D ( B ) respectively. We say that A is contained in B if D ( A ) ⊆ D ( B ) and A = B on D ( A ). We then write A ⊆ B .An operator A is Hermitian if for all x, y ∈ D ( A )( Ax | y ) = ( x | Ay ) , (5)where ( ·|· ) is the scalar product. An operator A is self-adjoint if A ∗ = A , where the star denotes the Hermitianadjoint. L α We will use the symbol L α for the expression (1), without specifying its domain. Operators that are given by thisexpression and have a concrete domain will be called realizations of L α .Following [2] (see Sec. 4 and App. A therein for details), we first introduce the maximal and minimal realizations of L α . The maximal realization, denoted L max α , has the domain D ( L max α ) = (cid:8) f ∈ L ( R + ) | L α f ∈ L ( R + ) (cid:9) . (6)and the minimal realization of L α , denoted L min α , is the closure of the restriction of L α to C ∞ c ( R + ). Obviously, L min α ⊂ L max α . Moreover, one can show that (cid:0) L min α (cid:1) ∗ = L max α . (7)Hence, L min α is Hermitian.(Note by R + we always denote the open positive half-line ]0 , ∞ [. Thus functions in C ∞ c ( R + ) are supported awayfrom 0.)One can show that for α ≥ L max α = L min α is self-adjoint. In what follows it will be denoted H m , where m := √ α is the positive square root of α .For α < L min α is strictly smaller than that of L max α and neither operator is self-adjoint. Therefore,both operators are of little use in physical applications: e.g. eigenvalues of L max α cover almost the whole complexplane and corresponding eigenvectors are not mutually orthogonal.Self-adjoint realisations of L α are self-adjoint operators H such that L min α ⊂ H = H ∗ ⊂ L max α . (8)They are constructed as follows. Firstly, one finds solutions of L α ψ = 0. These are of the form ψ ( x ) = (cid:26) a · x / m + b · x / − m , m = 0; a · √ x + b · √ x ln x, m = 0 . (9)Here, m := √ α . Note that m ∈ R or m ∈ i R is defined up to a sign.For α ≥ x / m with m positive is square integrable near zero. This is the key element of the proof ofthe essential self-adjointness of H m on C ∞ c ( R + ).For α < κ, ν be complex numbers. The followingspaces lie between D ( L min α ) and D ( L max α ): D ( H m,κ ) = { f ∈ D ( L max α ) | for c ∈ C f ( x ) − c ( κx / − m + x / m ) ∈ D ( L min α ) near 0 } , D ( H m, ∞ ) = { f ∈ D ( L max α ) | for c ∈ C f ( x ) − cx / − m ∈ D ( L min α ) near 0 } , D ( H ν ) = { f ∈ D ( L max0 ) | for c ∈ C f ( x ) − c ( x / ln x + νx / ) ∈ D ( L min0 ) near 0 } , D ( H ∞ ) = { f ∈ D ( L max0 ) | for c ∈ C f ( x ) − cx / ∈ D ( L min0 ) near 0 } . (10)3hey define realizations of L α denoted H m,κ and H ν . The above definition implies immediately that H m,κ = H − m,κ − . Moreover, Hermitian adjoints of these operators are ( H m,κ ) ∗ = H ¯ m, ¯ κ and ( H ν ) ∗ = H ¯ ν (with theconvention ¯ ∞ = ∞ ).To prove it, let us consider functions f ∈ D ( H m,κ ), g ∈ D ( H ¯ m, ¯ κ ). One has( H m,κ f | g ) − ( f | H ¯ m, ¯ κ g ) = lim x → (cid:16) ¯ f ( x ) ∂ x g ( x ) − g ( x ) ∂ x ¯ f ( x ) (cid:17) . By analysis of behaviour of f and g near x = 0 we conclude that the Wronskian at 0 vanishes. Calculations for H ν are analogous.Thus, self-adjoint extensions of L min m with m < • H m,κ , with m ∈ ]0 ,
1[ and κ ∈ R ∪ {∞} , • H m,κ , with m ∈ i R + and | κ | = 1, • H ν with ν ∈ R ∪ {∞} .Note that H and H − are the Laplacians with the Dirichlet, resp. Neumann boundary condition at 0. They willbe often denoted H D , resp. H N . After this discussion, we are able to identify the point spectra of the above Hamiltonians, following e.g. [1]. • For m ∈ ]0 ,
1[ and κ ∈ R ∪ {∞} : σ p ( H m,κ ) = (cid:26) − (cid:16) κ Γ( − m )Γ( m ) (cid:17) − /m (cid:27) for κ ∈ ] − ∞ , ,σ p ( H m,κ ) = ∅ for κ ∈ [0 , ∞ ] . (11) • For m = i m I ∈ i R + and | κ | = 1: σ p ( H i m I ,κ ) = (cid:26) − (cid:18) arg (cid:16) κ Γ( − i m I )Γ(i m I ) (cid:17) + 2 πnm I (cid:19) | n ∈ Z (cid:27) . (12)It implies that for α < L min α has infinitely many elements withaccumulation points at 0 and −∞ . • For ν ∈ R ∪ {∞} σ p ( H ν ) = {− ν − γ ) } ν ∈ R ,σ p ( H ∞ ) = ∅ . (13)In all cases, bound-state solutions to the eigenvalue problem H m,κ ψ = − k ψ or H ν ψ = − k ψ are of the form ψ ( x ) = √ kxK m ( kx ) or √ kxK ( kx ) , (14)where k > K m ( z ) is the MacDonald function of order m [19]. The proof can be found e.g. in [1], Sec. 5.2. Let us introduce the dilation (scaling) operator U τ . It is a unitary transformation on L ( R + ) acting on functionsin the following way: U τ f ( x ) = e τ/ f (e τ x ) . (15)We say that an operator B is homogenous of degree n if U τ BU − τ = e nτ B. max m and L min m are homogeneous of degree −
2. However, extensions H m,κ and H ν are homogeneous only for κ = 0or κ = ∞ and ν = ∞ . Moreover [1], U τ H m,κ U − τ = e − τ H m, e − mτ κ U τ H ν U − τ = e − τ H ν + τ . (16)Indeed, if f ∈ D ( H m,κ ), then U τ f ∈ D ( H m, e − mτ κ ), and if f ∈ D ( H ν ), then U τ f ∈ D ( H ν + τ ).For the purpose of this article, the action of R ∋ τ R τ defined in (4) can be called the renormalization group .In the set of self-adjoint realizations of L α we have the following behaviors of the renormalization group flow: • For α ≥ • for 0 < α < H m, ∞ and H m, . The former is attractive and the latter repulsive.(We choose m as the positive square root of α ). • for α = 0 there is only one fixed point: H ∞ . • for α < K and F correspond to the Krein and Friedrichs extensions of L min α .K=F FK K=F α “solid” “phasetransition” “liquid” “gas” In the literature, such as [11–15], the potential 1 /r is often considered in the 3-dimensional setting. It is straight-forward to reduce the 3-dimensional problem to 1-dimension.Every ψ ∈ L ( R ) can be written as ψ = ∞ X l =0 l X m = − l Y l,m ( θ, φ ) 1 r f l,m ( r ) , where Y l,m are spherical harmonics. We have4 π Z | ψ | d x = ∞ X l =0 l X m = − l Z ∞ | f l,m ( r ) | d r. Consider the 3-dimensional Laplacian ∆ . If ψ belongs to the domain of − ∆ , then − r ∆ ψ = ∞ X l =0 l X m = − l (cid:16) − ∂ r + (cid:16)(cid:0) l + 1 / (cid:1) − (cid:17) r (cid:17) f l,m , (17) f l,m ∼ r l +1 near 0 . (18)Thus − ∆ on the l degree spherical harmonics is equivalent to the operator H l + . In particular, on s-wave functions,it is equivalent to the Dirichlet Laplacian H = H D Now consider the following differential expression: L α, := − ∆ + gr . (19)5y the previous analysis, on l degree spherical harmonics it is equivalent to − ∂ r + (cid:16) ( l + 1 / + g − (cid:17) r , that is, to L α with α = ( l + 1 / + g , which can be viewed as H l + perturbed by gr . On s-wave functions, it is H D perturbed by gr . In this section we describe the approach to L α in the spirit of the Wilson renormalization. We will mostly followthe presentation of [11], preserving to a large extent its style and language. In particular, we use the 3-dimensionalformulation—as described in Subsect. 2.5, it is equivalent to the 1-dimensional case. Analysis of this problem inthe momentum representation can be found also in [12–15].A broader description of the Wilsonian renormalization procedure applied to quantum field theory is presentedin [21–23].The main object of the analysis of [11] is the formal expression˜ L α, ψ ( ~p ) = | ~p | ψ ( ~p ) + 14 π (cid:16) α − (cid:17) Z d ~q | ~p − ~q | ψ ( ~q ) . (20)Formally, it is L α, of Eq. (19) in the momentum representation. More precisely, if F f ( ~p ) is the Fourier transformof f ( ~x ), then ˜ L α, := F L α, F − . (21)For further convenience let us denote p := | ~p | . We will restrict our attention to s-wave functions, for which ψ ( ~p ) = ψ ( p ), since there is no angular dependence, and the operator Eq. (20) can be written in the following way(see Eqs. (1) and (4) in [11]):˜ L α ψ ( p ) = p ψ ( p ) + (cid:16) α − (cid:17) Z ∞ d q q (cid:16) θ ( p − q ) p − + θ ( q − p ) q − (cid:17) ψ ( q ) , (22)where θ ( x ) denotes the Heaviside step function and we dropped 3 from the subscript.Unfortunately, (22) does not define a Hermitian operator. To find a self-adjoint realisation, we introduce anultraviolet cutoff Λ and a counter-term f Λ Λ . Thus we consider a family of cutoff Hamiltonians˜ H α (Λ , f Λ ) ψ ( p ) := ( p ψ ( p ) + R Λ0 d q q (cid:16) V ( p, q ) + f Λ Λ (cid:17) ψ ( q ) , for | ~p | ≤ Λ0 , for | ~p | > Λ . (23)where V ( p, q ) = ( α −
14 ) (cid:16) θ ( p − q ) p − + θ ( q − p ) q − (cid:17) . (24)We would like to find out what kind of dependence of f Λ on Λ guarantees the existence of a limit H α (Λ , f Λ ) forΛ → ∞ as a self-adjoint operator. To this end we assume that we have a fixed eigenfunction ψ satisfying H α (Λ , f Λ ) ψ ( p ) = Eψ ( p ) , p < Λ . (25)Following the terminology of [11], we will say that two Hamiltonians ˜ H m (Λ , f Λ ) and ˜ H α (Λ ′ , f Λ ′ ) are equivalent if˜ H α (Λ , f Λ ) ψ ( p ) = ˜ H α (Λ ′ , f Λ ′ ) ψ ( p ) for p < min(Λ , Λ ′ ) . Two Hamiltonians are equivalent if the function γ (Λ) ≡ f Λ / Λ satisfies the equationd γ (Λ)dΛ = h γ (Λ) − / − α Λ i . (26)To prove it, we consider an infinitesimal reduction of the cut-off Λ → Λ − dΛ. Let ψ be a solution of˜ H α (Λ , f Λ ) ψ ( p ) = − k ψ ( p ) . (27)6y splitting the integral Z Λ0 d q q (cid:16) V ( p, q ) + γ (Λ) (cid:17) ψ ( q ) ≈ Z Λ − d Λ0 d q q (cid:16) V ( p, q ) + γ (Λ) (cid:17) ψ ( q ) + dΛ (cid:16) V (Λ , q ) + γ (Λ) (cid:17) ψ (Λ) , and using the relation resulting from Eqs. (23) and (27): ψ (Λ) = − k + Λ Z Λ0 d q q (cid:16) ( α −
14 )Λ − + γ (Λ) (cid:17) ψ ( q ) , assuming the large cut-off k ≪ Λ we conclude that the Hamiltonian ˜ H α (Λ , f Λ ) is equivalent to ˜ H α (Λ − dΛ , f Λ − dΛ )such that ˜ H α (Λ − dΛ , f Λ − dΛ ) ψ ( p ) := p ψ ( p ) + Z Λ − dΛ0 d q q (cid:16) V ( p, q ) + γ (Λ − dΛ) (cid:17) ψ ( q ) , where γ (Λ − dΛ) = γ (Λ) − dΛ h γ (Λ) − / − α Λ i . (28)Taking the limit dΛ → γ (Λ) = Λ − f Λ , where • for α >
0, we set m := √ α > f Λ = − m tanh (cid:18) − atanh (cid:16) f Λ + m + 1 / m (cid:17) + m log ΛΛ (cid:19) − m − . (29) • for α <
0, we set m I := √− α , f Λ = m I tan (cid:18) arctan (cid:16) f Λ − m I + 1 / m I (cid:17) + m I log ΛΛ (cid:19) + m I − . (30) • for α = 0 f Λ = f Λ + 1 / − ( f Λ + 1 /
4) log ΛΛ − . (31)Let ˜ U denote the dilation in the momentum representation, that is˜ U ( τ ) F = F U ( τ ) . (32)Note that in the momentum representation the dilation acts in the opposite way than in the position representation:˜ U ( τ ) ψ ( p ) = e − τ ψ (e − τ p ) . (33)It is easy to see that ˜ U ( τ ) ˜ H α (Λ , f ) ˜ U ( − τ ) = e − τ ˜ H α (e τ Λ , f ) . (34)Let us now choose a fixed cut-off scale Λ and analyze how these operators depend on the parameter f Λ : • For 0 < α < m = √ α , we have −| m | − m − / ≤ f Λ ≤ | m | − m − /
4. For these extreme values, theatanh function becomes ±∞ . For such an initial condition, the function f Λ is independent on Λ. It followsthat in this case there are two fixed points: f ± Λ = ±| m | − m − /
4, for which Hamiltonians ˜ H m (Λ , f ± Λ ) areequivalent for all values of Λ. • For α = 0 one has f Λ ∈ R ∪ {∞} and there is one fixed point corresponding to f Λ = − / • For α < m I := √− α we have f Λ ∈ R ∪ {∞} and there are no fixed points. Moreover, there is a discretescaling symmetry: for any n ∈ Z , Hamiltonians ˜ H (Λ , f Λ ) and ˜ H (Λ · exp( πn/m I ) , f Λ ) are equivalent.We thus reproduce the picture that we know from Sect. 2.7 Momentum approach R d Self-adjoint operators on L ( R d ) of the form H = − ∆ + V ( x ) (35)are usually called Schr¨odinger operators . They are typically studied by methods of the configuration space. However,they can be also approached from the momentum point of view.Let us recall the two most common conventions for the Fourier transformation:˜ f ( p ) := Z e − i xp f ( x )d x, (36) F f ( p ) := (2 π ) − d Z e − i xp f ( x )d x = (2 π ) − d ˜ f ( p ) . (37)Note that F is unitary. Let ˜ H denote the operator H in the momentum representation, that is˜ H := F H F − . (38)If the potential V is well-behaved, then ˜ H can be written as˜ Hψ ( p ) = p ψ ( p ) + (2 π ) − d Z ˜ V ( p − q ) ψ ( q )d q. (39) R + Often we consider Schr¨odinger operators on a subset of R d . Then the representation (39) becomes problematic,because it needs to take into account the boundary conditions. Besides, it is not obvious what should replace theFourier transformation.The case of a particular interest is R + . In this case we have two distinguished realizations of the 1-dimensionalLaplacian: the Dirichlet Laplacian ∆ D and the Neumann Laplacian ∆ N . Instead of the Fourier transformation itis natural to use the sine transformation F D or the cosine transformation F N :( F D f )( p ) := r π Z ∞ sin( px ) f ( x )d x, (40)( F N f )( p ) := r π Z ∞ cos( px ) f ( x )d x. (41)Both are involutions, that is, F ψ = F ψ = ψ, (42)they are unitary and diagonalize the Dirichlet/Neumann Laplacian: F D ( − ∆ D ) F D ψ ( p ) = p ψ ( p ) , (43) F N ( − ∆ N ) F N ψ ( p ) = p ψ ( p ) . (44)Consider now a Schr¨odinger operator on the half-line with a potential V . Let us first assume that V is sufficientlyregular, say, V ∈ L ( R + ). Then one can impose the Dirichlet or Neumann boundary conditions at 0, so that onehas two Schr¨odniger operators H D := − ∆ D + V ( x ) , (45) H N := − ∆ N + V ( x ) . (46)To obtain their momentum versions, it is natural to apply the sine transform to the first and the cosine transformto the second Hamiltonian: ˜ H D := F D H D F D , (47)˜ H N := F N H N F N . (48)8et us extend the potential V to an even function on the whole R , so that V ( − x ) = V ( x ). Let ˜ V ( p ) be theFourier transform of V , as in (36). Then the formula (39) has the following two generalizations to the half-line case:˜ H D ψ ( p ) = p ψ ( p ) + 12 π Z ∞ ( ˜ V ( p − q ) − ˜ V ( p + q ) (cid:1) ψ ( q )d q, (49)˜ H N ψ ( p ) = p ψ ( p ) + 12 π Z ∞ ( ˜ V ( p − q ) + ˜ V ( p + q ) (cid:1) ψ ( q )d q. (50) If the potential is singular at 0, say, non-integrable, we have several problems:1. We cannot use the standard Dirichlet/Neumann boundary conditions.2. We need to interpret V as an irregular distribution.3. The Fourier transform of V will grow at infinity, making the formulas (49) and (50) problematic.All these problems are present when want to analyze the Bessel operator on the half-line L α := − d d x + (cid:16) α − (cid:17) x (51)in the momentum representation.The function x has an obvious even extension to R . However this function is not integrable, and hence doesnot define a regular distribution. There exists a well-known one-parameter family of even distributions that outsideof 0 coincide with x . Their Fourier transforms are − π | p | + a, (52)where a is a constant. Let us set a = 0. (This constant will reappear in disguise of a counter-term anyway later.)A priori it is not obvious which transformation is more appropriate: the sine or the cosine transformation. Letus consider both, setting ˜ L α, D := F D L α F D , (53)˜ L α, N := F N L α F N , (54)(where, as before, L α is treated as a formal expression).Then, using − π | p | as the Fourier transform of x , (49) and (50) yield˜ L α, D ψ ( p ) = p ψ ( p ) + (cid:16) α − (cid:17) Z ∞ d q (cid:0) θ ( p − q ) q + θ ( q − p ) p (cid:1) ψ ( q ) , (55)˜ L α, N ψ ( p ) = p ψ ( p ) − (cid:16) α − (cid:17) Z ∞ d q (cid:0) θ ( p − q ) p + θ ( q − p ) q (cid:1) ψ ( q ) . (56)So far, we have treated ˜ L α, D and ˜ L α, N as formal expressions. We would like to make them well defined, first inthe “minimal sense”.In the position representation, an easy way to make the expression L α well defined was to restrict its domain to C ∞ c ( R + ), which after the closure led to the minimal realisation. Of course, it was not necessary to demand that thesupport is away from zero, however function in the domain should have vanished of an appropriate order at zero.In the momentum representation it is useful to restrict the domain to rapidly decaying functions—in the positionrepresentation this guarantees the smoothness. On the position side, this also guarantees that even/odd derivativesat zero vanish in the Dririchlet, resp. Neumann case, see (174). This does not suffice—in addition, we need toimpose conditions that on the position side yield vanishing first/zeroth derivative. Thus, we introduce the spaces L , ∞ j ( R + ) := (cid:26) ψ ∈ L , ∞ ( R + ) | Z ∞ | ψ ( p ) | p n < ∞ , n = 0 , , . . . ; Z ∞ ψ ( p ) p j d p = 0 (cid:27) , j = 0 , . (57)9 heorem 4.1
1. The operator ˜ L α, D given by the formula (55) restricted to L , ∞ ( R + ) is well defined and clos-able. Let us denote its closure by ˜ L min α, D . We then have ˜ L min α, D := F D L min α F D . (58)
2. The operator ˜ L α, N given by the formula (56) restricted to L , ∞ ( R + ) is well defined and closable. Let us denoteits closure by ˜ L min α, N . Then we have ˜ L min α, N := F N L min α F N . (59) Proof.
Consider first the Dirichlet case. Let ψ ∈ L , ∞ ( R + ) and x >
0. Consider φ ( p ) := Z ∞ d q (cid:0) θ ( p − q ) q + θ ( q − p ) p (cid:1) ψ ( q ) (60)= Z ∞ p ( p − q ) ψ ( q )d q. (61)Clearly, φ (0) = 0 and lim p →∞ φ ( p ) = 0. Moreover, φ ′ ( p ) = Z ∞ p ψ ( q )d q (62)and lim p →∞ φ ′ ( p ) = 0. We can integrate twice obtaining Z ∞ sin( px ) φ ( p )d p = 1 x Z ∞ cos( px ) φ ′ ( p )d p (63)= 1 x Z ∞ sin( px ) ψ ( p )d p. (64)By (175), Z ∞ sin( px ) p φ ( p )d p = − ∂ x Z ∞ sin( px ) φ ( p )d p. (65)Therefore, F D ˜ L α, D ψ = L α F D ψ. (66)Thus F D ˜ L α, D F D coincides with L α on F D L , ∞ ( R + ).Suppose now that φ = F D ψ with ψ ∈ L , ∞ ( R + ). We have φ ( n ) (0) = 0, n = 0 , , . . . (by (174), because ψ ∈ L , ∞ ( R + )). Moreover, φ ′ (0) = 0 (by (176), because ∞ R pψ ( p )d p = 0). Hence φ ( x ) = O ( x ).Let χ, ξ ∈ C ∞ ( R + ), χ = 1, ξ = 0 near 0 and χ = 0, ξ = 1 near ∞ . Set φ ǫ ( x ) := χ ( x/ǫ ) ξ ( xǫ ) ψ ( x ). Then weeasily show that k L α ( φ − φ ǫ ) k + k φ − φ ǫ k →
0. Clearly, φ ǫ ∈ C ∞ c ( R + ). Therefore F D L , ∞ ( R + ) ⊂ D ( L min α ).Clearly, C ∞ c ]0 , ∞ [ ⊂ F D L , ∞ ( R + ). Therefore, F D L , ∞ ( R + ) is dense in D ( L min α ).This proves that the closure of F D ˜ L α, D F D restricted to F D L , ∞ ( R + ) coincides with L min α .Consider next the Neumann case. Let ψ ∈ L , ∞ ( R + ). Consider φ ( p ) := Z ∞ d q (cid:0) θ ( p − q ) p + θ ( q − p ) q (cid:1) ψ ( q ) (67)= Z ∞ p ( q − p ) ψ ( q )d q. (68)Clearly, lim p →∞ φ ( p ) = 0. Moreover, φ ′ ( p ) = Z p ψ ( q )d q (69)10nd φ ′ ( p ) = 0, lim p →∞ φ ′ ( p ) = 0. We can integrate twice obtaining Z ∞ cos( px ) φ ( p )d p = − x Z ∞ sin( px ) φ ′ ( p )d p (70)= 1 x Z ∞ cos( px ) ψ ( p )d p. (71)Clearly, Z ∞ cos( px ) p φ ( p )d p = − ∂ x Z ∞ cos( px ) φ ( p )d p. (72)Therefore F N ˜ L α, N ψ = L α F N ψ. (73)Thus F N ˜ L α, N F N coincides with L α on F N L , ∞ ( R + ).Suppose now that φ = F N ψ with ψ ∈ L , ∞ ( R + ). We have φ ( n ) (0) = 0, n = 1 , , . . . (because ψ ∈ L , ∞ ( R + )).Moreover, φ (0) = 0 (because ∞ R ψ ( p )d p = 0). Hence φ ( x ) = O ( x ).The remaining arguments are the same as in the Dirichlet case. (cid:3) As we remember, elements of D ( L max α ) behave like c + x / m + c − x / − m as x →
0, where α = m . By (178) and(179), the sine and cosine transforms of x ± m are proportional to p − ∓ m . This suggests to enlarge the domain ofthe minimal operators by functions that behave like p − ∓ m as p → ∞ .More precisely, let ψ ± m, D and ψ ± m, N be functions in L ( R + ) satisfying the following condition: there existsΛ > p > Λ ⇒ ψ ± m, D ( p ) = p − ∓ m , Z Λ ψ ± m, D ( q ) q d q = Λ ∓ m ∓ m , (74) p > Λ ⇒ ψ log , D ( p ) = p − log( p ) , Z Λ ψ log , D ( q ) q d q = − + 2Λ log Λ , (75) p > Λ ⇒ ψ ± m, N ( p ) = p − ∓ m , Z Λ ψ ± m, N ( q )d q = Λ − ∓ m − ∓ m , (76) p > Λ ⇒ ψ log , N ( p ) = p − log( p ) , Z Λ ψ log , N ( q )d q = − − − − log Λ . (77)Note that (74) and (76) imply Λ > Λ ⇒ Z Λ0 ψ ± m, D ( q ) q d q = Λ ∓ m ∓ m , (78)Λ > Λ ⇒ Z Λ0 ψ log , D ( q ) q d q = − + 2Λ log Λ , (79)Λ > Λ ⇒ Z Λ0 ψ ± m, N ( q )d q = Λ − ∓ m − ∓ m , (80)Λ > Λ ⇒ Z Λ0 ψ log , N ( q )d q = − − − − log Λ . (81)11e set D ( ˜ L max α, D ) := D ( ˜ L min α, D ) + C ψ m, D + C ψ − m, D (82) D ( ˜ L max0 , D ) := D ( ˜ L min0 , D ) + C ψ , D + C ψ log , D (83) D ( ˜ L max α, N ) := D ( ˜ L min α, N ) + C ψ m, N + C ψ − m, N , (84) D ( ˜ L max0 , N ) := D ( ˜ L min0 , N ) + C ψ , N + C ψ log , N . (85)Clearly, the difference of two functions satisfying (74), (75), (76) or (77), belongs to L , ∞ ( R + ), resp to L , ∞ ( R + ).Therefore, (82), (84), (83) and (85) do not depend on the choices of ψ ± m, D and ψ ± m, N .The formulas (55) and (56) are in general ill defined on (82), (84), (83) and (85), because the integrals can bedivergent at ∞ . In order to define the maximal operators in the momentum representation, we note that every ψ in D ( ˜ L max α, D ) or in D ( ˜ L max α, N ) behaves like ψ ( p ) ∼ ˜ c + p − − m + ˜ c − p − + m , p → ∞ , (86)or ψ ( p ) ∼ ˜ c + p − log( p ) + ˜ c − p − , p → ∞ , (87)for some uniquely defined ˜ c + , ˜ c − . Then we set˜ L max α, D ψ ( p ) = p ψ ( p ) + (cid:16) α − (cid:17) lim Λ →∞ Z Λ0 d q (cid:0) θ ( p − q ) q + θ ( q − p ) p (cid:1) ψ ( q ) + ˜ c + p Λ − − m + m + ˜ c − p Λ − + m − m ! , (88)˜ L max0 , D ψ ( p ) = p ψ ( p ) −
14 lim Λ →∞ Z Λ0 d q (cid:0) θ ( p − q ) q + θ ( q − p ) p (cid:1) ψ ( q ) + ˜ c + p (cid:0) − + 2Λ − log Λ (cid:1) + 2˜ c − p Λ − ! , (89)˜ L max α, N ψ ( p ) = p ψ ( p ) − (cid:16) α − (cid:17) lim Λ →∞ Z Λ0 d q (cid:0) θ ( p − q ) p + θ ( q − p ) q (cid:1) ψ ( q ) − ˜ c + Λ − m − m − ˜ c − Λ + m + m ! , (90)˜ L max0 , N ψ ( p ) = p ψ ( p ) −
14 lim Λ →∞ Z Λ0 d q (cid:0) θ ( p − q ) p + θ ( q − p ) q (cid:1) ψ ( q ) + ˜ c + (cid:0) − log Λ (cid:1) − c − Λ ! . (91)(92)Note that in the Dirichlet case with α ≤ (and hence | Re( m ) | ≤ ) the counterterms in (88) are not necessary—they go to zero and the integrals are convergent. However in all other cases the counterterms are needed. Theorem 4.2
The operators ˜ L max α, D and ˜ L max α, N are well defined closed operators. We have ˜ L max α, D := F D L max α F D , (93)˜ L max α, N := F N L max α F N . (94) Proof.
Consider first the Dirichlet case. It is enough to consider ψ = ψ ± m, D p ψ ( p ) + (cid:16) α − (cid:17) lim Λ →∞ Z Λ0 d q (cid:0) θ ( p − q ) q + θ ( q − p ) p (cid:1) ψ ( q ) + p Λ − ∓ m ± m ! (95)= p ψ ( p ) + (cid:16) α − (cid:17) lim Λ →∞ Z Λ p d q ( p − q ) ψ ( q ) + p Λ − ∓ m ± m + Λ ∓ m ∓ m ! (96)= p (cid:0) ψ ( p ) − p − ∓ m (cid:1) + (cid:16) α − (cid:17) Z ∞ p d q ( p − q ) (cid:0) ψ ( q ) − q − ∓ m (cid:1) . (97)Thus ˜ L max α, D ψ ( p ) is well defined. 12sing Lemma A.2 1 and (96) we show that for x >
0, possibly in the sense of an oscillatory integral, r π Z ∞ sin( px ) (cid:0) ˜ L max α, D ψ (cid:1) ( p )d p (98)= − ∂ x + (cid:16) α − (cid:17) x !r π Z ∞ sin( px ) ψ ( p )d p, (99)which proves (93).The Neumann case is analogous. It is enough to consider ψ = ψ ± m, N . p ψ ( p ) − (cid:16) α − (cid:17) lim Λ →∞ Z Λ0 d q (cid:0) θ ( p − q ) p + θ ( q − p ) q (cid:1) ψ ( q ) − Λ ∓ m ∓ m ! (100)= p ψ ( p ) + (cid:16) α − (cid:17) lim Λ →∞ Z Λ p d q ( p − q ) ψ ( q ) + p Λ − ∓ m ± m + Λ ∓ m ∓ m ! (101)= p (cid:0) ψ ( p ) − p − ∓ m (cid:1) + (cid:16) α − (cid:17) Z ∞ p d q ( p − q ) (cid:0) ψ ( q ) − q − ∓ m (cid:1) . (102)Thus ˜ L max α, N ψ ( p ) is well defined.Using Lemma A.2 2 and (101) we show that for x > r π Z ∞ cos( px ) (cid:0) ˜ L max α, N ψ (cid:1) ( p )d p (103)= − ∂ x + (cid:16) α − (cid:17) x !r π Z ∞ cos( px ) ψ ( p )d p, (104)which proves (94). The special case m = 0 is proven in an analogous way. (cid:3) Let us remark that the “kinetic terms” for p − ± m , that is, p p − ± m , are never square integrable for | m | < L one needs to balance them with the integral terms. Let ˜ κ, ˜ ν ∈ C . Let us set D ( ˜ H m, ˜ κ, D / N ) := { g ∈ D ( ˜ L max m , D / N ) | for c ∈ C g ( p ) − c ( p − / − m + ˜ κp − / m ) ∈ D ( ˜ L min m , D / N ) for p near ∞} , D ( ˜ H m, ∞ , D / N ) := { g ∈ D ( ˜ L max m , D / N ) | for c ∈ C g ( p ) − cp − / m ∈ D ( ˜ L min m , D / N ) for p near ∞} , D ( ˜ H ˜ ν , D / N ) := { g ∈ D ( ˜ L max0 , D / N ) | for c ∈ C g ( p ) − c ( p − / ln p + ˜ νp − / ) ∈ D ( ˜ L min0 , D / N ) for p near ∞} , D ( ˜ H ∞ , D / N ) := { g ∈ D ( ˜ L max0 , D / N ) | for c ∈ C g ( p ) − cp − / ∈ D ( ˜ L min0 , D / N ) for p near ∞} . We define the operators ˜ H m, ˜ κ, D , ˜ H m, ˜ κ, N , ˜ H ˜ νm, D , ˜ H ˜ νm, N , to be the operators ˜ L max m , D , resp. ˜ L max m , N restricted to thedomains described above. Theorem 4.3
The above defined operators are self-adjoint in the following situations: • ˜ H m, ˜ κ, D , ˜ H m, ˜ κ, N for m ∈ ] − , \{ } and ˜ κ ∈ R ∪ {∞} , • ˜ H m, ˜ κ, D , ˜ H m, ˜ κ, N for m ∈ i R \{ } and | ˜ κ | = 1 , • ˜ H ˜ ν , D , ˜ H ˜ ν , N for ν ∈ R ∪ {∞} . esides, we have F D H m,κ F D = ˜ H m, D , ˜ κ D , ˜ κ D = sin( π + πm )Γ( − − m )sin( π − πm )Γ( − + m ) κ, (105) F N H m,κ F N = ˜ H m, N , ˜ κ N , ˜ κ N = cos( π + πm )Γ( − − m )cos( π − πm )Γ( − + m ) κ, (106) F D H ν F D = ˜ H ˜ ν D , D , ˜ ν D = − ν − γ + π , (107) F N H ν F N = ˜ H ˜ ν N , N , ˜ ν N = − ν − γ − π . (108) γ is the Euler constant. Equalities above follow from sine and cosine transforms (178) and (179). The limiting case m = 0 is obtained by considering the limit m → κ = − mν and ˜ κ D / N = − − m ˜ ν D / N . We will see now that the above operators can be defined as the limits as Λ → ∞ of cut-off operators:˜ L α, D (Λ) ψ ( p ) := θ (Λ − p ) (cid:18) p ψ ( p ) + (cid:16) α − (cid:17) Z Λ0 d q (cid:0) θ ( p − q ) q + θ ( q − p ) p (cid:1) ψ ( q ) (cid:19) , (109)˜ L α, N (Λ) ψ ( p ) := θ (Λ − p ) (cid:18) p ψ ( p ) − (cid:16) α − (cid:17) Z Λ0 d q (cid:0) θ ( p − q ) p + θ ( q − p ) q (cid:1) ψ ( q ) (cid:19) , (110)after adding an appropriate counterterm. The counterterms will be proportional to the following rank-one operators: K D (Λ) ψ ( p ) := θ (Λ − p ) Z Λ0 pqψ ( q )d q, (111) K N (Λ) ψ ( p ) := θ (Λ − p ) Z Λ0 ψ ( q )d q. (112)To gain the intuition about the above operators let us note that for a large class of functions ψ ,lim Λ →∞ ( F D φ | K D (Λ) F D ψ ) = 12 π | ψ ′ (0) | , (113)lim Λ →∞ ( F N φ | K N (Λ) F N ψ ) = 12 π | ψ (0) | . (114)We have already noted an important role played by the dilation in the position representation. In principle, wehave two kinds of momentum representation–obtained by the sine and cosine transformation. However, the dilationcoincides for both kinds of the momentum representation, and it will be denoted by ˜ U ( τ ):˜ U ( τ ) = F D U ( τ ) F − = F N U ( τ ) F − . (115)Note that it acts in the opposite way than in the position representation:˜ U ( τ ) ψ ( p ) = e − τ ψ (e − τ p ) . (116)It is easy to see that˜ U ( τ ) ˜ L α, D (Λ) ˜ U ( − τ ) = e − τ ˜ L α, D (e τ Λ) , ˜ U ( τ ) K D (Λ) ˜ U ( − τ ) = e − τ K D (e τ Λ); (117)˜ U ( τ ) ˜ L α, N (Λ) ˜ U ( − τ ) = e − τ ˜ L α, N (e τ Λ) , ˜ U ( τ ) K N (Λ) ˜ U ( − τ ) = e − τ K N (e τ Λ) . (118)Now we expect that for suitably chosen functions Λ f Λ , g Λ , the following operators approximate our Hamil-tonians: ˜ L α, D (Λ) + f Λ Λ K D (Λ) , (119)˜ L α, N (Λ) + Λ g Λ K N (Λ) . (120) Note that for m → − mν ) x − m + x m ∼ − m ( ν + log x ). f Λ and g Λ on the cut-off Λ. Assume that for large momenta ψ ( p ) ∼ p − − m +˜ κp − + m . For p < Λ we obtain the leading-order dependence on the cut-off: (cid:16) ˜ L α, D (Λ) + f Λ Λ K D (Λ) (cid:17) ψ ( p ) , (121) ∼ (cid:16) m − (cid:17)(cid:16) Λ − − m ( − − m ) + ˜ κ Λ − + m ( − + m ) (cid:17) + f Λ (cid:16) Λ − − m ( − m ) + ˜ κ Λ − + m ( + m ) (cid:17) , (122) (cid:0) ˜ L α, N (Λ) + Λ g Λ K N (Λ) (cid:1) ψ ( p ) (123) ∼ − (cid:16) m − (cid:17)(cid:16) Λ − m ( − m ) + ˜ κ Λ + m ( + m ) (cid:17) + g Λ (cid:16) Λ − m ( + m ) + ˜ κ Λ + m ( − m ) (cid:17) . (124)Demanding that this dependence vanishes yields f m, ˜ κ, Λ = − (cid:16) ( − m )Λ − m + ˜ κ ( + m )Λ m (cid:17)(cid:16) ( − m ) − Λ − m + ˜ κ ( + m ) − Λ m (cid:17) , (125)= − − m − m (cid:0) ( − m )˜ κ Λ m − ( + m )Λ − m (cid:1)(cid:0) ( − m )˜ κ Λ m + ( + m )Λ − m (cid:1) (126)= − − m − m tanh (cid:0) m log(Λ /λ D ) (cid:1) , λ − m D = ( − m )( + m ) ˜ κ ; (127) g m, ˜ κ, Λ = (cid:16) ( + m )Λ − m + ˜ κ ( − m )Λ m (cid:17)(cid:16) ( + m ) − Λ − m + ˜ κ ( − m ) − Λ m (cid:17) , (128)= 14 + m + m (cid:0) ( + m )˜ κ Λ m − ( − m )Λ − m (cid:1)(cid:0) ( + m )˜ κ Λ m + ( − m )Λ − m (cid:1) (129)= 14 + m + m tanh (cid:0) m log(Λ /λ N ) (cid:1) , λ − m D = ( + m )( − m ) ˜ κ. (130)In terms of an initial value at the scale Λ we can write f m, ˜ κ, Λ = − − m − m (cid:0) ( − m ) + f Λ (cid:1) Λ m + (cid:0) ( + m ) + f Λ (cid:1) Λ m − (cid:0) ( − m ) + f Λ (cid:1) Λ m + (cid:0) ( + m ) + f Λ (cid:1) Λ m ; (131) g m, ˜ κ, Λ = 14 + m + m (cid:0) ( + m ) − g Λ (cid:1) Λ m + (cid:0) ( − m ) − g Λ (cid:1) Λ m (cid:0) ( + m ) − g Λ (cid:1) Λ m + (cid:0) ( − m ) − g Λ (cid:1) Λ m . (132) f Λ and g Λ satisfy the differential equationsΛ ddΛ f Λ = (cid:16) f Λ + 14 + m (cid:17) − m ; (133)Λ ddΛ g Λ = − (cid:16) g Λ − − m (cid:17) + m . (134)For m = 0 the parameter ˜ κ does not work. Applying the de l’Hopital rule with ˜ κ = − − m ˜ ν we obtain f ˜ ν , Λ = − log Λ + ˜ ν + 2log Λ + ˜ ν − , (135) g ˜ ν , Λ = log Λ + ˜ ν − ν + 2 (136)The above heuristic analysis can be transformed into the following rigorous statement:15 heorem 4.4 The bounded self-adjoint operators ˜ L m , D (Λ) + f m, ˜ κ, Λ Λ K D (Λ) , (137)˜ L m , N (Λ) + Λ g m, ˜ κ, Λ K N (Λ) (138) converge as Λ → ∞ to the operators ˜ H m, ˜ κ, D , resp. ˜ H m, ˜ κ, N in the strong resolvent sense.Proof. The proof will be based on Theorem VIII.25 (a) from [24]: Let { A } ∞ n =1 and A be self-adjoint operators andsuppose that D is a common core for all A n and A . If A n φ → Aφ for each φ ∈ D , then A n → A in the strongresolvent sense.Consider first the Dirichlet case. Fix vectors ψ ± m, D , as in (74). We will use the space D = L , ∞ ( R + ) + C ( ψ m, D + ˜ κψ − m, D ) , (139)which is a core of ˜ H m, ˜ κ, D . If ψ min ∈ L , ∞ ( R + ), then clearly˜ L m , D (Λ) ψ min → Λ →∞ ˜ L min m , D ψ min = ˜ H m, ˜ κ, D ψ min . (140)Moreover, K D (Λ) ψ min ( p ) = − θ (Λ − p ) p Z ∞ Λ qψ min ( q )d q. (141)Now f m, ˜ κ, Λ is uniformly bounded, Z ∞ Λ qψ min ( q )d q = O (Λ −∞ ) , (cid:16) Z ∞ θ (Λ − p ) p d p (cid:17) = O (Λ ) . (142)Therefore, f m, ˜ κ, Λ Λ K D (Λ) ψ min → Λ →∞ . (143)Now consider ψ := ψ m, D + ˜ κψ − m, D . Remember that f Λ is chosen such that (122) is 0. Therefore, for largeenough Λ, f Λ Λ K D (Λ) ψ ( p ) = θ (Λ − p ) f Λ Λ p (cid:16) Λ − m ( − m ) + ˜ κ Λ + m ( + m ) (cid:17) (144)= θ (Λ − p ) (cid:16) m − (cid:17) p (cid:16) Λ − − m ( − − m ) + ˜ κ Λ − + m ( − + m ) (cid:17) . (145)Hence, (cid:16) ˜ L m , D (Λ) + f m, ˜ κ, Λ Λ (cid:17) ψ ( p ) (146)= θ (Λ − p ) p ψ ( p ) + (cid:16) m − (cid:17) Z Λ0 d q (cid:0) θ ( p − q ) q + θ ( q − p ) p (cid:1) ψ ( q ) + p Λ − − m + m + ˜ κp Λ − + m − m !! (147)= θ (Λ − p ) (cid:18)(cid:0) p ( ψ ( p ) − p − − m − ˜ κp − + m (cid:1) + (cid:16) m − (cid:17) Z ∞ p d q ( p − q ) (cid:0) ψ ( q ) − q − m − ˜ κq − + m (cid:1)(cid:19) . (148)We can drop θ (Λ − p ) from the expression above (remember that Λ is large enough) and we obtain (cid:16) ˜ L m , D (Λ) + f m, ˜ κ, Λ Λ (cid:17) ψ → Λ →∞ ˜ L max α, D ψ = ˜ H m, ˜ κ, D ψ. (149)The Neumann case is analogous. Fix vectors ψ ± m, N , as in (76). We will use the space D = L , ∞ ( R + ) + C ( ψ m, N + ˜ κψ − m, N ) , (150)16hich is a core of ˜ H m, ˜ κ, N . If ψ min ∈ L , ∞ ( R + ), then clearly˜ L m , N (Λ) ψ min → Λ →∞ ˜ L min m , N ψ min = ˜ H m, ˜ κ, N ψ min . (151)Moreover, K N (Λ) ψ min ( p ) = − θ (Λ − p ) Z ∞ Λ ψ min ( q )d q. (152)Now, g m, ˜ κ, Λ is uniformly bounded, Z ∞ Λ ψ min ( q )d q = O (Λ −∞ ) , (cid:16) Z ∞ θ (Λ − p )d p (cid:17) = O (Λ ) . (153)Therefore, Λ g m, ˜ κ, Λ K N (Λ) ψ min → Λ →∞ . (154)Now consider ψ := ψ m, N + ˜ κψ − m, N . Remember that g Λ is chosen such that (124) is 0. Therefore,Λ g Λ K N (Λ) ψ ( p ) = θ (Λ − p )Λ g Λ (cid:16) Λ − − m ( − − m ) + ˜ κ Λ − + m ( − + m ) (cid:17) (155)= θ (Λ − p ) (cid:16) m − (cid:17)(cid:16) Λ − m ( − m ) + ˜ κ Λ + m ( + m ) (cid:17) . (156)Hence, (cid:16) ˜ L m , N (Λ) + Λ g m, ˜ κ, Λ (cid:17) ψ ( p ) (157)= θ (Λ − p ) p ψ ( p ) − (cid:16) m − (cid:17) Z Λ0 d q (cid:0) θ ( p − q ) p + θ ( q − p ) q (cid:1) ψ ( q ) − Λ − m − m − ˜ κ Λ + m + m !! . (158)Taking into account (90), we obtain (cid:16) ˜ L m , N (Λ) + Λ g m, ˜ κ, Λ (cid:17) ψ → Λ →∞ ˜ L max α, N ψ = ˜ H m, ˜ κ, N ψ. (cid:3) (159)Let us observe, that the norm of the counter-terms converges to infinity both in the Dirichlet and Neumanncases. In particulat, we cannot neglect the counterterms even in the Dirichlet case with | Re( m ) | < (when one canomit the counter-terms in Eq. (88)).Note also that the above analysis of ˜ H m, ˜ κ, D is essentially an expanded version of the Wilsonian approach of [11],which we described in Section 3, translated from 3 dimensions to 1 dimension. Treatment of the Schr¨odinger equation with potential 1 /r requires an appropriate definition of the domain. This canbe achieved directly, in the position representation, as it was presented in the Section 2. It can be also equivalentlydone in the momentum representation, essentially following the Wilsonian renormalization scheme. Despite havingbeen devised as an approximate method, this scheme when rigorously implemented yields a construction of self-adjoint realizations of the Schr¨odinger operator with with potential 1 /r . Acknowledgements
O.G. would like to thank Prof. Stanis law G lazek for suggesting this research topic and his comments on thiswork. He also thanks Maciej Lebek and Ignacy Na lecz for discussions about the Wilsonian approach.J.D. acknowledges useful discussions about renormalization with Prof. Stanis law G lazek. The work of J.D.was supported by National Science Center (Poland) under the grant UMO-2019/35/B/ST1/01651. Fourier analysis on a halfline
Fourier analysis on the line is well-known. Somewhat less known is Fourier analysis on the half-line, where the roleof the Fourier transformation is played by two trasformations: the cosine and sine transformation, which are themain subject of this appendix.
A.1 Cosine and sine transformation
Let us start with recalling basic constructions elements of the Fourier analysis on the line. Let H , ∞ ( R ) := { φ ∈ C ∞ ( R ) | Z ∞−∞ | φ ( n ) ( x ) | d x < ∞} , (160) L , ∞ ( R ) := n ψ ∈ L ( R ) | Z ∞−∞ | ψ ( p ) | | p | n d p < ∞ , n = 0 , , . . . o (161)be the Sobolev space and the weighted space of the infinite order–two examples of Frechet spaces. The Fouriertransformation swaps these spaces: F L , ∞ ( R ) = H , ∞ ( R ) . (162)Functions on the line can be decomposed into even and odd functions L ± ( R ) := { ψ ∈ L ( R ) | ψ ( − x ) = ± ψ ( x ) } , L ( R ) = L ( R ) ⊕ L − ( R ) . (163)Even and odd functions are preserved by the Fourier transformation: F L ± ( R ) = L ± ( R ) , (164) F L , ∞ ( R ) = H , ∞ ( R ) . (165)Every function on the halfline can be extended to an even or odd function: J ± ψ ( p ) := (cid:26) ψ ( p ) p ≥ ± ψ ( − p ) p < . (166) J ± maps L ( R + ) onto L ± ( R ). The restriction to the positive halfline is the left inverse of J :( J ± ψ ) (cid:12)(cid:12)(cid:12) R + = ψ, ψ ∈ L ( R + ) . (167)The sine and cosine transformations can be defined as the composition of the Fourier transformation with the aboveextension and the restriction, more precisely, F D ψ = i( F J − ψ ) (cid:12)(cid:12)(cid:12) R + , F N ψ = ( F J + ψ ) (cid:12)(cid:12)(cid:12) R + . (168)Introduce also the Frechet spaces analogous to (160) and (161) corresponding to the halfline: H , ∞ ( R + ) := { φ ∈ C ∞ ( R + ) | Z ∞ | φ ( n ) ( x ) | d x < ∞} , (169) L , ∞ ( R + ) := n ψ ∈ L ( R + ) | Z ∞ | ψ ( p ) | p n < ∞ , n = 0 , , . . . o . (170)We will also need the following closed subspaces of H , ∞ ( R + ): H , ∞ + ( R + ) := { φ ∈ H , ∞ ( R + ) | φ ( n ) = 0 , n = 1 , , , . . . } , (171) H , ∞− ( R + ) := { φ ∈ H , ∞ ( R + ) | φ ( n ) = 0 , n = 0 , , , . . . } . (172)The following proposition is straightforward: 18 roposition A.1 J + H , ∞ + ( R + ) = H , ∞ ( R ) ∩ L ( R ) , J − H , ∞− ( R + ) = H , ∞ ( R ) ∩ L − ( R ) , (173) F N L , ∞ ( R + ) = H , ∞ + ( R + ) , F D L , ∞ ( R + ) = H , ∞− ( R + ) . (174) For ψ ∈ L , ∞ ( R + ) we have ∂ nx F D ψ = ( − n F D p n ψ, ∂ nx F N ψ = ( − n F N p n ψ, (175) ∂ n +1 x F D ψ = ( − n +1 F N p n +1 ψ, ∂ n +1 x F N ψ = ( − n F D p n +1 ψ. (176) A.2 Homogeneous functions on the halfline
Let f ∈ L ( R + ). We say that f possesses an oscillatory integral if for any φ ∈ C ∞ c ([0 , ∞ [) such that φ = 1 near 0lim Λ →∞ Z ∞ f ( p ) φ ( p/ Λ)d p (177)exists and does not depend on the choice of φ . The value (177) is then called the oscillatory integral of f .Note that the integrals that appear in the definitions of the cosine and sine transforms (40), (41) for functions,say, from L ( R + ) can always be understood as oscillatory (but not always in the usual Lebesgue sense).Note the following formulas, valid for x >
0, where one needs to use oscillatory integrals: r π Z ∞ sin( px ) x λ d x = − r π p − λ − sin( π λ )Γ( − λ ) , λ > − , (178) r π Z ∞ cos( px ) x λ d x = r π p − λ − cos( π λ )Γ( − λ ) , λ > − . (179) Lemma A.2
Suppose that ψ ∈ L ( R + ) such that for large p we have ψ ( p ) = p λ . Set ψ ( p ) := lim Λ →∞ (cid:16) − Z Λ p ψ ( q )d q + Λ λ +1 λ + 1 (cid:17) , (180) ψ ( p ) := lim Λ →∞ (cid:16) Z Λ p ( p − q ) ψ ( q )d q − p Λ λ +1 λ + 1 + Λ λ +2 λ + 2 (cid:17) . (181) (The above limits exist, because the functions after the limit sign are constant for large Λ ). Clearly, ψ ′ ( p ) = ψ ( p ) , ψ ′ ( p ) = ψ ( p ) , (182) for large p , ψ ( p ) = p λ +1 λ + 1 , ψ ( p ) = p λ +2 ( λ + 1)( λ + 2) . (183) and the following holds:1. Suppose that lim Λ →∞ (cid:16) − Z Λ0 qψ ( q )d q + Λ λ +2 λ + 2 (cid:17) = 0 . (184) Then Z ∞ sin( px ) ψ ( p )d p = − x Z ∞ cos( px ) ψ ( p )d p = − x Z ∞ sin( px ) ψ ( p )d p. (185)
2. Suppose that lim Λ →∞ (cid:16) Z Λ0 ψ ( q )d q − Λ λ +1 λ + 1 (cid:17) = 0 . (186) Then Z ∞ cos( px ) ψ ( p )d p = x Z ∞ sin( px ) ψ ( p )d p = − x Z ∞ cos( px ) ψ ( p )d p. (187)19 The above integrals are not always defined as Lebesgue integrals—they are always OK as oscillatory integrals)Proof.