Jan Dereziński
email: jan.derezinski@fuw.edu.pl

Jérémy Faupin
email: jeremy.faupin@univ-lorraine.fr

Nhat Quang

Serge Nguyen

Richard
email: richard@math.nagoya-u.ac.jp

Prof Hugon Franciszek

Szafraniec

On radial Schrödinger operators with a Coulomb potential: General boundary conditions

This paper presents the spectral analysis of 1-dimensional Schrödinger operator on the half-line whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r 2 . The coupling constants are allowed to be complex, and all possible boundary conditions at 0 are considered. The resulting closed operators are organized in three holomorphic families. These operators are closely related to the Whittaker equation. Solutions of this equation are thoroughly studied in a large appendix to this paper. Various special cases of this equation are analyzed, namely the degenerate, the Laguerre and the doubly degenerate cases. A new solution to the Whittaker equation in the doubly degenerate case is also introduced.

Introduction

This paper is devoted to 1-dimensional Schrödinger operators with Coulomb and centrifugal potentials. These operators are given by the differential expressions L β,α := -∂ 2

x + α -

1 4 1 x 2 - β x . (1.1)
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 [START_REF] Whittaker | An expression of certain known functions as generalized hypergeometric functions[END_REF]. For this reason, we call (1.1) the Whittaker operator.

This paper is a continuation of a series of papers [START_REF] Bruneau | Homogeneous Schrödinger operators on half-line[END_REF][START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF][START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF] devoted to an analysis of exactly solvable 1-dimensional Schrödinger operators. We follow the same philosophy as in [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF]. 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 [START_REF] Bruneau | Homogeneous Schrödinger operators on half-line[END_REF][START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF] we considered the operator

L α := -∂ 2 x + α - 1 4 1 x 2 . (1.2)
As is known, it is useful to set α = m 2 . In [START_REF] Bruneau | Homogeneous Schrödinger operators on half-line[END_REF] the following holomorphic family of closed realizations of (1.2) was introduced:

H m , with -1 < Re(m),
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 [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF], it is natural to organize them into two holomorphic families:

H m,κ , with -1 < Re(m) < 1, m = 0, κ ∈ C ∪ {∞}, defined by L m 2 with boundary conditions ∼ x Note that related investigations about these operators have also been performed in [START_REF] Smirnov | Eigenfunction expansions for the Schrödinger equation with inversesquare potential[END_REF][START_REF] Smirnov | Coupling constant dependence for the Schrödinger equation with an inverse-square potential[END_REF].

In [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF] and in the present paper we study closed realizations of the differential operator (1.1) on L 2 (R + ). Again, it is useful to set α = m 2 . In [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF] we introduced the family H β,m , with β ∈ C, -1 < Re(m), defined by L β,m 2 with boundary conditions ∼ x 1 2 +m 1 -β 1 + 2m

x .

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.

The above holomorphic families include all possible well-posed realizations of L β,m 2 in the region |Re(m)| < 1 with one exception: the special case (β, m, κ) = 0, - 1 2 , 0 which corresponds to the Neumann Laplacian H -

1 2 = H -1 2 ,0 = H 1 2 ,∞
, and which is already covered by the families H m and H m,κ .

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. [START_REF] Buchholz | The confluent hypergeometric function with special emphasis on its applications[END_REF].

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 [START_REF] Kurasov | On the Coulomb potential in one dimension[END_REF][START_REF] Fischer | Comment on 'On the Coulomb potential in one dimension' by P Kurasov[END_REF][START_REF] Kurasov | Response to "Comment on 'On the Coulomb potential in one dimension[END_REF] about self-adjoint extensions of the 1-dimensional Schrödinger operator on the real line with a Coulomb potential (without the centrifugal term). Papers [START_REF] Kostenko | On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators[END_REF][START_REF] Fulton | Sturm-Liouville Operators with Singularities and Generalized Nevanlinna Functions[END_REF][START_REF] Kurasov | An Operator Theoretic Interpretation of the Generalized Titchmarsh-Weyl Coefficient for a Singular Sturm-Liouville Problem Math[END_REF] discuss generalized Nevanlinna functions naturally appearing in the context of such operators, especially in the range of parameters |Re(m)| ≥ 1. See also [START_REF] Bulla | Deficiency indices and singular boundary conditions in quantum mechanics[END_REF][START_REF] Dollard | Asymptotic convergence and the Coulomb interaction[END_REF][START_REF] Gaspard | Connection formulas between Coulomb wave functions[END_REF][START_REF] Gesztesy | On the one-dimensional Coulomb Hamiltonian[END_REF][START_REF] Gesztesy | On the high-energy behaviour of scattering phase shifts for Coulomb-like potentials[END_REF][START_REF] Gitman | Self-adjoint extensions in quantum mechanics. General theory and applications to Schrödinger and Dirac equations with singular potentials[END_REF][START_REF] Guillot | Perturbation of the Laplacian by Coulomb like potentials[END_REF][START_REF] Herbst | On the connectedness structure of the Coulomb S-matrix[END_REF][START_REF] Humblet | Analytical structure and properties of Coulomb wave functions for real and complex energies[END_REF][START_REF] Marchesin | Time-dependent and time-independent potential scattering for asymptotically Coulomb potentials[END_REF][START_REF] Michel | Direct demonstration of the completeness of the eigenstates of the Schrödinger equation with local and nonlocal potentials bearing a Coulomb tail[END_REF][START_REF] Mukunda | Completeness of the Coulomb wave functions in quantum mechanics[END_REF][START_REF] Mulherin | Coulomb scattering. I. Single channel[END_REF][START_REF] Seaton | Coulomb functions for attractive and repulsive potentials and for positive and negative energies[END_REF] 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 [START_REF] Bruneau | Homogeneous Schrödinger operators on half-line[END_REF][START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF][START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF], 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

-∂ 2 z + m 2 - 1 4 1 z 2 - β z + 1 4 f (z) = 0.
Its solutions are called Whittaker functions, which can be expressed in terms of Kummer's confluent functions. The theory of the Whittaker equation is the second subject of the 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 [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF][START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]. 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 [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF], 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 [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF][START_REF] Buchholz | The confluent hypergeometric function with special emphasis on its applications[END_REF][START_REF] Slater | Confluent Hypergeometric Function[END_REF]. 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 and2, 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

R + =]0, ∞[, N = {0, 1, 2, . . . } and N × = {1, 2, . . . }. Likewise, we set C × = C \ {0} and R × = R \ {0}.
The Hilbert space L 2 (R + ) is endowed with the scalar product

(h 1 |h 2 ) = ∞ 0 h 1 (x) h 2 (x) dx.
We will also use the bilinear form defined by

h 1 |h 2 = ∞ 0 h 1 (x) h 2 (x) dx.
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

(h 1 |Ah 2 ) = (A * h 1 |h 2 ), h 1 |Ah 2 = A # h 1 |h 2 .
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

W (f, g; x) := f (x)g (x) -f (x)g(x).
(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 [START_REF] Georgescu | 1-dimensional Schroedinger operators with complex potentials[END_REF] 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

-∂ 2 x + m 2 - 1 4 1 x 2 - β x f = 0 (2.2)
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 + :

j β,m (x) := Γ(1 + 2m) √ π β -1 4 -m x 1/4 J 2m 2 βx ,
where J m is defined in Section B.4. For β = 0 we set

j 0,m (x) := x m+ 1 2 .
Then, the equation (2.2) is solved by the functions j β,m , see [7, Sec. 2.8] and Section B.5. For 2m ∈ Z, j β,m and j β,-m span the space of solutions of (2.2). They are square integrable around zero if and only if |Re(m)| < 1.

We still need to consider the special cases m ∈ -1 2 , 0, 1 2 . In fact, we shall not consider separately m = -1 2 because Equation (2.2) with m = -1 2 coincides with the case m = 1 2 . As companions to j β,0 and j β, 1 2

for β = 0 we introduce

y β,0 (x) := β -1 4 x 1/4 √ πY 0 2 βx - (ln(β) + 2γ) √ π J 0 2 βx , y β, 1 2 (x) := β 1 4 x 1/4 - √ πY 1 2 βx + (ln(β) + 2γ -1) √ π J 1 2 βx ,
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.

Then j β,0 , y β,0 and j β,

1 2 , y β, 1 2
span the space of solutions of (2.2) for m = 0 and for m = 1 2 respectively. Indeed, a short computation leads to

W (j β,0 , y β,0 ; x) = 1 and W (j β, 1 2 , y β, 1 2 ; x) = -1.
Since the solutions j β,0 , y β,0 and j β,

1 2 , y β, 1 2
are also square integrable around zero, for any

m ∈ C with |Re(m)| < 1 the space of solutions of L β,α f = 0 is 2-dimensional.
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 [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF]. For any m ∈ C with -2m ∈ N × one has

j β,m (x) = x 1 2 +m 1 - β 1 + 2m x + O x 2 . (2.3)
In the exceptional cases one has

j β,0 (x) = x 1 2 1 -βx + O x 5 2 , j β, 1 2 (x) = x 1 - β 2 x + O x 3 ,
together with

y β,0 (x) = x 1 2 ln(x) 1 -βx + O x 3 2 , y β, 1 2 (x) = 1 -βx ln(x) + O x 2 | ln(x)| .

Maximal and minimal operators

For any α and β ∈ C we consider the differential expression

L β,α := -∂ 2 x + α - 1 4 1 x 2 - β x
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 [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]Sec. 3.2] for the details. The domain of L max β,α is given by We say that f ∈ D(L min β,α ) around 0, (or, by an abuse of notation,

D(L max β,α ) = f ∈ L 2 (R + ) | L β,α f ∈ L 2 (R + ) , while L min β,α is the closure of the restriction of L β,α to C ∞ c ]0, ∞[ ,
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.

(a) f (x) = o x 3 2 | ln(x)| and f (x) = o x 1 2 | ln(x)| if α = 0, (b) f (x) = o x 3 2 and f (x) = o x 1 2 if α = 0. (iii.a) If |Re(m)| < 1 with m ∈ -1 2 , 0, 1 2 , then for any f ∈ D(L max β,m 2 ) there exists a unique pair a, b ∈ C such that f -a j β,m -b j β,-m ∈ D(L min β,m 2 ) around 0. (iii.b) If f ∈ D(L max β,0 ), then there exists a unique pair a, b ∈ C such that f -a j β,0 -b y β,0 ∈ D(L min β,0 ) around 0. (iii.c) If f ∈ D(L max β, 1 4
), then there exists a unique pair a, b ∈ C such that

f -a j β, 1 2 -b y β, 1 2 ∈ D(L min β, 1 4
) around 0.

(iv) If |Re(m)| < 1, then D(L min β,m 2 ) = f ∈ D(L max β,m 2 ) | W (f, g; 0) = 0 for all g ∈ D(L max β,m 2 ) = f ∈ D(L max β,m 2 ) | f (x) = o x 1 2 +|Re(m)| near 0 . (v) If |Re(m)| 1, then D(L min β,m 2 ) = D(L max β,m 2 ).
Proof. The statements (i)-(iii) and (v) are a reformulation of [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]Prop. 3.1] with the current notations. Only (iv) requires elaboration. The first equality in (iv) follows from [START_REF] Georgescu | 1-dimensional Schroedinger operators with complex potentials[END_REF]Thm. 3.4], given that W (f, g; ∞) = 0 for all f, g ∈ D(L max β,m 2 ) by (i). The inclusion

D(L min β,m 2 ) ⊂ f ∈ D(L max β,m 2 ) | f (x) = o x 1
2 +|Re(m)| near 0 is a consequence of (ii). To prove the converse inclusion, let f ∈ D(L max β,m 2 ). Assuming for instance that m / ∈ -1 2 , 0, 1 2 and applying (iii.a), one can write

f ζ = aj β,m ζ + bj β,-m ζ + f min , for some ζ ∈ C ∞ c [0, ∞[ such that ζ = 1 around 0, a, b ∈ C and f min ∈ D(L min β,m 2 ). From (2.3) and (ii), we deduce that if f (x) = o x 1 2 +|Re(m)| near 0 then, necessarily, a = b = 0. Hence we have proved that f ∈ D(L max β,m 2 ) | f (x) = o x 1 2 +|Re(m)| near 0 ⊂ D(L min β,m 2 ) in the case where m / ∈ -1 2 , 0, 1 2 .
The same argument applies if m = ± 1 2 or m = 0, using (iii.b) or (iii.c) instead of (iii.a).

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 ∪ {∞}:

D(H β,m,κ ) = f ∈ D(L max β,m 2 ) | for some c ∈ C, f -c j β,m + κ j β,-m ∈ D(L min β,m 2 ) around 0 , κ = ∞, D(H β,m,∞ ) = f ∈ D(L max β,m 2 ) | for some c ∈ C, f -c j β,-m ∈ D(L min β,m
2 ) around 0 . The second family corresponds to m = 0:

D(H ν β,0 ) = f ∈ D(L max β,0 ) | for some c ∈ C, f -c y β,0 + ν j β,0 ∈ D(L min β,0 ) around 0 , ν ∈ C, D(H ∞ β,0 ) = f ∈ D(L max β,0 ) | for some c ∈ C, f -c j β,0 ∈ D(L min β,0 ) around 0 .
Finally, in the special case m = 1 2 we have the third family:

D(H ν β, 1 2 ) = f ∈ D(L max β, 1 4
) | for some c ∈ C, f -c y β, 1 2 + ν j β, 1 2 ∈ D(L min β, 1 4
) around 0 , ν ∈ C, D(H ∞ β, 1 2 ) = f ∈ D(L max β, 1 4
) | for some c ∈ C, f -c j β, 1 2 ∈ D(L min β, 1 4
) around 0 .

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

j β,m (x) with x 1 2 +m 1 - β 1 + 2m x if -1 < Re(m) ≤ - 1 2 , j β,m (x) with x 1 2 +m if - 1 2 < Re(m) < 1, y β, 1 2 (x) with 1 -βx ln(x), y β,0 (x) with x 1 2 ln(x).
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 ∪ {∞}

H β,-1 2 ,κ := H ∞ β, 1 2 , H β,0,κ := H ∞ β,0 , and H β, 1 2 ,κ := H ∞ β, 1 2 .
An invariance property follows directly from the definition: with the convention that 1 0 = ∞ and 1 ∞ = 0.

It is also convenient to introduce another two-parameter family of operators, which cover only special boundary conditions, which we call pure:

H β,m := H β,m,0 = H β,-m,∞ . (2.4)
With this notation, for any β ∈ C, one has

H β,-1 2 = H ∞ β, 1 2 , H β,0 = H ∞ β,0 , and H β, 1 2 = H ∞ β, 1 2 .
Remark 2.4. The family H β,m is essentially identical to the family denoted by the same symbol introduced and studied in [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]. 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 2 := H 0, 1 2 .
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 [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF][START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF].

H 0,m,κ = H m,κ |Re(m)| < 1, m ∈ -1 2 , 1 2 , κ ∈ C ∪ {∞}, H ν 0, 1 2 = H -1 2 ,ν = H 1 2 , 1 ν ν ∈ C ∪ {∞}, H ν 0,0 = H ν 0 ν ∈ C ∪ {∞},
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

H β=0,m=-1 2 = H β=0,m= 1 2 = H ∞ 0, 1 2 = H m= 1 2 ,κ=0 = H m=-1 2 ,κ=∞
while the Neumann Laplacian is given by

H 0 0, 1 2 = H m=-1 2 ,κ=0 = H m= 1 2 ,κ=∞ .
Note that the former operator was also described in [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF] 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 .

H β,m,κ * = H β, m,κ H β,m,κ # = H β,m,κ κ ∈ C ∪ {∞}, H ν β, 1 2 * = H ν β, 1 2 H ν β, 1 2 # = H ν β, 1 2 ν ∈ C ∪ {∞}, H ν β,0 * = H ν β,0 H ν β,0 # = H ν β,0 , ν ∈ C ∪ {∞}.
Proof. Let us prove the first statement, the other ones can be obtained similarly. Recall from Proposition 2.1 (see also [START_REF] Bruneau | Homogeneous Schrödinger operators on half-line[END_REF]Prop. A.2]) that for any f ∈ D(L max β,m 2 ) and g ∈ D(L max β, m2 ), the functions f, f , g, g are continuous on R + . In addition, the Wronskian of f and g, as introduced in (2.1), possesses a limit at zero, and we have the equality

(L max β,m 2 f |g) -(f |L max β, m2 g) = -W ( f , g; 0). In particular, if f ∈ D(H β,m,κ ) one infers that (H β,m,κ f |g) = (f |L max β, m2 g) -W ( f , g; 0).
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, m2 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(H β, m,κ ). One then deduces that (H β,m,κ ) * = H β, 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:

(i) β ∈ R, m ∈] -1, 1[ and κ ∈ R ∪ {∞}, (ii) β ∈ R, m ∈ iR × and |κ| = 1.
The operators H ν β, 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.

Proposition 2.7. For m ∈ C with |Re(m)| < 1 one has U τ H β,m,κ U -τ = e -2τ H e τ β,m,e -2τ m κ κ ∈ C ∪ {∞}, U τ H ν β, 1 2
U -τ = e -2τ H e τ (ν-βτ ) e τ β, 1 2 ν ∈ C ∪ {∞}, U τ H ν β,0 U -τ = e -2τ H ν+τ e τ β,0 ν ∈ C ∪ {∞}, with the conventions α ∞ = ∞ for any α ∈ C \ {0} and ∞ + τ = ∞.

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

L β,m 2 f = -k 2 f
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 [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF], 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.

Theorem 3.1. 1. Let β ∈ C × , |Re(m)| < 1 with m ∈ -1 2 , 0, 1 2
, and let κ ∈ C ∪ {∞}. Then the operator H β,m,κ possesses an eigenvalue λ ∈ C in the following cases:

(i) λ = -k 2 , Re(k) > 0, β 2k + m -1 2 / ∈ N and κ = (2k) -2m Γ(2m) Γ(-2m) Γ 1 2 -m -β 2k Γ 1 2 + m -β 2k , (3.1)
(ii) λ = µ 2 , 0 < µ < ±Im(β) and κ = e ±iπm (2µ) -2m Γ(2m) Γ(-2m) Γ 1 2 -m ∓ i β 2µ Γ 1 2 + m ∓ i β 2µ , (iii) λ = 0, β ∈ [0, ∞[, and
κ = Γ(2m) Γ(-2m) (-β) 2m . 2. Let β ∈ C × and ν ∈ C∪{∞}. Then H ν β, 1 2
possesses an eigenvalue λ in the following cases:

(i) λ = -k 2 , Re(k) > 0, β 2k / ∈ N and ν = -β ψ - β 2k + 2γ -1 - k β + ln(2k) , (ii) λ = µ 2 , 0 < µ < ±Im(β), and
ν = -β ψ ∓ i β 2µ + 2γ -1 ∓ i π 2 ± i µ β + ln(2µ) , (iii) λ = 0, ±Im( √ β) > 0, and ν = -β ln(β) + 2γ -1 ∓ iπ .
3. Let β ∈ C × and ν ∈ C∪{∞}. Then H ν β,0 possesses an eigenvalue λ in the following cases:

(i) λ = -k 2 , Re(k) > 0, β 2k -1 2 / ∈ N and ν = ψ 1 2 - β 2k + 2γ + ln(2k), (ii) λ = µ 2 , 0 < µ < ±Im(β), and
ν = ψ 1 2 ∓ i β 2µ ∓ i π 2 + 2γ + ln(2µ), (iii) λ = 0, ±Im( √ β) > 0, and
ν = ln(β) + 2γ + 2 ln(2) ∓ iπ.
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

x → h ± β,m (x) := x 1/4 H ± 2m 2 βx , (3.2)
with H ± m the Hankel function for dimension 1, see [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF]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

H ± m (z) =          ±i √ 2 √ π z 1 2 ln(z) + γ ∓ i π 2 + O |z| 5 2 ln(|z|) if m = 0, ∓i 1 √ π z 2 -1 2 ± i 2 √ π ln z 2 + γ -1 2 ∓ i π 2 z 2 3 2 + O |z| 7 2 ln(|z|) if m = 1, ∓i √ π sin(πm) z 2 1 2 1 Γ(1-m) z 2 -m -e ∓iπm Γ(1+m) z 2 m + O(|z| 5 2 -|Re(m)| ) if m ∈ Z.
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

H ± m (z) = e ±i(z-1 2 πm-1 4 π) 1 + O(|z| -1 ) . Since | arg(2 √ βx)| ≤ π/2, it follows that for ±Im( √ β) > 0 one has h ± β,m (x) = x 1/4 e ±i(2 √ βx-πm-1
and h ± β,m belongs to L 2 near infinity. Note that for ±Im( √ β) > 0 one and only one of the two solutions of (3.2) is square integrable at infinity. It only remains to check in which domain of the operators

H β,m,κ , H ν β, 1 2
, or H ν β,0 does h ± β,m belong to. This can easily be deduced from the expansion provided above, and yields to the statements 1.(iii), 2.(iii) and 3.(iii).

Let us now prove the statements 1.(ii), 2.(ii) and 3.(ii). We consider the equation L β,m 2 f = µ 2 f for some µ > 0. Two linearly independent solutions are provided by the functions x → H ± β 2µ ,m (2µx) introduced in [7, Sec. 2.7], see also (A.29). From the asymptotic expansion near infinity given by

H ± β 2µ ,m (2µx) = e ∓i π 2 ( 1 2 +m) e πβ 4µ (2µx) ±i β 2µ e ±iµx 1 + O(x -1 ) , (3.3)
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 + (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

I δ,m (z) = z 1 2 +m Γ(1 + 2m) 1 - δ 1 + 2m z + O(z 2 ) (3.4)
and

H ± δ,m (z) = ∓ ie ∓iπm Γ(-2m) Γ 1 2 -m ∓ iδ z 1 2 +m 1 - δ 1 + 2m z ∓ i Γ(2m) Γ 1 2 + m ∓ iδ z 1 2 -m 1 - δ 1 -2m z + o z 3 2 .
For 2m ∈ Z one has to consider the expression for K δ, 1 2 and K δ,0 provided in (A.18) and (A. [START_REF] Humblet | Analytical structure and properties of Coulomb wave functions for real and complex energies[END_REF]) respectively. Then, by considering the Taylor expansion near 0 of these functions one gets

K δ, 1 2 (z) = 1 Γ(1 -δ) + 1 Γ(-δ) z ln(z) + 1 Γ(-δ) ψ(-δ) + 2γ -1 - 1 2δ z + o z 3 2 , (3.5)
and

K δ,0 (z) = - 1 Γ 1 2 -δ z 1 2 ln(z) + ψ 1 2 -δ + 2γ z 1 2 -δz 3 2 ln(z) + O z 3 2 . (3.6)
From Equation (A.29) one finally deduces the relations

H ± δ, 1 2 (z) = ∓ i 1 Γ(1 ∓ iδ) - 1 Γ(∓iδ) z ln(z) - 1 Γ(∓iδ) ψ(∓iδ) + 2γ -1 ∓ i π 2 ± i 1 2δ z + o z 3 2
and

H ± δ,0 (z) = ±i 1 Γ 1 2 ∓ iδ z 1 2 ln(z) + ψ 1 2 ∓ iδ ∓ i π 2 + 2γ z 1 2 -δz 3 2 ln(z) + O z 3 2 .
To show 1.(ii) we consider the function

x → H + β 2µ ,m (2µx) if Im(β) > 0 and x → H - β 2µ ,m (2µx)
if Im(β) < 0, and check for which κ these functions belong to D(H β,mκ ).

For |Re(m)| < 1 and m ∈ -1 2 , 0, 1 2 one has

H ± β 2µ ,m (2µx) = ∓ ie ∓iπm Γ(-2m) Γ 1 2 -m ∓ i β 2µ (2µx) 1 2 +m 1 - β 1 + 2m x ∓ i Γ(2m) Γ 1 2 + m ∓ i β 2µ (2µx) 1 2 -m 1 - β 1 -2m x + o x 3 2 = ∓ ic j β,m + κj β,-m (x) + o x 3 2 with c := e ∓iπm Γ(-2m) Γ( 1 2 -m∓i β 2µ ) (2µ) 1 2 +m and κ := 1 c Γ(2m) Γ 1 2 + m ∓ i β 2µ (2µ) 1 2 -m = e ±iπm (2µ) -2m Γ(2m) Γ(-2m) Γ 1 2 -m ∓ i β 2µ Γ 1 2 + m ∓ i β 2µ .
Note that the conditions ±Im(β) > 0, |Re(m)| < 1, and µ < ±Im(β) imply that ±i β 2µ + m -1 2 ∈ N. The proof of 2.(ii) and 3.(ii) can be obtained similarly once the following expressions are taken into account:

H ± β 2µ , 1 2 (2µx) = 2µ β 1 Γ ∓ i β 2µ 1 -βx ln(x) - 2µ Γ ∓ i β 2µ ψ ∓ i β 2µ + 2γ -1 ∓ i π 2 ± i µ β + ln(2µ) x + o x 3 2
and

H ± β 2µ ,0 (2µx) = ± i (2µ) 1 2 Γ 1 2 ∓ i β 2µ x 1 2 ln(x) + ψ 1 2 ∓ i β 2µ ∓ i π 2 + 2γ + ln(2µ) x 1 2 -βx 3 2 ln(x) + O x 3 2 .
We shall now turn to the generic case (statements 1.(i), 2.(i) and 3.(i)), namely the equation L β,m 2 f = -k 2 f for some k ∈ C with Re(k) > 0. In the non-degenerate case, solutions of this equation are provided by the functions

x → K β 2k ,m (2kx) and x → I β 2k ,±m (2kx).
(3.7)

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 I δ,±m (z) = 1 Γ 1 2 ± m -δ z -δ e z 2 1 + O(z -1 )
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 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 -1 2 ∈ N
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) that for m ∈ -1 2 , 0,

1 2 K β 2k ,m (2kx) = - π sin(2πm) I β 2k ,m (2kx) Γ 1 2 -m -β 2k - I β 2k ,-m (2kx) Γ 1 2 + m -β 2k =(2k) 1 2 +m Γ(-2m) Γ 1 2 -m -β 2k x 1 2 +m 1 - β 1 + 2m x + (2k) 1 2 -m Γ(2m) Γ 1 2 + m -β 2k x 1 2 -m 1 - β 1 -2m x + o(x 3
2 ).

Similarly, it follows from (A.18) that

K β 2k , 1 2 (2kx) = - 1 β 2k Γ -β 2k 1 -βx ln(x) + 2k Γ -β 2k ψ - β 2k + 2γ -1 - k β + ln(2k) x + o x 3 2 , (3.8)
while for m = 0 one infers from (A. [START_REF] Humblet | Analytical structure and properties of Coulomb wave functions for real and complex energies[END_REF]) that

K β 2k ,0 (2kx) = - (2k) 1 2 Γ 1 2 -β 2k x 1 2 ln(x)+ ψ 1 2 - β 2k +2γ+ln(2k) x 1 2 -βx 3 2 ln(x) +O x 3 2 .
(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). Remark 3.3. Self-adjoint operators that are included in the families H β,m,κ , H ν β, 1 2 and H ν β,0 do not have eigenvalues in ]0, ∞[. Indeed, in Theorem 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.

Theorem 3.4. (i) If |Re(m)| < 1, m ∈ -1
2 , 0, 1 2 and κ ∈ C ∪ {∞}, the eigenvalues of the operator H 0,m,κ are of the form -k 2 with Re(k) > 0, where

κ = k 2 -2m Γ(m) Γ(-m) , (ii) If ν ∈ C ∪ {∞},
ν = γ + ln k 2 .
Remark 3.5. Note that Theorem 3.4 can be derived from Theorem 3.1. Indeed, for m ∈ -1 2 , 0, 1 2 we infer from the Legendre duplication formula m) .

Γ(z)Γ 1 2 + z = 2 1-2z √ π Γ(2z), that (2k) -2m Γ(2m) Γ(-2m) Γ 1 2 -m -β 2k Γ 1 2 + m -β 2k β=0 = k 2 -2m Γ(m) Γ(-
For m = 1 2 , we first note that

Γ 1 2 = √ π and Γ -1 2 = -2 √ π.
Then we use the relations ψ(1 + z) = ψ(z) + 1 z and ψ(1) = -γ, and infer that

lim β→0 -β ψ - β 2k + 2γ -1 - k β + ln(2k) = k 2 Γ -1 2 Γ 1 2 = -k.
Finally for m = 0, from the equality ψ 1 2 = -2 ln(2) -γ one gets

ψ 1 2 - β 2k + 2γ + ln(2k) β=0 = γ + ln k 2 .
As a consequence of the expressions provided in Theorem 3.1, the discreteness of the spectra of all operators can be inferred in C \ [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 [START_REF] Bruneau | Homogeneous Schrödinger operators on half-line[END_REF] or in [START_REF] Georgescu | 1-dimensional Schroedinger operators with complex potentials[END_REF]. 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

D(L max ) := f ∈ L 2 (R + ) ∩ AC 1 (R + ) | -∂ 2 x + V f ∈ L 2 (R + ) L max f := -∂ 2 x + V f, f ∈ D(L max ).
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 :

R + → C belongs to L 2 around 0 (respectively around ∞) if there exists ζ ∈ C ∞ c [0, ∞[ with ζ = 1 around 0 such that f ζ ∈ L 2 (R + ) (respectively f (1 -ζ) ∈ L 2 (R + )).
The following statement contains several results proved in [START_REF] Georgescu | 1-dimensional Schroedinger operators with complex potentials[END_REF].

Proposition 3.6. Let V ∈ L 1 loc (R + ). Let k ∈ C and suppose that u(k, •), v(k, •) ∈ AC 1 (R + ) solve -∂ 2 x + V u(k, •) = -k 2 u(k, •), -∂ 2 x + V v(k, •) = -k 2 v(k, •). 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 R(-k 2 ; x, y) := 1 W (k) u(k, x) v(k, y) for 0 < x < y, u(k, y) v(k, x) for 0 < y < x, 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

D(H) = f ∈ D(L max ), f -u(k, •) ∈ D(L min ) around 0 , = f ∈ D(L max ), f -v(k, •) ∈ D(L min ) around ∞ , Hf = -∂ 2 x + V f, f ∈ D(H).
Moreover -k 2 belongs to the resolvent set of H and R(-

k 2 ) = (H + k 2 ) -1 .
By using such a statement, it has been proved in [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF] that, for k ∈ C such that Re(k) > 0 and

β 2k -1 2 -m / ∈ N, we have that -k 2 / ∈ σ(H β,m ) and R β,m (-k 2 ) := (k 2 + H β,m ) -1 has the integral kernel R β,m (-k 2 ; x, y) = 1 2k Γ 1 2 + m -β 2k    I β 2k ,m (2kx)K β 2k ,m (2ky) for 0 < x < y, I β 2k ,m (2ky)K β 2k ,m (2kx) for 0 < y < x.
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.

(i) For m ∈ -1 2 , 0, 1 2 and κ = ∞ set ω β,m,κ (k) := (2k) -m Γ 1 2 + m -β 2k Γ(1 -2m) + κ (2k) m Γ 1 2 -m -β 2k Γ(1 + 2m) . (3.10) If ω β,m,κ (k) = 0, then -k 2 ∈ σ(H β,m,κ ) and the integral kernel of R β,m,κ (-k 2 ) := (H β,m,κ + k 2 ) -1 is given by R β,m,κ (-k 2 ; x, y) = 1 ω β,m,κ (k) (2k) -m Γ(1 -2m)Γ 1 2 + m -β 2k R β,m (-k 2 ; x, y) + κ (2k) m Γ(1 + 2m)Γ 1 2 -m -β 2k R β,-m (-k 2 ; x, y) = R β,m (-k 2 ; x, y) + κ(2k) m Γ 1 2 + m -β 2k 2kω β,m,κ (k)Γ(1 + 2m) K β 2k ,m (2ky)K β 2k ,m (2kx).
(3.11)

If κ = ∞ and β 2k + m -1 2 ∈ N, then -k 2 ∈ σ(H β,m,∞ ) and R β,m,∞ (-k 2 ) = R β,-m (-k 2 ) (ii) For m = 1 2 , ν = ∞, and β 2k ∈ N × set ω ν β, 1 2 (k) := βψ 1 - β 2k + 2βγ + β ln(2k) + k -β + ν. If ω ν β, 1 2 (k) = 0, then -k 2 ∈ σ(H ν β, 1 2
) and the integral kernel of

R ν β, 1 2
(-k 2 ) := (H ν β, 1 2 + k 2 ) -1 is given by R ν β, 1 2 (-k 2 ; x, y) = R β, 1 2
(-k 2 ; x, y) + Γ 1 -β 2k 2 ω ν β, 1 2 (k) K β 2k , 1 2 (2kx)K β 2k , 1 2 (2ky). (3.12) If ν = ∞ and β 2k ∈ N × , then -k 2 ∈ σ(H ∞ β, 1 2 ) and R ∞ β, 1 2
(-k 2 ) = R β, 1 2
(-k 2 ).
(iii) For m = 0, ν = ∞, and

β 2k -1 2 ∈ N set ω ν β,0 (k) := ψ 1 2 - β 2k + 2γ + ln(2k) -ν. If ω ν β,0 (k) = 0, then -k 2 ∈ σ(H ν β,0 ) and the integral kernel of R ν β,0 (-k 2 ) := (H ν β,0 + k 2 ) -1 is given by R ν β,0 (-k 2 ; x, y) = R β,0 (-k 2 ; x, y) + Γ 1 2 -β 2k 2 2kω ν β,0 (k) K β 2k ,0 (2kx)K β 2k ,0 (2ky).
(3.13)

If ν = ∞ and β 2k -1 2 ∈ N, then -k 2 / ∈ σ(H ∞ β,0 ) and R ∞ β,0 (-k 2 ) = R β,0 (-k 2 ).
For the proof of this theorem, we shall mainly rely on a similar statement which was proved in [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]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

R β,m,κ (-k 2 ; x, y) = 1 2kω β,m,κ (k) ×      (2k) -m Γ(1-2m) I β 2k ,m (2kx) + κ (2k) m Γ(1+2m) I β 2k ,-m (2kx) K β 2k ,m (2ky) for 0 < x < y, ( 2k
) -m Γ(1-2m) I β 2k ,m (2ky) + κ (2k) m Γ(1+2m) I β 2k ,-m (2ky) K β 2k ,m (2kx) for 0 < y < x. (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) that

(2k) -m Γ(1 -2m) I β 2k ,m (2kx) + κ (2k) m Γ(1 + 2m) I β 2k ,-m (2kx) = (2k) 1 2 Γ(1 + 2m)Γ(1 -2m) x 1 2 +m 1 - β 1 + 2m x + κx 1 2 -m 1 - β 1 -2m x + O x 5 2 -|Re(m)| .
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 [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]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

R ν β, 1 2 (-k 2 ; x, y) = Γ 1 -β 2k 2 ω ν β, 1 2 (k) (3.15) ×        ω ν β, 1 2 (k) 2kΓ(1-β 2k ) I β 2k , 1 2 (2kx) + K β 2k , 1 2 (2kx) K β 2k , 1 2 (2ky) for 0 < x < y, ω ν β, 1 2 (k) 2kΓ(1-β 2k ) I β 2k , 1 2 (2ky) + K β 2k , 1 2 (2ky) K β 2k , 1 2 (2kx) for 0 < y < x.
We deduce from (3.4) and (3.8) that The proof of (iii) is analogous. We use first (3.13) for the boundedness. Then we rewrite Green's function as

ω ν β, 1 2 (k) 2kΓ 1 -β 2k I β 2k , 1 2 (2kx) + K β 2k , 1 2 (2kx) = 1 Γ 1 -β 2k 1 -βx ln(x) + νx + o x
R ν β,0 (-k 2 ; x, y) = Γ 1 2 -β 2k 2 2kω ν β,0 (k) ×      ω ν β,0 (k) Γ( 1 2 -β 2k ) I β 2k ,0 (2kx) + K β 2k ,0 (2kx) K β 2k ,0 (2ky) for 0 < x < y, ω ν β,0 (k) Γ( 1 2 -β 2k ) I β 2k ,0 (2ky) + K β 2k ,0 (2ky) K β 2k ,0 (2kx) for 0 < y < x. (3.16)
We check that

ω ν β,0 (k) Γ 1 2 -β 2k I β 2k ,0 (2kx) + K β 2k ,0 (2kx) = - (2k) 1 2 Γ 1 2 -β 2k x 1 2 ln(x) + νx 1 2 + o x 1 2
, by (3.4) and (3.9).

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: Proposition 3.8. Let k ∈ C with Re(k) > 0. We have the following properties.

(ii') For m = 1 2 , ν = ∞ and β 2k ∈ N × , set ζ ν β, 1 2 (k) := (-1) β 2k +1 2kΓ β 2k + 1 β ln(2k) + ψ 1 + β 2k + 2γ -1 -k + ν . Then -k 2 ∈ σ(H ν β, 1 2
) and the integral kernel of R ν β, 1 2 (-k 2 ) is given by

R ν β, 1 2 (-k 2 ; x, y) = (-1) 1+ β 2k 2kΓ 1 + β 2k ×      -Γ 1 + β 2k X β 2k , 1 2 (2kx) + ζ ν β, 1 2 (k)K β 2k , 1 2 (2kx) K β 2k , 1 2 (2ky) for 0 < x < y, -Γ 1 + β 2k X β 2k , 1 2 (2ky) + ζ ν β, 1 2 (k)K β 2k , 1 2 (2ky) K β 2k , 1 2 (2kx) for 0 < y < x. (iii') For m = 0, ν = ∞, and β 2k -1 2 ∈ N, set ζ ν β,0 (k) := (-1) 1 2 + β 2k ψ 1 2 + β 2k + 2γ + ln(2k) -ν Γ 1 2 + β 2k .
Then -k 2 ∈ σ(H ν β,0 ) and the integral kernel of R ν β,0 (-k 2 ) is given by R ν β,0 (-k 2 ; x, y) = (-1)

1 2 + β 2k 2kΓ 1 2 + β 2k ×      -Γ 1 2 + β 2k X β 2k ,0 (2kx) + ζ ν β,0 (k)K β 2k ,0 (2kx) K β 2k ,0 (2ky) for 0 < x < y, -Γ 1 2 + β 2k X β 2k ,0 (2ky) + ζ ν β,0 (k)K β 2k ,0 (2ky) K β 2k ,0 (2kx) for 0 < y < x.
Proof. (ii ) is proved similarly as (ii) of Theorem 3.7, by using for m = 1 2 , ν = ∞ and

β 2k ∈ N × that -Γ 1 + β 2k X β 2k , 1 2 (2kx) + ζ ν β, 1 2 (k)K β 2k , 1 2 (2kx) = 1 -βx ln x + νx + o(x).
This follows from (A.24), (A.20), and (3.5).

(iii ) is proved similarly as (iii) of Theorem 3.7. In particular, using (A.24), (A.21), and (3.6) one verifies that

-Γ 1 2 + β 2k X β 2k ,0 (2kx) + ζ ν β,0 (k)K β 2k ,0 (2kx) = (2k) 1 2 x 1 2 ln(x) + νx 1 2 + o x 1 2 .

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 [START_REF] Kato | Perturbation theory for linear operators[END_REF], see also [START_REF] Dereziński | Continuous and holomorphic functions with values in closed operators[END_REF]. 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.

Let H be a complex Banach space. Let {H(z)} z∈Θ be a family of closed operators on H with nonempty resolvent set, where Θ is an open subset of C d . {H(z)} z∈Θ is called holomorphic on Θ if for any z 0 ∈ Θ, there exist λ ∈ C and a neighborhood Θ 0 ⊂ Θ of z 0 such that, for all z ∈ Θ 0 , λ belongs to the resolvent set of H(z) and the map Θ 0 z → (H(z) -λ) -1 ∈ B(H) is holomorphic on Θ 0 . Note that if Θ 0 z → (H(z) -λ) -1 ∈ B(H) is locally bounded on Θ 0 and if there exists a dense subset D ⊂ H such that, for all f, g ∈ D, the map Θ 0 z → (f |(H(z) -λ) -1 g) is holomorphic on Θ 0 , then Θ 0 z → (H(z) -λ) -1 ∈ B(H) is holomorphic on Θ 0 . Besides, by Hartog's theorem, z → (f |(H(z) -λ) -1 g) is holomorphic if and only if it is separately analytic in each variable.

This definition naturally generalizes to families of operators defined on (C ∪ {∞}) d instead of C d , recalling that a map ϕ :

C ∪ {∞} → C is called holomorphic in a neigh- borhood of ∞ if the map ψ : C → C defined by ψ(z) = φ(1/z) if z = 0 and ψ(0) = φ(∞) is holomorphic in a neighborhood of 0.
Recall that the family H β,m has been defined on C × {m ∈ C | Re(m) > -1} in [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF], see also (2.4). However, it is not holomorphic on the whole domain. The following has been proved in [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF].

Theorem 3.9. The family of closed operators (β, m) → H β,m is holomorphic on

C × {m ∈ C | Re(m) > -1}\ 0, -1 2 .
However, it cannot be extended by continuity to include the point 0, - m → H m has been introduced, and satisfies H m = H 0,m for m = -1 2 . Note also that for any β we have H β,-

1 2 = H β, 1 2 . It then turns out that lim β→0 H β,-1 2 = H 1 2 = H -1 2 = lim m→-1 2 H 0,m ,
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

Π := {m ∈ C | -1 < Re(m) < 1}.
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: (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 .

Theorem 3.10. (i) The family of closed operators {H β,m,κ } is holomorphic on C × Π × C ∪ {∞} except for 0, -1 2 × C ∪ {∞} ∪ 0, 1 2 × C ∪ {∞} ∪ C × (0, -1). (3.17) (ii) The family of closed operators {H ν β,0 } is holomorphic on C × C ∪ {∞} . (iii) The family of closed operators H ν β, 1 2 is holomorphic on C × C ∪ {∞} . 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
If m 0 / ∈ -1 2 , 0, 1 2 and κ 0 = ∞, the statement directly follows from the equality H β,m,∞ = H β,-m,0 .

Suppose now that m 0 = 0 and that κ 0 ∈ C \ {-1}. We extend by continuity the definition of ω β,m,κ (k) in (3.10) for m = 0 by setting

ω β,0,κ (k) := 1 + κ Γ 1 2 -β 2k .
We also choose k ∈ C with Re(k) > 0 such that β 0 2k -1 2 ∈ N. This latter requirement implies that ω β 0 ,m 0 ,κ 0 (k) = 0, and by continuity of the map (β, m, κ) → ω β,m,κ (k), there exists a neighborhood U 0 of (β 0 , 0, κ 0 ) such that for all (β, m, κ) in this neighborhood, ω β,m,κ (k) = 0. In particular, by Theorem 3.7, one verifies that, for all f, g ∈

L 2 (R + ), the map (β, m, κ) → (f |(H β,m,κ + k 2 ) -1 g) is well-defined and holomorphic on U 0 provided that (3.14) is extended to U 0 ∩ {(β, 0, κ) | β ∈ C, κ ∈ C} by R β,0,κ (-k 2 ; x, y) = Γ 1 2 -β 2k 2k      I β 2k ,0 (2kx)K β 2k ,0 (
2ky) for 0 < x < y, I β 2k ,0 (2ky)K β 2k ,0 (2kx) for 0 < y < x.

Note that this corresponds to the integral kernel of (H β,0,0

+ k 2 ) -1 = (H ∞ β,0 + k 2 ) -1
. This shows that {H β,m,κ } is holomorphic on U 0 (provided that U 0 is chosen small enough so that (β, 0, -1) ∈ U 0 ).

If m 0 = 0 and κ 0 = ∞, the argument is similar once it is observed that

(H β,0,∞ + k 2 ) -1 = (H ∞ β,0 + k 2 ) -1 = (H β,0,0 + k 2 ) -1 .
It remains to consider the cases m 0 = ± 1 2 and β 0 = 0. Assume for instance that m 0 = -1 2 , β 0 = 0, and κ 0 ∈ C. We extend by continuity the definition of ω β,m,κ (k) in (3.10) for m = -1 2 by setting

ω β,-1 2 ,κ (k) := (2k) 1 2 Γ -β 2k .
We also choose k ∈ C with Re(k) > 0 such that β 0 2k / ∈ N. Then we have ω β 0 ,-1 2 ,κ 0 (k) = 0, and by continuity of (β, m, κ) → ω β,m,κ (k) there exists a neighborhood U 0 of (β 0 , -1 2 , κ 0 ) such that ω β,m,κ (k) = 0 for all (β, m, κ) in U 0 . By Theorem 3.7, one then verifies that for all f, g ∈ L 2 (R + ), the map (β, m, κ)

→ (f |(H β,m,κ + k 2 ) -1 g) is well-defined and holomorphic on U 0 provided that (3.14) is extended to U 0 ∩ β, -1 2 , κ | β ∈ C, κ ∈ C by R β,-1 2 ,κ (-k 2 ; x, y) = 1 2kω β,-1 2 ,κ (k)      (2k) 1 2 I β 2k ,-1 2 (2kx)K β 2k ,-1 2 (2ky) for 0 < x < y, (2k)
1 2 I β 2k ,-1 2 (2ky)K β 2k ,-1 2 (2kx) for 0 < y < x, = Γ 1 -β 2k 2k      I β 2k , 1 2 (2kx)K β 2k , 1 2
(2ky) for 0 < x < y,

I β 2k , 1 2 (2ky)K β 2k , 1 2
(2kx) for 0 < y < x.

Note that this corresponds to the integral kernel of

H β, 1 2 ,0 + k 2 -1 = H ∞ β, 1 2 + k 2 -1 .
This shows that {H β,m,κ } is holomorphic on U 0 . The argument easily adapts to the case m 0 = 1 2 and β 0 = 0. As before, if m 0 = ± 1 2 , β 0 = 0, and κ 0 = ∞, the statement follows from the equalities

H β,± 1 2 ,∞ + k 2 -1 = H ∞ β, 1 2 + k 2 -1 = H β,± 1 2 ,0 + k 2 -1 .
The second part of the statement (i) follows directly from [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]Thm. 3.5]. To prove (ii) and (iii), the argument is analogous and simpler: it suffices to use the formulas (3.15) to prove (ii) and (3.16) to prove (iii).

The following statement shows that the domains of holomorphy obtained in Theorem 3.10 are maximal for m ∈ Π. In particular, we will prove that (3.17

(β, m) → H β,m,κ defined on C × Π \ {(0, - 1
2 ), (0, 1 2 )} cannot be extended by continuity at (0, - 1 2 ) and (0, 1 2 ). If κ = 0, the family (β, m) → H β,m,0 defined on C×Π\{(0, -1 2 )} cannot be extended by continuity at (0, - 1 2 ), and for κ = ∞ the family (β, m) → H β,m,∞ defined on C × Π \ {(0, 1 2 )} cannot be extended by continuity at (0, 1 2 ).

(ii) For any fixed β ∈ C, the family (m, κ) → H β,m,κ defined on Π× C∪{∞} \{(0, -1)} cannot be extended by continuity at (0, -1).

Proof. (i) Let us first consider β = 0. Recall that in [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF] the family of closed operators Π×(C∪{∞}) (m, κ) → H m,κ has been introduced, and that this family is holomorphic on Π × (C ∪ {∞}) \ {0} × (C ∪ {∞}). Here is its relationship to the families from the present article:

H m,κ :=      H 0,m,κ if m / ∈ {-1 2 , 1 2 } H κ -1 0, 1 2 if m = 1 2 H κ 0, 1 2 if m = -1 2
Let us now focus on m = -1 2 and on m = 1 2 . We have for any κ ∈ C ∪ {∞}

H β,-1 2 ,κ = H β, 1 2 ,κ = H β, 1 2 = H ∞ β, 1 2 .
Therefore, for κ = 0,

lim β→0 H β, 1 2 ,κ = H ∞ 0, 1 2 = H κ -1 0, 1 2 = lim m→ 1 2 H 0,m,κ . (3.18)
Similarly, for κ = ∞,

lim β→0 H β,-1 2 ,κ = H ∞ 0, 1 2 = H κ 0, 1 2 = lim m→-1 2 H 0,m,κ . (3.19)
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

H β,0,κ = H β,0 = H ∞ β,0 ,
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

R β,m,-1 (-k 2 ; x, y) = 1 2kω β,m,-1 (k) ×      (2k) -m Γ(1-2m) I β 2k ,m (2kx) -(2k) m Γ(1+2m) I β 2k ,-m (2kx) K β 2k ,m (2ky) for 0 < x < y, ( 2k
) -m Γ(1-2m) I β 2k ,m (2ky) -(2k) m Γ(1+2m) I β 2k ,-m (2ky) K β 2k ,m (2kx) for 0 < y < x,
where ω β,m,-1 (k) is defined in (3.10). One then infers that

g β,k,x (m) := 1 ω β,m,-1 (k) (2k) -m Γ(1 -2m) I β 2k ,m (2kx) - (2k) m Γ(1 + 2m) I β 2k ,-m (2kx) = (2k) 1 2 Γ(1-2m)Γ(1+2m) x 1 2 +m -x 1 2 -m (2k) -m Γ( 1 2 +m-β 2k )Γ(1-2m) - (2k) m Γ( 1 2 -m-β 2k )Γ(1+2m)
+ O(x 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

g β,k,x (0) = - (2k) 1 2 Γ 1 2 -β 2k ln(2k) + ψ 1 2 -β 2k + 2γ x 1 2 ln(x) + o(x 1
2 ), x → 0.

Thus, the family of operators { Hβ,m,-1 } defined by

Hβ,m,-1 = H β,m,-1 if m = 0 H 0 β,0 if m = 0, is holomorphic for m ∈ Π. It thus follows that lim κ→-1 H β,0,κ = H ∞ β,0 = H 0 β,0 = lim m→0 H β,m,-1 , (3.20)
which concludes the proof.

Eigenprojections

Let us now describe a family of projections {P β,m (λ)}, which is closely related to the Whittaker operator. We shall define it by specifying its integral kernel P β,m (λ; x, y).

We first define the projections for λ ∈ C\[0, ∞[. As usual, we write λ = -k 2 with Re(k) > 0. We define:

(i) For -1 < Re(m) < 1, m ∈ -1 2 , 0, 1 2 ,
P β,m (-k 2 ; x, y)

:= k sin(2πm)Γ 1 2 + m -β 2k Γ 1 2 -m -β 2k π 2m + β 2k ψ 1 2 + m -β 2k -β 2k ψ 1 2 -m -β 2k K β 2k ,m (2kx)K β 2k ,m (2ky).
(ii) For m = 0,

P β,0 (-k 2 ; x, y) := kΓ 1 2 -β 2k 2 1 + β 2k ψ 1 2 -β 2k K β 2k ,0 (2kx)K β 2k ,0 (2ky).
(iii) For m = 1 2 ,

P β, 1 2
(-k 2 ; x, y) := - kΓ -β 2k Γ 1 -β 2k 1 + k β + β 2k ψ 1 -β 2k K β 2k , 1 2 (2kx)K β 2k , 1 2 (2ky).
(iv

) For m = -1 2 , P β,-1 2 (-k 2 ; x, y) := P β, 1 2
(-k 2 ; x, y).

Then we consider λ ∈]0, ∞[. We shall distinguish between points coming from the upper and lower half-plane by writing λ ± i0 = -(∓iµ) 2 , where µ > 0. Let 0 < µ < ±Im(β). We define:

(i) For -1 < Re(m) < 1, m ∈ -1 2 , 0, 1 2 P β,m (µ 2 ± i0; x, y) := µ sin(2πm)Γ 1 2 + m ∓ i β 2µ Γ 1 2 -m ∓ i β 2µ e ∓iπm π 2m ± i β 2µ ψ 1 2 + m ∓ i β 2µ ∓ i β 2k ψ 1 2 -m ∓ i β 2µ H ± β 2µ ,m (2µx)H ± β 2µ ,m (2µy).
(ii) For m = 0,

P β,0 (µ 2 ± i0; x, y) := µΓ 1 2 ∓ i β 2µ 2 1 ± i β 2µ ψ 1 2 ∓ i β 2µ H ± β 2µ ,0 (2µx)H ± β 2µ ,0 (2µy).
(iii) For m = 1 2 ,

P β, 1 2 (µ 2 ± i0; x, y) := µΓ ∓ i β 2µ Γ 1 ∓ i β 2µ ±i + µ β -β 2µ ψ 1 ∓ i β 2µ H ± β 2µ , 1 2 (2µx)H ± β 2µ , 1 2 (2µy).
(iv

) For m = -1 2 , P β,-1 2 (µ 2 ± i0; x, y) := P β, 1 2 (µ 2 ± i0; x, y).
Finally, if λ = 0, we set, for any -1 < Re(m) < 1 and ±Im( √ β) > 0,

P β,m (0; x, y) = 3β sin(2πm) me ∓iπ2m (4m 2 -1) (βx) 1 4 H ± 2m (2 βx)(βy) 1 4 H ± 2m (2 βy).
Note that for m ∈ -1 2 , 0, 1 2 the extension by continuity gives P β,0 (0; x, y) = -6πβ(βx)

1 4 H ± 0 (2 βx)(βy) 1 4 H ± 0 (2 βy). and P β, 1 2 (0; x, y) = P β,- 1 2
(0; x, y) = 3πβ(βx)

1 4 H ± 1 (2 βx)(βy) 1 4 H ± 1 ( 2
βy). In the next proposition, we study the regularity of the map (β, m, k) → P β,m (-k 2 ) in a suitable region. Proposition 3.12. The function (β, m, k) → P β,m (-k 2 ), defined on the set

C × Π × {k ∈ C | Re(k) > 0} ∪ {(β, m, ∓iµ) | β ∈ C, m ∈ Π, 0 < µ < ±Im(β)} ∪ {(β, m, 0) | β ∈ C, m ∈ Π, 0 < ±Im( β)}, (3.21)
has values in bounded projections. Moreover, it is continuous on

C × Π × {k ∈ C | Re(k) > 0} ∪ {(β, m, ∓iµ) | β ∈ C, m ∈ Π, 0 < µ < ±Im(β)}, (3.22)
and holomorphic in C × Π × {Re(k) > 0}. It satisfies

P β,m (-k 2 ) = P β,-m (-k 2 ), (3.23)
P β,m (-k 2 ) # = P β,m (-k 2 ), (3.24)
P β,m (-k 2 ) * = P β, m(-k 2 ), (3.25)
for all (β, m, k) in the set (3.21).

Proof. The fact that P β,m (-k 2 ) are rank-one projections follows directly from their expressions together with Corollaries A. 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.

Theorem 3.13. Let β ∈ C, m ∈ Π \ -1 2 , 0, 1 2 , κ ∈ C ∪ {∞} and ν ∈ C ∪ {∞}. Let λ ∈ C be an eigenvalue of H β,m,κ , H ν β,0 or H ν β, 1 2
respectively. Then the corresponding self-transposed eigenprojection is P β,m (λ), P β,0 (λ) or P β,

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

K β 2k ,m (2k•) | K β 2k ,m (2k•) = π sin(2πm) 2m + β 2k ψ 1 2 + m -β 2k -β 2k ψ 1 2 -m -β 2k kΓ 1 2 + m -β 2k Γ 1 2 -m -β 2k .
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 ,

lim k→0 f |P β,m (-k 2 )f = f |P β,m (0)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

g β,m,k (x) := ∓i Γ 1 2 + m -β 2k √ π β 2k 1 2 -m K β 2k ,m (2kx)
, 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

± arg(β) -arg(k) ∈]ε, π -ε[ with ε > 0, β 2k 1 2 +m I β 2k ,m (2kx) = (βx) 1 2 +m e -kx ∞ j=0 1 2 + m -β 2k j Γ(1 + 2m + j) (2kx) j j! ≤ |βx| 1 2 +m ∞ j=0 c j x j |Γ(1 + 2m + j)| ,
for some constant c > 0 depending on β and m but independent of k and x. Using that

g β,m,k (x) = ∓i √ π sin(2πm) β 2k 1 2 -m - Γ 1 2 + m -β 2k Γ 1 2 -m -β 2k I β 2k ,m (2kx) + I β 2k ,-m (2kx) ,
together with Lemma B.2, one then deduces that

g β,m,k (x) ≤ c 1 e c 2 x ,
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

f |P β,m (-k 2 )f = k sin(2πm)Γ 1 2 + m -β 2k Γ 1 2 -m -β 2k π 2m + β 2k ψ 1 2 + m -β 2k -β 2k ψ 1 2 -m -β 2k K β 2k ,m (2k•)|f 2 = - k sin(2πm) 2m + β 2k ψ 1 2 + m -β 2k -β 2k ψ 1 2 -m -β 2k Γ 1 2 -m -β 2k Γ 1 2 + m -β 2k β 2k 2m-1 g β,m,k |f 2 = 2k 2 sin(2πm) β β 2k 2k β 3 m 6 (-1 + 4m 2 ) + o(1) Γ 1 2 -m -β 2k Γ 1 2 + m -β 2k β 2k 2m g β,m,k |f 2 → k→0 3β sin(2πm) m(4m 2 -1) e ±iπ2m g β,m,0 |f 2 = f |P β,m (0)f ,
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,

P β,m (-k 2 ) -P β,m (0) f | P β,m (-k 2 ) -P β,m (0) f = P β,m (-k 2 )f |f + P β,m (0)f |f -2 P β,m (0)f |P β,m (-k 2 )f .
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

I β,m (z) = z 1 2 +m Γ 1 2 + m + β Γ 1 2 + m -β 1 0 e ±z(s-1 2 ) s m∓β-1 2 (1 -s) m±β-1 2 ds.
Based on (A.3) one easily gets

W I β,m , I β,-m ; x = - sin(2πm) π (A.4)
as well as the following identity

I β,m (z) = e ∓iπ( 1 2 +m) I -β,m e ±iπ z . (A.5)
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:

K β,m = π sin(2πm) - I β,m Γ 1 2 -m -β + I β,-m Γ 1 2 + m -β . (A.6)
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:

K β,m (z) = z β e -z 2 2 F 0 1 2 + m -β, 1 2 -m -β; -; -z -1 .
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:

K β,m (z) = z 1 2 ∓m e -z 2 Γ 1 2 -β ∓ m ∞ 0 e -zs s -1 2 -β∓m (1 + s) -1 2 +β∓m ds.
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

K β,m (z) = z β e -z 2 1 + O(z -1 ) . (A.7)
By using the relation (A.6) one also obtains that

W I β,m , K β,m ; x = - 1 Γ 1 2 + m -β . (A.8)
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.

The function K β,m is distinguished by the fact that it decays exponentially, while the solutions I β,±m (z) explode exponentially, see [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]Eq. (2.14) & (2.22)]. This is also the case for the analytic continuations of K -β,m by the angles ±π, which by the symmetry (A.2) are also solutions of (A.1). It will be convenient to introduce a name for a solution constructed from these two analytic continuations. There is some arbitrariness for this choice, but we have decided on: X β,m (z) := 1 2 e -iπ( 1 2 +m) K -β,m e iπ z + e iπ( 1 2 +m) K -β,m e -iπ z .

(A.9)

As a consequence of this definition and of (A.5) one gets the relations

X β,m (z) = - π sin(2πm) I β,m (z) Γ 1 2 -m + β - cos(2πm)I β,-m (z) Γ 1 2 + m + β , (A.
I β,m = 1 sin(π(m + β)) cos(2πm) Γ 1 2 + m + β K β,m - 1 Γ 1 2 + m -β X β,m . (A.12)
By taking formulas (A.6), (A.10), and (A.11) into account, one infers that the Wronskian is provided by

W (K β,m , X β,m ; x) = -sin π(m + β) .
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

X β,m (z) = Γ 1 2 -m -β Γ 1 2 -m + β K β,m (z).
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.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

I β,-p 2 = -β - p -1 2 p I β, p 2 , or equivalently, 1 Γ 1+p 2 -β I β,-p 2 = 1 Γ 1-p 2 -β I β, p 2 .
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

K β, p 2 (z) = (-1) p+1 ln(z) I β, p 2 (z) Γ 1-p 2 -β + (-1) p+1 e -z 2 z 1+p 2 Γ 1-p 2 -β ∞ k=0 1+p 2 -β k z k (p + k)! k! × ψ 1+p 2 -β + k -ψ(p + 1 + k) -ψ(1 + k) + (-1) p+1 e -z 2 z 1+p 2 Γ 1-p 2 -β p j=1 1+p 2 -β -j (-1) j-1 (j -1)!z -j (p -j)! , (A.17)
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: A.18) and to p = 0:

K β, 1 2 (z) = ln(z) I β, 1 2 (z) Γ -β + e -z 2 Γ(1 -β) + e -z 2 Γ -β ∞ k=0 1 -β k z 1+k (1 + k)! k! ψ 1 -β + k -ψ(2 + k) -ψ(1 + k) , (
K β,0 (z) = - ln(z) I β,0 (z) Γ 1 2 -β - e -z 2 Γ 1 2 -β ∞ k=0 1 2 -β k z 1 2 +k (k!) 2 ψ 1 2 -β + k -2ψ(1 + k) . (A.19)
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) Γ 1-p 2 + β + (-1) p+1 e z 2 z 1+p 2 Γ 1-p 2 + β ∞ k=0 1+p 2 + β k (-1) k z k (p + k)! k! × ψ 1+p 2 + β + k -ψ(p + 1 + k) -ψ(1 + k) - (-1) p+1 e z 2 z p+1 2 Γ 1-p 2 + β p j=1 1+p 2 + β -j (j -1)!z -j (p -j)! .
In particular, the following expansion will be useful:

X β, 1 2 (z) = - 1 Γ(1 + β) + 1 Γ(β) z ln(z) + 1 Γ(β) ψ(1 + β) + 2γ -1 - 1 2β z + o(z) (A.20) and X β,0 (z) = - 1 Γ 1 2 + β z 1 2 ln(z) -ψ 1 2 + β + 2γ z 1 2 + o z 1 2 (A.21)
Note also that the following identity holds:

X β,-p 2 = (-1) p X β, p 2
, as a consequence of (A.9).

β m -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 4 3 2 1 1 2 3 4
Figure 1: The vertical lines correspond to the degenerate cases, the lines with slope 1 to the decaying Laguerre case, the lines with slope -1 with the exploding Laguerre 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.14) reduces to

I p+1 2 +n, p 2 = (-1) n (p + n)! K p+1 2 +n, p 2
.

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.

β m I - I + II + II - -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 4

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

√ z∂ z + -1 2 -m √ z - √ z 2 I β,m (z) = - 1 2 -m -β I β+ 1 2 ,m+ 1 2 (z), √ z∂ z + -1 2 + m √ z + √ z 2 I β,m (z) = I β-1 2 ,m-1 2 (z), √ z∂ z + -1 2 + m √ z - √ z 2 I β,m (z) = I β+ 1 2 ,m-1 2 (z), √ z∂ z + -1 2 -m √ z + √ z 2 I β,m (z) = 1 2 + m -β I β-1 2 ,m+ 1 2 (z), z∂ z + β - z 2 I β,m (z) = 1 2 + m + β I β+1,m (z), z∂ z -β + z 2 I β,m (z) = 1 2 + m -β I β-1,m (z).
By using the relation between the functions K β,m and the functions I β,m provided in (A.6), one infers from the above relations the following ones:

√ z∂ z + -1 2 -m √ z - √ z 2 K β,m (z) = -K β+ 1 2 ,m+ 1 2 (z), √ z∂ z + -1 2 + m √ z + √ z 2 K β,m (z) = - 1 2 + m + β K β-1 2 ,m-1 2 (z), √ z∂ z + -1 2 + m √ z - √ z 2 K β,m (z) = -K β+ 1 2 ,m-1 2 (z), √ z∂ z + -1 2 -m √ z + √ z 2 K β,m (z) = - 1 2 -m + β K β-1 2 ,m+ 1 2 (z), z∂ z + β - z 2 K β,m (z) = -K β+1,m (z), z∂ z -β + z 2 K β,m (z) = 1 2 + m -β 1 2 -m -β K β-1,m (z).

A.6 Integral identities

Let us start with a general fact about 1-dimensional Schrödinger operators, see for example [5, Eq. (3.24)].

Lemma A.1. For i ∈ {1, 2}, suppose that v i ∈ D(L max β,α ) satisfies L β,α v i = λ i v i for some λ i ∈ C. Then, for all a, b ∈]0, ∞[, (λ 1 -λ 2 ) b a v 1 (x)v 2 (x)dx = W (v 1 , v 2 ; b) -W (v 1 , v 2 ; a), (A.25)
where W is the Wronskian introduced in (2.1).

As a consequence of this lemma one has:

Proposition A.2. Let k, p ∈ C with Re(k) > 0 and Re(p) > 0.

and

H ± β,m (z) =e ∓i π 2 ( 1 2 +m) K ±iβ,m (e ∓i π 2 z) = ±iπ sin(2πm) e ∓iπm J β,m (z) Γ 1 2 -m ∓ iβ - J β,-m (z) Γ 1 2 + m ∓ iβ , (A.29)
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

L β,m 2 H ± β 2µ ,m (2µx) = µ 2 H ± β 2µ ,m (2µx)
valid for any µ > 0.

Proposition A.4. Let µ, η > 0 with µ < ±Im β) and η < ±Im β).

(i) If -1 < Re(m) < 1, m ∈ -1 2 , 0, 1 2 , then (µ 2 -η 2 ) ∞ 0 H ± β 2µ ,m (2µx)H
± β 2η ,m (2ηx)dx
= πe ∓iπm sin(2πm) 4µη µ m η -m Γ 1 2 + m ∓ i β 2η Γ 1 2 -m ∓ i β 2µ - η m µ -m Γ 1 2 + m ∓ i β 2µ Γ 1 2 -m ∓ i β 2η . (ii) If m = 0, then (µ 2 -η 2 ) ∞ 0 H ± β 2µ ,0 (2µx)H ± β 2η ,0 (
2ηx)dx = 4µη ψ 1 2 ∓ i β 2µ -ψ 1 2 ∓ i β 2η + ln(µ) -ln(η) Γ 1 2 ∓ i β 2µ Γ 1 2 ∓ i β 2η . (iii) If m = 1 2 , then (µ 2 -η 2 ) ∞ 0 H ± β 2µ , 1 2 (2µx)H ± β 2η , 1 2 (2ηx)dx = β ψ 1 ∓ i β 2µ -ψ 1 ∓ i β 2η ∓ i µ β ± i η β + ln(µ) -ln(η) Γ 1 ∓ i β 2η Γ 1 ∓ i β 2µ . Corollary A.5. Let 0 < µ < ±Im(β). (i) For -1 < Re(m) < 1, m ∈ -1 2 , 0, 1 2 one has ∞ 0 H ± β 2µ ,m (
2µx) 2 dx = πe ∓iπm sin(2πm) To make our presentation of the hyperbolic Bessel equation as much parallel to that of the Whittaker equation as possible, we introduce the function X m (z) := 1 2 e -iπ( 1 2 +m) K m e iπ z + e iπ( 1 2 +m) K m e -iπ z .

2m ± i β 2µ ψ 1 2 + m ∓ i β 2µ ∓ i β 2k ψ 1 2 -m ∓ i β 2µ µΓ 1 2 + m ∓ i β 2µ Γ 1 2 -m ∓ i β
Then the following relations hold:

X m = - 1
K 0,m (z) = K m z 2 , X 0,m (z) = X m z 2 .

B.2 The degenerate case

For m ∈ Z the following relation holds:

I -m (z) = I m (z).
Assuming that m ∈ N, we also have .

I m (z) = z 2

B.3 The half-integer case

The half-integer case of the hyperbolic Bessel equation is a special case of the doubly degenerate case of the Whittaker equation. However, it is worthwhile to discuss it separately. In particular, for n ∈ N the function I -1 2 -n is not proportional to the function I 0,-1 2 -n , which is identically 0 by (B.4).

B.5 The zero eigenvalue Whittaker equation

The zero eigenvalue Whittaker equation is provided by the equation 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.

L β,m 2 f := -∂ 2 z + m 2 -
Proof. Using the definition of Pochhammer's symbol recalled in Section A.1, one infers that lim k→0 1 2 + m ∓ β 2k j (±2k) j = (-β) j .

In addition, for all k ∈ C with |k| < 1, one has

1 2 +m + κx 1 2

 11 -m , and H ν 0 , with ν ∈ C ∪ {∞}, defined by L 0 with boundary conditions ∼ x 1 2 ν + ln(x) .

  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

Proposition 2 . 1 .

 21 Let α, β, m ∈ C. (i) If f ∈ D(L max β,α ), then f and f are continuous functions on R + and converge to 0 at infinity. (ii) If f ∈ D(L min β,α ), then near 0 one has:

Proposition 2 . 3 .

 23 For any β ∈ C, |Re(m)| < 1 and κ ∈ C ∪ {∞} the following relation holds H β,m,κ = H β,-m,κ -1

Proposition 2 . 5 .

 25 For m ∈ C with |Re(m)| < 1 one has

2

  and H ν β,0 are self-adjoint if and only if β ∈ R and ν ∈ R ∪ {∞}.

  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 x → H - β 2µ ,m (2µx) never belongs to L 2 near infinity. Conversely, 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 x → H + β 2µ ,m

1 2

 1 the eigenvalues of the operator H ν 0, are of the form -k 2 with Re(k) > 0, where ν = -k, (iii) If ν ∈ C ∪ {∞}, the eigenvalues of the operator H ν 0,0 are of the form -k 2 with Re(k) > 0, where

3 2 ,, 1 2 .

 322 which belongs to L 2 around 0 and corresponds to the boundary condition defining H ν β

1 2 . 2 .

 22 Let us sketch what happens at 0, -1 Recall that in [2, 6] a holomorphic family m ∈ C | Re(m) > -1

  ) are sets of nonremovable singularities of the family (β, m, κ) → {H β,m,κ }. Proposition 3.11. (i) For any fixed κ ∈ C × , the family of closed operators

  3 and A.5 and Proposition B.4. Continuity on the domain (3.22) and holomorphy on C × Π × {Re(k) > 0}, as well as the relations (3.23)-(3.25), follow again from the expressions involved in the definitions of P β,m (-k 2 ).

10 ) and e ∓iπ( 1 2

 102 +m) K -β,m e ±iπ z = X β,m (z) ∓ iπI β,-m (z) Γ 1 2 + m + β .In addition, by using the equalitiescos π(m -β) = cos 2πm -π(m + β) = cos(2πm) cos π(m + β) + sin(2πm) sin π(m + β) (A.11) one infers from (A.6) and (A.10) that cos(2πm)K β,m cos π(m -β) -cos(2πm) cos π(m + β) I β,m = sin π(m + β) I β,m , which finally leads to the relation

Figure 2 :

 2 Figure 2: Solutions for the doubly degenerate case: Region I -: the decaying Laguerre and the exploding Laguerre solutions. Region I + : any of the three solutions. Region II + : the decaying Laguerre and the exploding logarithmic solutions. Region II -: the exploding Laguerre and the decaying logarithmic solutions.

1 4 1 z 2 2 )

 1422 The modified Bessel equationThe modified (or hyperbolic type) Bessel equation for dimension 1-∂ 2 z + m 2 -1 + 1 f = 0, (B.1)is up to a trivial rescaling, a special case of the Whittaker equation with β = 0. Its theory was discussed at length in [6, App. A]. Nevertheless, we briefly discuss some of its elements here, explaining the parallel elements to the theory of the Whittaker equation, as well as the differences.Let the modified Bessel function for dimension 1 beI m (z) =The equation (B.1) is invariant with respect to m → -m. At the level of the function (B.2) this property is reflected byI m (z) = e ∓iπ( 1 2 +m) I m (e ±iπ z).For the Wronskian we haveW (I m , I -m ; z) = -sin(πm).The function K m can be introduced for m ∈ Z byK m (z) = 1 sin(πm) -I m (z) + I -m (z) .For m ∈ Z the definition is extended by continuity. Note that the relation K m (z) = K -m (z) holds, and that W (K m , I m ; z) = 1.

  sin(πm) I m -cos(2mπ)I -m ,I m = 1 2 sin(mπ) cos(2mπ)K m -X m . (B.3)The precise relations between the Whittaker functions for β = 0 and Bessel-type functions are of the form I 0,m (z)

8 ) 1 4 v( 2 √

 812 It is easy to see that if v solves the trigonometric Bessel equation of dimension 1 (B.1) with parameter 2m, then the function f defined by f (x) := (βx) βx) solves the equation (B.8). 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 For any fixed x ∈ R + , any m ∈ Π and β ∈ C × , one has lim

∞ j=0 1 2Proposition B. 3 . 2 ,

 132 j ≤ c j j! 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 getslim k→0 + m ∓ β 2k j (±2kx) j Γ(1 + 2m + j)j! = ∞ j=0 (-βx) j Γ(1 + 2m + j)j! ,which leads directly to the equality (B.9). The equality (B.10) can then be deduced from (B.9) by using the relation (A.28) between the functions I β,m and J β,m .For (B.11), by using successively (A.6), (B.13), (B.9), and [6, App. A.5] one gets 2m (2 βx) + o(1),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. Lemma B.2. Let a, b ∈ C. For |z| → ∞ with | arg(z)| < π -ε and ε > 0 one has lim z→∞ Γ(a + z) Γ(b + z) z b-a = 1. (B.13) Proof. Recall first the logarithmic version of Stirling formula [1, Eq. 6.1.41] : ln Γ(z) = z ln(z) -This readily implies that ln Γ(a + z) -ln Γ(b + z) + (b -a) ln(z) → z→∞ 0. After exponentiation it leads to the statement. B.6 Integrals for zero eigenvalue solutions of the Whittaker equation Based on the results of the previous sections and on Lemma A.1, one easily gets: Let k ∈ C with Re(k) > 0 and let β ∈ C with ±Im( √ β) > 0. (i) If m ∈ C with |Re(m)| < 1 and m ∈ -1

  holds, it follows from relation (A.8) that the functions K β

	2k ,m (2k•) 2k ,m (2k•) does not belong to L 2 near 2k ,-m (2k•) are linearly dependent, but still I β and I β 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.

π) 1 + O(|x| -1 2 ) ,

-|m| ), x → 0.

Supported by the grantTopological invariants through scattering theory and noncommutative geometry from Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328, and on leave of absence from Univ.

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 The Whittaker equation A.1 General theory

In this section we collect basic information about the Whittaker equation. This should be considered as a supplement to [START_REF] Richard | On radial Schroedinger operators with a Coulomb potential[END_REF]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

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

We shall call this situation the decaying Laguerre case. In this case the relation (A.12) reduces to

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.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. [START_REF] Kurasov | On the Coulomb potential in one dimension[END_REF] 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.

In order to analyze the doubly degenerate case more precisely, let us divide (A.22) into 4 distinct regions (see Figure 2).

which follows for example from (A.23). By setting

is the decaying Laguerre solution, see (A.13), and

is the exploding Laguerre solution, see (A.15).

, and the equality (A.23) can be rewritten as

corresponds to the exploding Laguerre solution. In this region, the space of solutions can also be spanned by the pair K β,m and I β,m , or by the pair I β,m and X β,m .

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.

Proof. The proof consists in an application of Lemma A.1. Consider k, p ∈ C with Re(k) > 0, Re(p) > 0 and set λ 1 = -k 2 and λ 2 = -p 2 . As shown in the proof of Theorem 3.1 the functions v i defined by

belong to D(L max β,m 2 ) and are eigenfunctions of L β,m 2 associated with the eigenvalues λ i . Let us then set W (v 1 , v 2 ; 0) := lim x 0 W (v 1 , v 2 ; x) and observe that lim x→+∞ W (v 1 , v 2 ; x) = 0, as a consequence of Proposition 2.1. This yields directly to

Let us now set

Then, the identity (A.6) leads to

and with the expansion provided in A.3 one directly infers that

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. [START_REF] Humblet | Analytical structure and properties of Coulomb wave functions for real and complex energies[END_REF].

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

In [7, Sec. 2.6 & 2.7] we introduced the functions

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.

Figure 3: The two regions in the half-integer case Note that unlike for the Whittaker equation, in both regions I -and I + the functions I m , I -m and K m are well defined and distinct, and any two of them form a basis of solutions of (B.1). In this case all solutions are elementary functions: For n ∈ N and m = ±( 12 + n) one has

Note also that (B.5) and (B.6) are special cases of (B.3), namely

B.4 The standard Bessel equation

The standard (or trigonometric-type) Bessel equation for dimension 1

is up to a trivial rescaling, a special case of the trigonometric-type Whittaker equation with β = 0. One can introduce the following functions which solve this equation (see [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF]App. A] for more information) :

and

In the next proposition, we consider the integral of ((βx)

By a contour integration with a vanishing contribution at infinity, one gets that for ±Im(

This leads to the statement of the proposition.

Remark B.5. Curiously, a naive computation suggests incorrectly that ∞ 0 (βx)

Indeed, for m ∈ -1 2 , 0, 1 2 and k ∈ C with Re(k) > 0, and such that ± arg(β)arg

-e ∓iπ2m .

By taking a limit as k → 0, one obtains from Lemma B.2 that

-e ∓iπ2m = 0. (B.17

Although by (B.11) the term in the square braket of (B.16) converges pointwise to (βx)

1 4 H ± 2m (2 √ βx), a limit lim k→0 and the integral in (B.16) can certainly not be exchanged, since otherwise it would lead to a contradiction.

To conclude, we give a lemma which was used in the proof of Proposition 3.14. which leads directly to the statement.