On radial Schroedinger operators with a Coulomb potential

This paper presents a thorough analysis of 1-dimensional Schroedinger operators whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. We allow both coupling constants to be complex. Using natural boundary conditions at 0, a two parameter holomorphic family of closed operators is introduced. We call them the Whittaker operators, since in the mathematical literature their eigenvalue equation is called the Whittaker equation. Spectral and scattering theory for Whittaker operators is studied. Whittaker operators appear in quantum mechanics as the radial part of the Schroedinger operator with a Coulomb potential.


Introduction
This paper is devoted to 1-dimensional Schrödinger operators with Coulomb and centrifugal potentials. These operators are given by the differential expressions The parameters α and β are allowed to be complex valued. We shall study realizations of L β,α as closed operators on L 2 (R + ), and consider general boundary conditions. The operator given in (1.1) is one of the most famous and useful exactly solvable models of Quantum Mechanics. It describes the radial part of the Hydrogen Hamiltonian. In the mathematical literature, this operator goes back to Whittaker, who studied its eigenvalue equation in [33]. For this reason, we call (1.1) the Whittaker operator. This paper is a continuation of a series of papers [2,6,7] devoted to an analysis of exactly solvable 1-dimensional Schrödinger operators. We follow the same philosophy as in [6]. We start from a formal differential expression depending on complex parameters. Then we look for closed realizations of this operator on L 2 (R + ). We do not restrict ourselves to self-adjoint realizations -we look for realizations that are well-posed, that is, possess non-empty resolvent sets. This implies that they satisfy an appropriate boundary condition at 0, depending on an additional complex parameter. We organize those operators in holomorphic families.
Before describing the holomorphic families introduced in this paper, let us recall the main constructions from the previous papers of this series. In [2,6] we considered the operator As is known, it is useful to set α = m 2 . In [2] the following holomorphic family of closed realizations of (1.2) was introduced: defined by L m 2 with boundary conditions ∼ x 1 2 +m .
It was proved that for Re(m) ≥ 1 the operator H m is the only closed realization of L m 2 .
In the region −1 < Re(m) < 1 there exist realizations of L m 2 with mixed boundary conditions. As described in [6], it is natural to organize them into two holomorphic families: Note that related investigations about these operators have also been performed in [31,32].
It was noted in this reference that this family is holomorphic except for a singularity at (β, m) = 0, − 1 2 , which corresponds to the Neumann Laplacian. For Re(m) ≥ 1 the operator H β,m is also the only closed realization of L β,m 2 . In the region −1 < Re(m) < 1 there exist other closed realizations of L β,m 2 . The boundary conditions corresponding to H β,m are distinguished-we will call them pure. The goal of the present paper is to describe the most general well-posed realizations of L β,m 2 , with all possible boundary conditions, including the mixed ones.
We shall show that it is natural to organize all well-posed realizations of L β,m 2 for −1 < Re(m) < 1 in three holomorphic families: The generic family H β,m,κ , with β ∈ C, −1 < Re(m) < 1, m ∈ − 1 2 , 0, 1 2 , κ ∈ C ∪ {∞}, defined by L β,m 2 with boundary conditions ∼ x 1 2 +m 1 − β 1 + 2m x + κx After having introduced these families and describing a few general results, we provide the spectral analysis of these operators and give the formulas for their resolvents. We also describe the eigenprojections onto the eigenfunctions of these operators. They can be organized into a single family of bounded 1-dimensional projections P β,m (λ), where λ satisfies L max β,m P β,m (λ) = λP β,m (λ). Here L max β,m denotes the maximal operator which is introduced in Section 2.3.
There exists a vast literature devoted to Schrödinger operators with Coulomb potentials, including various boundary conditions. Let us mention, for instance, an interesting dispute in Journal of Physics A [22,10,23] about self-adjoint extensions of the 1-dimensional Schrödinger operator on the real line with a Coulomb potential (without the centrifugal term). Papers [21,11,24] discuss generalized Nevanlinna functions naturally appearing in the context of such operators, especially in the range of parameters |Re(m)| ≥ 1. See also [4,9,12,13,14,15,17,18,19,25,26,27,28,29] and references therein. However, essentially all these references are devoted to real parameters β, m and self-adjoint realizations of Whittaker operators. The philosophy of using holomorphic families of closed operators, which we believe should be one of the standard approaches to the study of special functions, seems to be confined to the series of paper [2,6,7], which we discussed above.
The main reason why we are able to analyze the operator (1.1) so precisely is the fact that it is closely related to an exactly solvable equation, the so-called Whittaker equation paper. It is extensively developed in a large appendix to this paper. It can be viewed as an extension of the theory of Bessel and Whittaker equation presented in [6,7]. We discuss in detail various special cases: the degenerate, the Laguerre and the doubly degenerate cases. Besides the well-known Whittaker functions I β,m and K β,m , described for example in [7], we introduce a new kind of Whittaker functions, denoted X β,m . It is needed to fully describe the doubly degenerate case.
The Whittaker equation and its close cousin, the confluent equation, are discussed in many standard monographs, including [1,3,30]. Nevertheless, it seems that our treatment contains a number of facts about the Whittaker equation, which could not be found in the literature. For example, we have never seen a satisfactory detailed treatment of the doubly degenerate case. The function X β,m seems to be our invention. Without this function it would be difficult to analyze the doubly degenerate case. Figures 1  and 2, which illustrate the intricate structure of the degenerate, Laguerre and doubly degenerate cases, apparently appear for the first time in the literature. Another result that seems to be new is a set of explicit formulas for integrals involving products of solutions of the Whittaker equation. These formulas are related to the eigenprojections of the Whittaker operator.

The Whittaker operator
In this section we define the main objects of our paper: the Whittaker operators H β,m,κ , H ν β, 1 2 and H ν β,0 on the Hilbert space L 2 ]0, ∞[ .

Notations
We shall use the notations The Hilbert space L 2 (R + ) is endowed with the scalar product We will also use the bilinear form defined by The Hermitian conjugate of an operator A is denoted by A * . Its transpose is denoted by A # . If A is bounded, then A * and A # are defined by the relations The definition of A * has the well-known generalization to the unbounded case. The definition of A # in the unbounded case is analogous.
The following holomorphic functions are understood as their principal branches, that is, their domain is C\] − ∞, 0] and on ]0, ∞[ they coincide with their usual definitions from real analysis: ln(z), √ z, z λ . We set arg(z) := Im ln(z) . The extensions of these functions to ] − ∞, 0] or to ] − ∞, 0[ are from the upper half-plane.
The Wronskian of two continuously differentiable functions f and g on R + is denoted by W (f, g; ·) and is defined for x ∈ R + by (2.1)

Zero-energy eigenfunctions of the Whittaker operator
In order to study the realizations of the Whittaker operator L β,α one first needs to find out what are the possible boundary conditions at zero. The general theory of 1dimensional Schrödinger operators says that there are two possibilities: (i) there is a 1-parameter family of boundary conditions at zero, (ii) there is no need to fix a boundary condition at zero.
One can show that (i)⇔(i') and (ii)⇔(ii'), where (i') for any λ ∈ C the space of solutions of (L β,α − λ)f = 0 which are square integrable around zero is 2-dimensional, (ii') for any λ ∈ C the space of solutions of (L β,α − λ)f = 0 which are square integrable around zero is at most 1-dimensional.
We refer to [5] and references therein for more details.
In the above criterion one can choose a convenient λ. In our case the simplest choice corresponds to λ = 0. Therefore, we first discuss solutions of the zero eigenvalue Whittaker equation for m and β in C. As analyzed in more details in Section B.5, solutions of (2.2) can be constructed from solutions of the Bessel equation. More precisely, for β = 0, let us define the following function for x ∈ R + : where J m is defined in Section B.4. For β = 0 we set Then, the equation ( for β = 0 we introduce where γ is Euler's constant and Y m is defined in Section B.4. For β = 0 we set y 0,0 (x) := x 1 2 ln(x) and y 0, 1 2 (x) := 1.
Let us describe the asymptotics of these solutions near zero. The following results can be computed based on the expressions provided in the appendix of [6]. For any m ∈ C with −2m ∈ N × one has In the exceptional cases one has together with

Maximal and minimal operators
For any α and β ∈ C we consider the differential expression acting on distributions on R + . The corresponding maximal and minimal operators in L 2 (R + ) are denoted by L max β,α and L min β,α , see [7,Sec. 3.2] for the details. The domain of L max β,α is given by while L min β,α is the closure of the restriction of L β,α to C ∞ c ]0, ∞[ , the set of smooth functions with compact supports in R + . The operators L min β,α and L max β,α are closed and we have L min We say that f ∈ D(L min β,α ) around 0, (or, by an abuse of notation, f (x) ∈ D(L min β,α ) around 0) if there exists ζ ∈ C ∞ c [0, ∞[ with ζ = 1 around 0 such that f ζ ∈ D(L min β,α ). The following result follows from the theory of one-dimensional Schrödinger operators. (i) If f ∈ D(L max β,α ), then f and f are continuous functions on R + and converge to 0 at infinity.

Families of Whittaker operators
We can now provide the definition of three families of Whittaker operators. The first family covers the generic case. The Whittaker operator H β,m,κ is defined for any β ∈ C, for any m ∈ C with |Re(m)| < 1 and m ∈ − 1 2 , 0, 1 2 , and for any κ ∈ C ∪ {∞}: f − c j β,−m ∈ D(L min β,m 2 ) around 0 . The second family corresponds to m = 0: Finally, in the special case m = 1 2 we have the third family: Remark 2.2. Observe that the above boundary conditions could be described with the help of simpler functions. For example, in the above definitions we can replace The three families H β,m,κ , H ν β, 1 2 and H ν β,0 cover all possible well-posed extensions of L β,m 2 with |Re(m)| < 1. As already mentioned, we do not introduce a special family for m = − 1 2 , since it is covered by the family corresponding to m = 1 2 . For convenience, we also extend the definition of the first family to the exceptional cases by setting for β ∈ C and any κ ∈ C ∪ {∞} It is also convenient to introduce another two-parameter family of operators, which cover only special boundary conditions, which we call pure: (2.4) With this notation, for any β ∈ C, one has Remark 2.4. The family H β,m is essentially identical to the family denoted by the same symbol introduced and studied in [7]. The only difference with that reference is that the operator corresponding to (β, m) = 0, − 1 2 was left undefined, since it corresponds to a singularity. In the current paper we have decided to set H 0,− 1 Here is a comparison of the above families with the families H m,κ , H ν 0 introduced in [6] when β = 0. In the first column we put one of the newly introduced family, in the second column we put the families from [6,7].
with the convention that 1 0 = ∞ and 1 ∞ = 0. For completeness, let us also mention two special operators which are included in these families (for clarity, the indices are emphasized). The Dirichlet Laplacian on R + is given by while the Neumann Laplacian is given by Note that the former operator was also described in [6] by H m= 1 2 while the latter operator was described by H m=− 1 2 . We now gather some easy properties of the operators H β,m,κ .
Proof. Let us prove the first statement, the other ones can be obtained similarly. Recall from Proposition 2.1 (see also [2,Prop. A.2]) that for any f ∈ D(L max β,m 2 ) and g ∈ D(L max β,m 2 ), the functions f, f , g, g are continuous on R + . In addition, the Wronskian off and g, as introduced in (2.1), possesses a limit at zero, and we have the equality Thus, g ∈ D (H β,m,κ ) * if and only if W (f , g; 0) = 0, and then (H β,m,κ ) * g = L max β,m 2 g. By taking into account the explicit description of D(H β,m,κ ), straightforward computations show that W (f , g; 0) = 0 if and only if g ∈ D(Hβ ,m,κ ). One then deduces that (H β,m,κ ) * = Hβ ,m,κ . The property for the transpose of H β,m,κ can be proved similarly.
By combining Propositions 2.3 and 2.5 one easily deduces the following characterization of self-adjoint operators contained in our families: Corollary 2.6. The operator H β,m,κ is self-adjoint if and only if one of the following sets of conditions is satisfied: (ii) β ∈ R, m ∈ iR × and |κ| = 1. Let us finally mention some equalities about the action of the dilation group. For that purpose, we recall that the unitary group {U τ } τ ∈R of dilations acts on f ∈ L 2 (R + ) as U τ f (x) = e τ /2 f (e τ x). The proof of the following lemma consists in an easy computation.

Spectral theory
In this section we investigate the spectral properties of the Whittaker operators.

Point spectrum
The point spectrum is obtained by looking at general solutions of the equation for k ∈ C with Re(k) ≥ 0, and by considering only the solutions which are in the domain of the operators H β,m,κ , H ν β, 1 2 , or H ν β,0 . In the following statement, the convention 1 0 = ∞ and 1 ∞ = 0 is still used, Γ stands for the usual gamma function, ψ is the digamma function defined by ψ(z) = Γ (z)/Γ(z) and γ = −ψ(1). Since the special case β = 0 has already been considered in [6], we assume that β = 0 in the following statement, and recall in Theorem 3.4 the results obtained for β = 0. It is also useful to note that the condition β ∈ [0, ∞[ guarantees that either +Im( √ β) > 0 or −Im( √ β) > 0, due to our definition of the square root. 1 2 , and let κ ∈ C ∪ {∞}. Then the operator H β,m,κ possesses an eigenvalue λ ∈ C in the following cases: 2. Let β ∈ C × and ν ∈ C∪{∞}. Then H ν β, 1 2 possesses an eigenvalue λ in the following cases: 3. Let β ∈ C × and ν ∈ C∪{∞}. Then H ν β,0 possesses an eigenvalue λ in the following cases: Proof. We start with the special case λ = −k 2 = 0. The two solutions of the equation L β,m 2 f = 0 are provided by the functions with H ± m the Hankel function for dimension 1, see [6, App. A.5]. We then infer from [6, App. A.5] that for any z with −π < arg(z) ≤ π, one has as z → 0 For |Re(m)| < 1, this implies that the two functions h ± β,m belong to L 2 (R + ) near 0. On the other hand, for large z and | arg(∓iz)| < π − ε, ε > 0, one has and h ± β,m belongs to L 2 near infinity. Note that for ±Im( one infers that at most one of these functions is in L 2 near infinity, depending on the sign of Im(β). More precisely, for Im(β) > 0, the map x → H + β 2µ ,m (2µx) belongs to L 2 near infinity if µ < Im(β) and does not belong to L 2 near infinity otherwise. Under the same condition Im(β) > 0, the map infinity if µ < −Im(β) and does not belong to L 2 near infinity otherwise. Under the same condition Im(β) < 0, the map x → H + β 2µ ,m (2µx) never belongs to L 2 near infinity.
Finally, for Im(β) = 0, none of these functions belongs to L 2 near infinity. For the asymptotic expansion near 0, the information on H ± δ,m provided in [7, Eq. (2.31)] is not sufficient. However, the appendix of the current paper contains all the necessary information on these special functions. By taking into account the Taylor expansion of I δ,m near 0 provided in (A.3) and the equality Γ(α)Γ(1 − α) = π sin(πα) one infers that for |Re(m)| < 1 and m ∈ − 1 2 , 0, 1 2 one has and For 2m ∈ Z one has to consider the expression for K δ, 1 2 and K δ,0 provided in (A.18) and (A.19) respectively. Then, by considering the Taylor expansion near 0 of these functions one gets and From Equation (A.29) one finally deduces the relations To show 1.(ii) we consider the function if Im(β) < 0, and check for which κ these functions belong to D(H β,mκ ).
We refer again to the appendix for an introduction to these functions. The behaviour for large z of the function K δ,m (z) has been provided in (A.7), from which one infers that the first function in (3.7) is always in L 2 near infinity. On the other hand, since for | arg(z)| < π 2 one has it follows that the remaining two functions in (3.7) do not belong to L 2 near infinity as long as β 2k ∓ m − 1 2 ∈ N. Still in the non-degenerate case and when the condition β 2k + m − 1 2 ∈ N holds, it follows from relation (A.8) that the functions K β 2k ,m (2k·) and I β 2k ,−m (2k·) are linearly dependent, but still I β 2k ,m (2k·) does not belong to L 2 near infinity. Similarly, when β 2k − m − 1 2 ∈ N it is the function I β 2k ,−m (2k·) which does not belong to L 2 near infinity.
Let us now turn to the degenerate case, when m ∈ − 1 2 , 0, 1 2 . In this situation the two functions I δ,m and I δ,−m are no longer independent, as a consequence of (A.4). In the non-doubly degenerate case (see the appendix for more details), which means for β 2k , m ∈ Z, ± 1 2 or for β 2k , m ∈ Z + 1 2 , 0 , the above arguments can be mimicked, and one gets that only the function K β 2k ,m (2k·) belongs to L 2 near infinity. In the doubly degenerate case, the function X δ,m , introduced in (A.9), has to be used. This function is independent of the function K δ,m , as shown in (A.24). However, this function explodes exponentially near infinity, which means that X β 2k ,m (2k·) does not belong to L 2 near infinity. Once again, only the function K β 2k ,m (2k·) plays a role. As a consequence of these observations, it will be sufficient to concentrate on the function K β 2k ,m (2k·) and to check for which κ or ν does this function belong to the domain of the operators H β,m,κ , H ν β, 1 2 , or H ν β,0 respectively. For the behavior of this function near 0 one infers from (A.6) and (3.4 Similarly, it follows from (A.18) that (3.9) The statements 1.(i), 2.(i), and 3.(i) follow then straightforwardly.
Remark 3.2. A special feature of positive eigenvalues described in Theorem 3.1 is that the corresponding eigenfunctions have an inverse polynomial decay at infinity, and not an exponential decay at infinity, as it is often expected. This property can be directly inferred from the asymptotic expansion provided in (3.3). 1 a necessary condition for the existence of strictly positive eigenvalues is that Im(β) = 0. This automatically prevents these operators to be self-adjoint, as a consequence of Corollary 2.6.
For completeness let us recall the results already obtained in [6,Sec. 5] for β = 0.

Green's functions
Let us now turn our attention to the continuous spectrum. We shall first look for an expression for Green's function. We will use the well-known theory of 1-dimensional Schrödinger operators, as presented for example in the appendix of [2] or in [5]. We begin by recalling a result on which we shall rely.
Let AC(R + ) denote the set of absolutely continuous functions from R + to C, that is functions whose distributional derivative belongs to L 1 loc (R + ). Let also AC 1 (R + ) be the set of functions from R + to C whose distributional derivatives belong to AC(R + ). If V ∈ L 1 loc (R + ), it is not difficult to check that the operator −∂ 2 x + V can be interpreted as a linear map from AC 1 (R + ) to L 1 loc (R + ). The maximal operator associated to −∂ 2 x + V is then defined as The minimal operator L min is the closure of L max restricted to compactly supported functions. Note that L max = (L min ) # . As before, we say that a function f : The following statement contains several results proved in [5].
Assume that u(k, ·), v(k, ·) are linearly independent and that u(k, ·) ∈ L 2 around 0, v(k, ·) ∈ L 2 around ∞. Let W (k) := W u(k, ·), v(k, ·); x be the Wronskian of these two solutions. Set and assume that R(−k 2 ; x, y) is the integral kernel of a bounded operator R(−k 2 ). Then there exists a unique closed realization H of −∂ 2 x + V with the boundary condition at 0 given by u(k, ·) and at ∞ given by v(k, ·) in the sense that Moreover −k 2 belongs to the resolvent set of H and R(−k 2 ) = (H + k 2 ) −1 .
Let us now describe the integral kernel of the resolvent of all operators under investigation. We recall that our parameters are β ∈ C, κ ∈ C ∪ {∞}, ν ∈ C ∪ {∞}, and m ∈ C satisfying −1 < Re(m) < 1. Note also that the convention 1 0 = ∞ and 1 ∞ = 0 is still used.
Theorem 3.7. Let k ∈ C with Re(k) > 0. We have the following properties.
For the proof of this theorem, we shall mainly rely on a similar statement which was proved in [7,Sec. 3.4]. The context was less general, but some of the estimates turn out to be still useful.
Proof of Theorem 3.7. The proof consists in checking that all conditions of Proposition 3.6 are satisfied.
For (i) we need to show that the integral kernel R β,m,κ (−k 2 ; x, y) defines a bounded operator on L 2 (R + ). This follows from (3.11), because all numerical factors are harmless and because by [7, Thm. 3.5] R β,m (−k 2 ; x, y) and R β,−m (−k 2 ; x, y) are the kernels defining bounded operators.
Moreover, we can write (3.14) Since K β 2k ,m (2k·) belongs to L 2 (R + ), this solution is L 2 around ∞. For the other solution, one verifies by (3.4 Therefore, this function belongs to L 2 around 0 and satisfies the same boundary condition at 0 as j β,m, + κj β,−m . By Proposition 3.6, this proves (i) when κ = ∞. Note that in the special case κ = ∞, it is enough to observe that H β,m,∞ = H β,−m,0 and to apply the previous result.
To prove (ii), consider first ν = ∞ and β 2k ∈ N × . It has been proved in [7,Thm. 3.5] that the first kernel of (3.12) defines a bounded operator. The second kernel corresponds to a constant multiplied by a rank one operator defined by the function K β 2k ,m (2k·) ∈ L 2 (R + ) and therefore this operator is also bounded. Next we write We deduce from (3.4) and (3.8) that which belongs to L 2 around 0 and corresponds to the boundary condition defining H ν β, 1 2 .
The proof of (iii) is analogous. We use first (3.13) for the boundedness. Then we rewrite Green's function as (3. 16) We check that Strictly speaking, the formulas of Thm 3.7 are not valid in doubly degenerate points, where the functions K β,m and I β,m are proportional to one another. To obtain well defined formulas one needs to use the function X β,m , as described in the following proposition: We have the following properties.

Holomorphic families of closed operators
In this section we show that the families of operators introduced before are holomorphic for suitable values of the parameters. A general definition of a holomorphic family of closed operators can be found in [20], see also [8]. Actually, we will not need its most general definition. For us it is enough to recall this concept in the special case where the operators possess a nonempty resolvent set. Recall that the family H β,m has been defined on C × {m ∈ C | Re(m) > −1} in [7], see also (2.4). However, it is not holomorphic on the whole domain. The following has been proved in [7]. where these limits have to be understood as weak resolvent limits. Note that in the sequel and in particular in (3.18), (3.19), and (3.20), the limits should be understood in such a sense. Let us consider now the families of operators involving mixed boundary conditions. To this end, it will be convenient to introduce the notation Recall that (β, m, κ) → {H β,m,κ } has been defined on C × Π × (C ∪ {∞}). However, it is not holomorphic on this whole set: Proof. (i) Let (β 0 , m 0 , κ 0 ) belong to the domain C × Π × C ∪ {∞} . First assume that m 0 / ∈ − 1 2 , 0, 1 2 and that κ 0 ∈ C. Let k ∈ C with Re(k) > 0 such that ω β 0 ,m 0 ,κ 0 (k) = 0, where ω β,m,κ (k) is defined in (3.10). By continuity of the map (β, m, κ) → ω β,m,κ (k), there exists a neighborhood U 0 of (β 0 , m 0 , κ 0 ) such that for all (β, m, κ) in this neighborhood, we have ω β,m,κ (k) = 0. Hence, by Theorem 3.7, we infer that −k 2 / ∈ σ(H β,m,κ ), and the resolvent (H β,m,κ +k 2 ) −1 ∈ B L 2 (R + ) is the operator whose kernel is given by (3.14). It then easily follows from the analyticity properties of the maps (β, m, κ) → I β 2k ,±m (2kx) and (β, m, κ) → K β 2k ,m (2kx) (for fixed x > 0 and k) that, for all f, g ∈ L 2 (R + ), the map (β, m, κ) → (f, (H β,m,κ + k 2 ) −1 g) is holomorphic on U 0 . Hence {H β,m,κ } is holomorphic on U 0 .
Proof. (i) Let us first consider β = 0. Recall that in [6] the family of closed operators Π×(C∪{∞}) (m, κ) → H m,κ has been introduced, and that this family is holomorphic Here is its relationship to the families from the present article: Let us now focus on m = − 1 2 and on m = 1 2 . We have for any κ ∈ C ∪ {∞} Therefore, for κ = 0, This proves (i) when κ ∈ {0, ∞}. The proof in these special cases is similar.
(ii) Let us first consider a fixed parameter β ∈ C and m = 0. By definition we have independently of κ ∈ C ∪ {∞}. We now consider a fixed parameter β ∈ C and κ = −1.
Choosing k ∈ C with Re(k) > 0 such that β 2k − 1 2 ∈ N, it follows from (3.14) that for any m = 0 in a complex neighborhood of 0, the integral kernel of the resolvent of H β,m,−1 is given by where ω β,m,−1 (k) is defined in (3.10). One then infers that By using this expression, one can verify that the map m → g β,k,x (m), defined in a punctured complex neighborhood of 0, can be analytically extended at 0 with Thus, the family of operators {H β,m,−1 } defined bỹ is holomorphic for m ∈ Π. It thus follows that which concludes the proof.
Proposition 3.12. The function (β, m, k) → P β,m (−k 2 ), defined on the set has values in bounded projections. Moreover, it is continuous on We recall from Proposition 2.5 that the operators H β,m,κ , H ν β,0 and H ν β, 1 2 are selftransposed. Moreover, it follows from Theorem 3.1 and its proof that all eigenvalues of these operators are simple. If λ is a simple eigenvalue of a self-transposed operator H associated to an eigenvector u such that u|u = 1, we define the self-transposed eigenprojection associated to λ as P = u|· u.
In the case where λ is in addition an isolated point of the spectrum, it is then easy to see that the self-transposed eigenprojection P coincides with the usual Riesz projection corresponding to λ.
Proof. We prove the theorem in the case where λ = −k 2 with Re(k) > 0 and m / ∈ − 1 2 , 0, 1 2 . The other cases are similar. From the proof of Theorem 3.1, we know that if λ is an eigenvalue of H β,m,κ , then a corresponding eigenstate is given by x → K β 2k ,m (2kx). Corollary A.3 shows that This proves that P β,m (−k 2 ) is the self-transposed eigenprojection corresponding to λ, as claimed.
In the next proposition, we show that the family P β,m (−k 2 ) is not continuous at k = 0. Proposition 3.14. Let m ∈ Π and β ∈ C such that ±Im( √ β) > 0. Then the map k → P β,m (−k 2 ) is not continuous at k = 0.
Proof. Let us first consider the case m / ∈ {− 1 2 , 0, 1 2 }. We claim that, for all continuous and compactly supported function f , where k ∈ C is chosen such that Re(k) > 0 and ± arg(β) − arg(k) ∈]ε, π − ε[ with ε > 0. To shorten the expressions below, we set in this proof and g β,m,0 (x) := (βx) 1 4 H ± 2m (2 βx). We show that g β,m,k is uniformly bounded, for k satisfying the conditions above, by a locally integrable function. From the definition (A.3) of I β,m and proceeding as in the proof of Proposition B.1, we obtain that, for k ∈ C such that Re(k) > 0, |k| < 1, and for some constant c > 0 depending on β and m but independent of k and x. Using that together with Lemma B.2, one then deduces that for some positive constants c 1 , c 2 independent of k and x.
The previous bound together with the dominated convergence theorem and Proposition B.1 show that lim k→0 g β,m,k |f = g β,m,0 |f , for all continuous and compactly supported function f , and for k satisfying the conditions exhibited above. We then have that where we used Lemma B.6 in the third equality. Now, we claim that P β,m (−k 2 ) is not continuous at k = 0 for the strong operator topology. Indeed, using that P β,m (−k 2 ) is a self-transposed projection, we infer that, for f continuous and compactly supported,

A similar computation as above gives
for suitably chosen compactly supported functions f . By contradiction, this proves that P β,m (−k 2 ) is not strongly continuous at k = 0. The cases m ∈ − 1 2 , 0, 1 2 can be treated similarly. The only difference is that Lemma B.6 is no more necessary, but has to be replaced by the expansion as z → ∞.

A.1 General theory
In this section we collect basic information about the Whittaker equation. This should be considered as a supplement to [7,Sec. 2]. The Whittaker equation is represented by the equation We observe that the equation does not change when we replace m with −m. It has also another symmetry: Solutions of (A.1) are provided by the functions z → I β,±m (z) which are defined by where (a) k := a(a + 1) · · · (a + k − 1) and (a) 0 = 1 are the usual Pochhammer's symbols and 1 F 1 is Kummer's confluent hypergeometric function. For Re(m) > − 1 2 and Re m ∓ β + 1 2 > 0 the function I β,m has also an integral representation given by Based on (A.3) one easily gets as well as the following identity Another solution of (A.1) is provided by the function z → K β,m (z). For m ∈ 1 2 Z it can be defined by the following relation: For the remaining m we can extend the definition of K β,m by continuity, see Subsect. A.3. Note that K β,−m = K β,m , and that the function K β,m can also be expressed in terms of the function 2 F 0 , namely: An alternative definition of K β,m can be provided by an integral representation valid for Re − β ∓ m + 1 2 > 0 and Re(z) > 0: Note that the function K β,m decays exponentially for large Re(z), more precisely, if ε > 0 and |arg(z)| < π − ε, then one has By using the relation (A.6) one also obtains that We would like to treat I β,m , I β,−m and K β,m as the principal solutions of the Whittaker equation (A.1). There are however cases for which this is not sufficient. Therefore, we introduce below a fourth solution, which we denote by X β,m . To the best of our knowledge, this function has never appeared elsewhere in the literature.
Hence for m + β ∈ Z the solutions K β,m and X β,m are proportional to one another. In fact, for such β, m, we have Note that this corresponds to the lines m + β = n ∈ Z. However in our applications, we need X β,m on the lines m+β − 1 2 = n ∈ Z, where K β,m and X β,m are linearly independent.

A.2 The Laguerre cases
Let us now consider two special cases, namely when − 1 2 − m + β := n ∈ N and when − 1 2 − m − β := n ∈ N. In the former case, observe that the Wronskian of I β,m and K β,m vanishes, see (A.8). It means that in such a case these two functions are proportional to one another. In order to deal with this situation we define, for p ∈ C and n ∈ N, the Laguerre polynomials by the formulas Then, by setting 2m = p, we get Note that this solution can also be expressed in terms of the K β,m function, namely and more generally for ∈ Z one has In the special case − 1 2 − m − β := n ∈ N a similar analysis with p = 2m leads to and to We shall call this situation the exploding Laguerre case. In this case the relation (A.12) reduces to and more generally for ∈ Z one has

A.3 The degenerate case
In this section we consider the special case m ∈ 1 2 Z, which will be called the degenerate case, see Figure 1. In this situation the Wronskian of I β,m and I β,−m vanishes, see (A.4). More precisely, for any p ∈ N one has the identity Based on this equality and by a limiting procedure, an expression for the functions K β, p 2 has been provided in [7,Thm. 2.2], namely where ψ is the digamma function defined by ψ(z) = Γ (z) Γ(z) . Note that the equality (or definition) (a) j = Γ(a+j) Γ(a) has also been used for arbitrary j ∈ Z. For our applications the most important functions correspond to p = 1: 18) and to p = 0: Let us still provide the expression for the function X β, p

2
. Starting from its definition in (A.9) and by using the expansion (A.17) as well as the identity provided in (A.5) one gets X β, p 2 (z) = (−1) p+1 ln(z) I β, p 2 (z) In particular, the following expansion will be useful: and Note also that the following identity holds: as a consequence of (A.9).

A.4 The doubly degenerate case
We shall now consider the region In other words, we consider m ∈ Z, β ∈ Z + 1 2 , or m ∈ Z + 1 2 , β ∈ Z. This situation will be called the doubly degenerate case. We will again set m = p 2 with p ∈ Z. Note that for (m, β) in (A.22) we have the identity which is a special case of (A.12). In this case we also have Hence K β,m and X β,m always span the space of solutions in the doubly degenerate case.
Region II − . β + m ∈ − N + 1 2 , −β + m ∈ N + 1 2 . By setting n := −β − m − 1 2 ∈ N, then the equality (A.16) reduces to Thus I β,m is proportional to X β,m and corresponds to the exploding Laguerre case. The second solution is K β,m . It decays exponentially and has a logarithmic singularity at zero, therefore we call this function the decaying logarithmic solution.
Region II + . β + m ∈ N + 1 2 , −β + m ∈ − N + 1 2 . By setting n := β − m − 1 2 ∈ N, then the equality (A.14) reduces to Thus I β,m is proportional to K β,m and corresponds to the decaying Laguerre case. The second solution is X β,m . It explodes exponentially and has a logarithmic singularity at zero, therefore we call this function the exploding logarithmic solution.
The results of this section are summarized in Figure 2.

A.5 Recurrence relations
Solutions of the Whittaker equation satisfy interesting recurrence relations. These relations can be checked by using the series provided in (A.3). The computations are straightforward, but rather lengthy. These relations read .
Then, the identity (A.6) leads to As a consequence of these equalities one gets This proves (i). The equalities (ii) and (iii) can be proved similarly by using (A.18) and (A. 19).
By using the L'Hospital's rule one directly obtains:

A.7 The trigonometric type Whittaker equation
Along with the standard Whittaker equation (A.1), sometimes called hyperbolic type, it is natural to consider the trigonometric type Whittaker equation which solve (A. 27). Note that the function H ± β,m has been used in the proof of Theorem 3.1 when dealing with positive eigenvalues of the Whittaker operators.

A.8 Integral identities in the trigonometric case
Here are the analogues of Proposition A.2 and Corollary A.3 in the trigonometric case. The approach can be mimicked from Section A.6 because of the identity valid for any µ > 0.
By analogy of the presentation of Section A.4 we can divide the half-integer case into two regions, namely Region I − . m ∈ − 1 2 − N, and Region I + . m ∈ 1 2 + N. The following schematic diagram of various special cases for the Bessel equation is an analog of Fig. 2. One can also obtain solutions of (B.8) by rescaling solutions of the hyperbolic-type or trigonometric-type Whittaker equation: Proposition B.1. For any fixed x ∈ R + , m ∈ Π and β ∈ C × , one has lim k→0 1 2k where the first limit is taken such that ± arg(β) − arg(k) ∈]ε, π − ε[ with ε > 0, and the second limit is taken with µ > 0 and is valid if Re(β) > 0.
Proof. Using the definition of Pochhammer's symbol recalled in Section A.1, one infers that lim In addition, for all k ∈ C with |k| < 1, one has for some constant c independent of k and j. Hence, by an application of the version of the Lebesgue dominated convergence theorem for series, one gets lim k→0 ∞ j=0 1 2 + m ∓ β 2k j (±2kx) j Γ(1 + 2m + j)j! = where we have used that ± arg β 2k ∈]0, π] and that arg − β 2k < π − ε for ε > 0. The equality (B.12) can then be deduced from (B.11) by using the relation (A.29) between the functions K β,m and H ± β,m .
The following lemma plays a key role in the above proof. (iii) If m = 1 2 , then ∞ 0 (βx) In the next proposition, we consider the integral of ((βx)