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 term1/r2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/r^2$$\end{document}. 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 b;a :¼ Ào 2 x þ a À 1 4 1 The parameters a and b are allowed to be complex valued. We shall study realizations of L b;a as closed operators on L 2 ðR þ Þ, 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 [32]. 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 ðR þ Þ. We do not restrict ourselves to self-adjoint realizations-we look for realizations that are wellposed, 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 L a :¼ Ào 2 x þ a À 1 4 1 x 2 : ð1:2Þ As is known, it is useful to set a ¼ m 2 . In [2] the following holomorphic family of closed realizations of (1.2) was introduced: H m ; with À 1\ReðmÞ; defined by L m 2 with boundary conditions $ x 1 2 þm : It was proved that for ReðmÞ ! 1 the operator H m is the only closed realization of L m 2 . In the region À1\ReðmÞ\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: defined by L m 2 with boundary conditions $ x Note that related investigations about these operators have also been performed in [30,31].
In [7] and in the present paper we study closed realizations of the differential operator (1.1) on L 2 ðR þ Þ. Again, it is useful to set a ¼ m 2 . In [7] we introduced the family H b;m ; with b 2 C; À1\ReðmÞ; defined by L b;m 2 with boundary conditions $ x It was noted in this reference that this family is holomorphic except for a singularity at ðb; mÞ ¼ À 0; À 1 2 Á , which corresponds to the Neumann Laplacian. For ReðmÞ ! 1 the operator H b;m is also the only closed realization of L b;m 2 . In the region À1\ReðmÞ\1 there exist other closed realizations of L b;m 2 . The boundary conditions corresponding to H b;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 b;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 b;m 2 for À1\ReðmÞ\1 in three holomorphic families: The generic family H b;m;j ; with b 2 C; À1\ReðmÞ\1; m 6 2 È À ; with b 2 C; m 2 C [ f1g defined by L b; 1 4 with boundary conditions $ 1 À bx lnðxÞ þ mx: The above holomorphic families include all possible well-posed realizations of L b;m 2 in the region jReðmÞj\1 with one exception: the special case ðb; m; jÞ ¼ À 0; À 1 2 ; 0 Á which corresponds to the Neumann Laplacian H À 1 2 ¼ H À 1 2 ;0 ¼ H1 2 ;1 , and which is already covered by the families H m and H m;j .
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 eigenfunctions of these operators. They can be organized into a single family of bounded 1-dimensional projections P b;m ðkÞ such that L max b;m P b;m ðkÞ ¼ kP b;m ðkÞ. Here L max b;m denotes the maximal operator which is introduced in Sect. 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 [10,21,22] about self-adjoint extensions of the 1-dimensional Schrödinger operator on the real line with a Coulomb potential (without the centrifugal term). Papers [11,20,23] discuss generalized Nevanlinna functions naturally appearing in the context of such operators, especially in the range of parameters jReðmÞj ! 1. See also [4,9,[12][13][14][15][16][17][18][24][25][26][27][28] and references therein. However, essentially all these references are devoted to real parameters b; 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 f ðzÞ ¼ 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 [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 b;m and K b;m , described for example in [7], we introduce a new kind of Whittaker functions, denoted X b;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 [1,3,29]. 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 b;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 b;m;j , H m b; 1 2 and H m b;0 on the Hilbert space L 2 À 0; 1½ Á .
The Hilbert space L 2 ðR þ Þ 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 Ã . Its transpose is denoted by A # . If A is bounded, then A Ã and A # are defined by the relations The definition of A Ã 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 Cn À 1; 0 and on 0; 1½ they coincide with their usual definitions from real analysis: lnðzÞ, ffiffi z p , z k . We set argðzÞ :¼ Im À lnðzÞ Á . Sometimes it will be convenient to include in the domain of our functions two copies of À 1; 0½, describing the limits from the upper and lower half-plane. They correspond to the limiting cases argðzÞ ¼ AEp.
The Wronskian of two continuously differentiable functions f and g on R þ is denoted by Wðf ; g; ÁÞ and is defined for x 2 R þ by Wðf ; g; xÞ :¼ f ðxÞg 0 ðxÞ À f 0 ðxÞgðxÞ: ð2:1Þ

Zero-energy eigenfunctions of the Whittaker operator
In order to study the realizations of the Whittaker operator L b;a one first needs to find out what are the possible boundary conditions at zero. The general theory of 1-dimensional 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 0 ) and (ii),(ii 0 ), where (i 0 ) for any k 2 C the space of solutions of ðL b;a À kÞf ¼ 0 which are square integrable around zero is 2-dimensional, (ii 0 ) for any k 2 C the space of solutions of ðL b;a À kÞf ¼ 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 k. In our case the simplest choice corresponds to k ¼ 0. Therefore, we first discuss solutions of the zero eigenvalue Whittaker equation for m and b in C. As analyzed in more details in Sect. B.7, solutions of (2.2) can be constructed from solutions of the Bessel equation. More precisely, for b 6 ¼ 0, let us define the following function for x 2 R þ : where J m is defined in Sect. B.6. For b ¼ 0 we set j 0;m ðxÞ :¼ x mþ
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 2 C with À2m 6 2 N Â one has In the exceptional cases one has

Maximal and minimal operators
For any a and b 2 C we consider the differential expression acting on distributions on R þ . The corresponding maximal and minimal operators in L 2 ðR þ Þ are denoted by L max b;a and L min b;a , see [7,Sec. 3.2] for the details. The domain of L max b;a is given by while L min b;a is the closure of the restriction of L b;a to C 1 c À 0; 1½ Á , the set of smooth functions with compact supports in R þ . The operators L min b;a and L max b;a are closed and we have We say that f 2 DðL min b;a Þ around 0, (or, by an abuse of notation, f ðxÞ 2 The following result follows from the theory of one-dimensional Schrödinger operators.
, then for any f 2 DðL max b;m 2 Þ there exists a unique pair a; b 2 C such that f À aj b;m À b j b;Àm 2 DðL min b;m 2 Þ around 0: Þ, then there exists a unique pair a; b 2 C such that Þ around 0: (iv) If jReðmÞj\1, then Proof The statements (i)-(iii) and (v) are a reformulation of [7,Prop. 3.1] with the current notations. Only (iv) requires elaboration. The first equality in (iv) follows from [5,Thm. 3.4], given that Wðf ; g; 1Þ ¼ 0 for all f ; g 2 DðL max b;m 2 Þ by (i).

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 b;m;j is defined for any b 2 C, for any m 2 C with jReðmÞj\1 and m 6 2 È À 1 2 ; 0; 1 2 É , and for any j 2 C [ f1g: The second family corresponds to m ¼ 0: Finally, in the special case m ¼ 1 2 we have the third family: Þ j for some c 2 C; n 2 ðxÞ with 1 À bx lnðxÞ: Note that this can be seen directly, without passing through Bessel functions. We describe this approach below, and refer to [5] for the general theory.
The idea is to look for elements of DðL max b;m 2 Þ with a nontrivial behavior near 0. First we consider the general case and observe that It remains to find a second element of DðL max b;m 2 Þ when m ¼ 0 or when m ¼ 1 2 (as already mentioned we disregard m ¼ À 1 2 ). Firstly, we try to find the simplest possible elements of DðL max b;0 Þ with a logarithmic behavior near 0. We add more and more terms: Þ with a logarithmic behavior near 0: and H m b;0 cover all possible well-posed extensions of L b;m 2 with jReðmÞj\1. As already mentioned, we do not introduce a special family for m ¼ À 1 2 , since it is covered by the family corresponding to m ¼ 1 2 . For convenience, we also extend the definition of the first family to the exceptional cases by setting for b 2 C and any j 2 C [ f1g An invariance property follows directly from the definition: Proposition 2.3 For any b 2 C, jReðmÞj\1 and j 2 C [ f1g the following relation holds It is also convenient to introduce another two-parameter family of operators, which cover only special boundary conditions, which we call pure: ð2:11Þ With this notation, for any b 2 C, one has Remark 2.4 The family H b;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 ðb; mÞ ¼ À 0; À 1 2 Á was left undefined in [7]. This point corresponds to a singularity, nevertheless in the current paper we have decided to set H 0;À 1 2 :¼ H 0; 1 2 . Here is a comparison of the above families with the families H m;j , H m 0 introduced in [6] when b ¼ 0. In the first column we put one of the newly introduced family, in the second column we put the families from [6,7].
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 þ is given by while the Neumann Laplacian is given by ;j¼1 : Note that the former operator was also described in [6] by H m¼ 1 2 while the latter operator was described by H m¼À 1 2 . We now gather some easy properties of the operators H b;m;j .
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 2 DðL max b;m 2 Þ and g 2 DðL max b; m 2 Þ, the functions f ; f 0 ; g; g 0 are continuous on R þ . In addition, the Wronskian of f and g, as introduced in (2.1), possesses a limit at zero, and we have the equality In particular, if f 2 DðH b;m;j Þ one infers that j . The property for the transpose of H b;m;j can be proved similarly. h 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 b;m;j is self-adjoint if and only if one of the following sets of conditions is satisfied: Let us finally mention some equalities about the action of the dilation group. For that purpose, we recall that the unitary group fU s g s2R of dilations acts on f 2 L 2 ðR þ Þ as À U s f Á ðxÞ ¼ e s=2 f ðe s xÞ. The proof of the following lemma consists in an easy computation.
with the conventions a 1 ¼ 1 for any a 2 C n f0g and 1 þ s ¼ 1.

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 , or H m b;0 . In the following statement, C stands for the usual gamma function, w is the digamma function defined by wðzÞ ¼ C 0 ðzÞ=CðzÞ and c ¼ Àwð1Þ. Since the special case b ¼ 0 has already been considered in [6], we assume that b 6 ¼ 0 in the following statement, and recall in Theorem 3.4 the results obtained for b ¼ 0. It is also useful to note that the condition b 6 2 ½0; 1½ guarantees that either Þ [ 0, due to our definition of the square root.
Then the operator H b;m;j possesses an eigenvalue k 2 C in the following cases: CðÀ2mÞ ðÀbÞ 2m : possesses an eigenvalue k in the following cases: Then H m b;0 possesses an eigenvalue k in the following cases: (ii) k ¼ l 2 , 0\l\ AE ImðbÞ, and Proof We start with the special case k ¼ Àk with H AE 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 Àp\ argðzÞ p, one has as z ! 0 one infers that at most one of these functions is in L 2 near infinity, depending on the sign of ImðbÞ. More precisely, for ImðbÞ [ 0, the map x7 !H þ b 2l ;m ð2lxÞ belongs to L 2 near infinity if l\ImðbÞ and does not belong to L 2 near infinity otherwise. Under the same condition ImðbÞ [ 0, the map x7 !H À b 2l ;m ð2lxÞ never belongs to L 2 near infinity. Conversely, for ImðbÞ\0, the map x7 !H À b 2l ;m ð2lxÞ belongs to L 2 near infinity if l\ À ImðbÞ and does not belong to L 2 near infinity otherwise. Under the same condition ImðbÞ\0, the map x7 !H þ b 2l ;m ð2lxÞ never belongs to L 2 near infinity. Finally, for ImðbÞ ¼ 0, none of these functions belongs to L 2 near infinity.
For the asymptotic expansion near 0, the information on H AE d;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 d;m near 0 provided in (A.3) and the equality CðaÞCð1 À aÞ ¼ p sinðpaÞ one infers that for jReðmÞj\1 and m 6 2 For 2m 2 Z one has to consider the expression for K d; 1 2 and K d;0 provided in (A.18) and (A.19) respectively. Then, by considering the Taylor expansion near 0 of these functions one gets ð3:5Þ ð3:6Þ From Equation (A.29) one finally deduces the relations ;m ð2lxÞ if ImðbÞ\0, and check for which j these functions belong to DðH b;mj Þ. For jReðmÞj\1 and m 6 2 1 2 þm and Note that the conditions AEImðbÞ [ 0, jReðmÞj\1, and l\ AE ImðbÞ imply that AEi b 2l þ m À 1 2 6 2 N. The proof of 2.(ii) and 3.(ii) can be obtained similarly once the following expressions are taken into account: We shall now turn to the generic case (statements 1. We refer again to the appendix for an introduction to these functions. The behavior for large z of the function K d;m ðzÞ 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 j argðzÞj\ p 2 one has it follows that the remaining two functions in (3.7) do not belong to L 2 near infinity as long as b 2k Ç m À 1 2 6 2 N. Still in the non-degenerate case and when the condition b 2k þ m À 1 2 2 N holds, it follows from relation (A.8) that the functions K b 2k ;m ð2kÁÞ and I b 2k ;Àm ð2kÁÞ are linearly dependent, but still I b 2k ;m ð2kÁÞ does not belong to L 2 near infinity. Similarly, when b 2k À m À 1 2 2 N it is the function I b 2k ;Àm ð2kÁÞ which does not belong to L 2 near infinity.
Let us now turn to the degenerate case, when m 2 È À 1 2 ; 0; 1 2 É . In this situation the two functions I d;m and I d;Àm 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 , the above arguments can be mimicked, and one gets that only the function K b 2k ;m ð2kÁÞ belongs to L 2 near infinity. In the doubly degenerate case, the function X d;m , introduced in (A.9), has to be used. This function is independent of the function K d;m , as shown in (A.24). However, this function explodes exponentially near infinity, which means that X b 2k ;m ð2kÁÞ does not belong to L 2 near infinity. Once again, only the function K b 2k ;m ð2kÁÞ plays a role. As a consequence of these observations, it will be sufficient to concentrate on the function Similarly, it follows from (A.18) and (A. 19) that ð3:9Þ The statements 1.(i), 2.(i), and 3.(i) follow then straightforwardly. h 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). 1½. Indeed, in Theorem 3.1 a necessary condition for the existence of strictly positive eigenvalues is that ImðbÞ 6 ¼ 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 b ¼ 0. we infer from the Legendre duplication formula Cð2zÞ; that ð2kÞ À2m Cð2mÞ CðÀ2mÞ CðÀmÞ : . Then we use the relations wð1 þ zÞ ¼ wðzÞ þ 1 z and wð1Þ ¼ Àc, and infer that As a consequence of the expressions provided in Theorem 3.1, the discreteness of the spectra of all operators can be inferred in C n ½0; 1½.

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 ACðR þ Þ denote the set of absolutely continuous functions from R þ to C, that is functions whose distributional derivative belongs to L 1 loc ðR þ Þ. Let also AC 1 ðR þ Þ be the set of functions from R þ to C whose distributional derivatives belong to ACðR þ Þ. If V 2 L 1 loc ðR þ Þ, it is not difficult to check that the operator Ào 2 x þ V can be interpreted as a linear map from AC 1 ðR þ Þ to L 1 loc ðR þ Þ. The maximal operator associated to Ào 2 x þ 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 ¼ ðL min Þ # . As before, we say that a function f : The following statement contains several results proved in [5].
Let k 2 C and suppose that uðk; ÁÞ; vðk; ÁÞ 2 & and assume that RðÀk 2 ; x; yÞ is the integral kernel of a bounded operator RðÀk 2 Þ. Then there exists a unique closed realization H of Ào 2 x þ V with the boundary condition at 0 given by uðk; ÁÞ and at 1 given by vðk; ÁÞ in the sense that Moreover Àk 2 belongs to the resolvent set of H and RðÀk 2 Þ ¼ ðH þ k 2 Þ À1 .
By using such a statement, it has been proved in [7] that, for k 2 C such that for 0\x\y; for 0\y\x: x; yÞ þ jc b;Àm ðkÞR b;Àm ðÀk 2 ; x; yÞ À Á 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 b;m;j ðÀk 2 ; x; yÞ defines a bounded operator on L 2 ðR þ Þ. This follows from (3.11), because all numerical factors are harmless and because by [7, Thm. 3.5] R b;m ðÀk 2 ; x; yÞ and R b;Àm ðÀk 2 ; x; yÞ are the kernels defining bounded operators.
Moreover, we can write Therefore, this function belongs to L 2 around 0 and satisfies the same boundary condition at 0 as j b;m; þ jj b;Àm . By Proposition 3.6, this proves (i) when j 6 ¼ 1.
Note that in the special case j ¼ 1, it is enough to observe that H b;m;1 ¼ H b;Àm;0 and to apply the previous result.
To prove (ii), consider first m 6 ¼ 1 and b 2k 6 2 N Â . 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 b 2k ;m ð2kÁÞ 2 L 2 ðR þ Þ and therefore this operator is also bounded. Next we write which belongs to L 2 around 0 and corresponds to the boundary condition defining The proof of (iii) is analogous. We use first (3.13) for the boundedness. Then we rewrite Green's function as We check that by (3.4) and (3.9), see also (A.19). h Strictly speaking, the formulas of Theorem 3.7 are not valid in doubly degenerate points, when the functions K b;m and I b;m are proportional to one another, and the operator H b;m has an eigenvalue. To obtain well defined formulas one needs to use the function X b;m defined in (A.9), 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 [19], 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. Let H be a complex Banach space. Let fHðzÞg z2H be a family of closed operators on H with nonempty resolvent set, where H is an open subset of C d . fHðzÞg z2H is called holomorphic on H if for any z 0 2 H, there exist k 2 C and a neighborhood H 0 & H of z 0 such that, for all z 2 H 0 , k belongs to the resolvent set of HðzÞ and the map H 0 3 z7 !ðHðzÞ À kÞ À1 2 BðHÞ is holomorphic on H 0 . Note that if H 0 3 z7 !ðHðzÞ À kÞ À1 2 BðHÞ is locally bounded on H 0 and if there exists a dense subset D & H such that, for all f ; g 2 D, the map H 0 3 z7 !ðf jðHðzÞ À kÞ À1 gÞ is holomorphic on H 0 , then H 0 3 z7 !ðHðzÞ À kÞ À1 2 BðHÞ is holomorphic on H 0 . Besides, by Hartog's theorem, z7 !ðf jðHðzÞ À kÞ À1 gÞ is holomorphic if and only if it is separately analytic in each variable.
This definition naturally generalizes to families of operators defined on ðC [ f1gÞ d instead of C d , recalling that a map u : C [ f1g ! C is called holomorphic in a neighborhood of 1 if the map w : C ! C, defined by wðzÞ ¼ /ð1=zÞ if z 6 ¼ 0 and wð0Þ ¼ /ð1Þ, is holomorphic in a neighborhood of 0.
Recall that the family H b;m has been defined on C Â fm 2 C j ReðmÞ [ À 1g in [7], see also (2.11). 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 (3.19), (3.20), and (3.21), 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 P :¼ fm 2 C j À1\ReðmÞ\1g: Recall that ðb; m; jÞ7 !fH b;m;j g has been defined on C Â P Â ðC [ f1gÞ. However, it is not holomorphic on this whole set: Proof This expression appears in the numerator of (3.10) and plays an important role in the expression (3.14) for the resolvent of H b;m;j .

<
: ð2kyÞ for 0\x\y; for 0\y\x: ( Note that this corresponds to the integral kernel of This shows that fH b;m;j g is holomorphic on U 0 . The argument easily adapts to the case m 0 ¼ 1 2 and b 0 6 ¼ 0. As before, if m 0 ¼ AE 1 2 , b 0 6 ¼ 0, and j 0 ¼ 1, the statement follows from the equalities À The second part of the statement (i) follows directly from [7,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). h The following statement shows that the domains of holomorphy obtained in Theorem 3.10 are maximal for m 2 P. In particular, we will prove that (3.17) are sets of non-removable singularities of the family ðb; m; jÞ7 !fH b;m;j g.
Proof (i) Let us first consider b ¼ 0. Recall that in [6] the family of closed operators P Â ðC [ f1gÞ 3 ðm; jÞ7 !H m;j has been introduced, and that this family is holomorphic on P Â ðC [ f1gÞ n f0g Â ðC [ f1gÞ. Here is its relationship to the families from the present article: Let us now focus on m ¼ À 1 2 and on m ¼ 1 2 . We have for any j 2 C [ f1g Therefore, for j 6 ¼ 0, Similarly, for j 6 ¼ 1, This proves (i) when j 6 2 f0; 1g. The proof in these special cases is similar.
(ii) Let us first consider a fixed parameter b 2 C and m ¼ 0. By definition we have independently of j 2 C [ f1g. We now consider a fixed parameter b 2 C and j ¼ À1. Choosing k 2 C with ReðkÞ [ 0 such that b 2k À 1 2 6 2 N, it follows from (3.14) that for any m 6 ¼ 0 in a complex neighborhood of 0, the integral kernel of the resolvent of H b;m;À1 is given by By using this expression, one can verify that the map m7 !g b;k;x ðmÞ, defined in a punctured complex neighborhood of 0, can be analytically extended at 0 with g b;k;x ð0Þ ¼ À ð2kÞ Thus, the family of operators fH b;m;À1 g defined bỹ ( is holomorphic for m 2 P. It thus follows that lim j!À1 which concludes the proof.
Let us show that, for fixed ðb; mÞ such that b 2k À 1 2 6 2 N, the map is holomorphic for m near 0. It is clearly holomorphic in a punctured neighborhood of 0. Hence it suffices to show that it is continuous at m ¼ 0. Recall from (3.10) that : ð3:25Þ Then, by inserting j ¼ j ð0Þ ðm; mÞ for m 6 ¼ 0 into (3.25) we obtain Similarly, let us show that, for fixed ðb; mÞ such that b 2k 6 2 N, the map is holomorphic for m near 1 2 . By inserting j ¼ j ð 1 2 Þ ðb; m; mÞ for m 6 ¼ 1 2 into (3.25) we obtain which proves that (3.26) is holomorphic for m near 1 2 . The remaining restrictions on the domain of holomorphy are inferred directly from Theorem 3.10. h

Eigenprojections
Let us now describe a family of projections fP b;m ðkÞg which is closely related to the Whittaker operator. We will define it by specifying its integral kernel. We first introduce a holomorphic function for m 6 2 fÀ 1 2 ; 0; 1 2 g by One easily observes that f b;m ðkÞ ¼ f b;Àm ðkÞ. We can extend this function continuously to m 2 fÀ 1 2 ; 0; 1 2 g by : We now consider k 2 Cn½0; 1½, and as usual we write k ¼ Àk 2 with ReðkÞ [ 0. We then define the integral kernel P b;m ðk; x; yÞ: The definition (3.27) naturally extends to k 20; 1½, where we distinguish between points coming from the upper and lower half-plane by writing k AE i0 ¼ ÀðÇilÞ 2 with l [ 0. Thus, let us set k ¼ Çil and which can be naturally extended to m 2 fÀ 1 2 ; 0; 1 2 g by : For k ¼ Çil we can then rewrite (3.27) as 2l ;m ð2lyÞ: Finally, to handle k ¼ 0 we shall use the function sinð2pmÞ mð4m 2 À1Þ extended to fÀ 1 2 ; 0; 1 2 g by sinð2pmÞ mð4m 2 À 1Þ m¼0 ¼ À2p and sinð2pmÞ mð4m 2 À 1Þ The integral kernel P b;m ðÀk 2 ; x; yÞ defines an operator-valued map ðb; m; kÞ7 ! P b;m ðÀk 2 Þ described in the following proposition. are selftransposed. Moreover, it follows from Theorem 3.1 and its proof that all eigenvalues of these operators are simple. If k is a simple eigenvalue of a self-transposed operator H associated to an eigenvector u such that hujui ¼ 1, we define the selftransposed eigenprojection associated to k as P ¼ hujÁiu: In the case where k 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 k. . The other cases are similar. From the proof of Theorem 3.1, we know that if k is an eigenvalue of H b;m;j , then a corresponding eigenstate is given by x7 !K b 2k ;m ð2kxÞ. Corollary A.3 shows that This proves that P b;m ðÀk 2 Þ is the self-transposed eigenprojection corresponding to k, as claimed. h The point k ¼ 0 is rather special for the family P b;m ðÀk 2 Þ, as shown in next proposition.
Then the map k7 !P b;m ðÀk 2 Þ is not continuous at k ¼ 0.
Proof We consider the case where m 6 2 fÀ 1 2 ; 0; 1 2 g. The other cases are similar. First, we claim that for all continuous and compactly supported function f, where k 2 C is chosen such that ReðkÞ [ 0 and AE À argðbÞ À argðkÞ Á 2e; p À e½ with e [ 0. To shorten the expressions below, we set in this proof g b;m;k ðxÞ :¼ Çi We show that g b;m;k is uniformly bounded, for k satisfying the conditions above, by a locally integrable function. From the definition (A.3) of I b;m and proceeding as in the proof of Proposition B.2, we obtain that, for k 2 C such that ReðkÞ [ 0, jkj\1, and AE À argðbÞ À argðkÞ Á 2e; p À e½ with e [ 0, b 2k for some constant c [ 0 depending on b and m but independent of k and x. Using that 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.7 in the third equality. Now, we claim that P b;m ðÀk 2 Þ is not continuous at k ¼ 0 for the strong operator topology. Indeed, using that P b;m ðÀk 2 Þ is a self-transposed projection, we infer that, for f continuous and compactly supported, A similar computation as above gives Therefore, for suitably chosen compactly supported functions f. This proves that P b;m ðÀk 2 Þ is not continuous at k ¼ 0. h Acknowledgements S. R. was supported by the grant Topological invariants through scattering theory and noncommutative geometry from Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.

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 Àm. It has also another symmetry: Solutions of (A.1) are provided by the functions z7 !I b;AEm ðzÞ which are defined by

Á
[ 0 the function I b;m has also an integral representation given by Another solution of (A.1) is provided by the function z7 !K b;m ðzÞ. For m 6 2 1 2 Z it can be defined by the following relation: For the remaining m we can extend the definition of K b;m by continuity, see Sect. A.3. Note that K b;Àm ¼ K b;m , and that the function K b;m can also be expressed in terms of the function 2 F 0 , namely: An alternative definition of K b;m can be provided by an integral representation valid for Re [ 0 and ReðzÞ [ 0: e Àzs s À 1 2 ÀbÇm ð1 þ sÞ À 1 2 þbÇm ds: Note that the function K b;m decays exponentially for large ReðzÞ, more precisely, if e [ 0 and argðzÞ j j\ 3 2 p À e, then one has By using the relation (A.6) one also obtains that We would like to treat I b;m , I b;Àm and K b;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 b;m . To the best of our knowledge, this function has never appeared elsewhere in the literature. The function K b;m is distinguished by the fact that it decays exponentially, while the solutions I b;AEm ðzÞ explode exponentially, see [7,Eq. (2.14) and (2.22)]. This is also the case for the analytic continuations of K Àb;m by the angles AEp, 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 b;m ðzÞ :¼ 1 2 e Àipð 1 2 þmÞ K Àb;m À e ip z Á þ e ipð 1 2 þmÞ K Àb;m À e Àip z Á : ðA:9Þ As a consequence of this definition and of (A.5) one gets the relations X b;m ðzÞ ¼ À p sinð2pmÞ ðA:10Þ and e Çipð 1 2 þmÞ K Àb;m À e AEip z Á In addition, by using the equalities one infers from (A.6) and (A.10) that which finally leads to the relation By taking formulas (A.6), (A.10), and (A.11) into account, one infers that the Wronskian is provided by Hence for m þ b 2 Z the solutions K b;m and X b;m are proportional to one another. In fact, for such b; m, we have Note that this corresponds to the lines m þ b ¼ n 2 Z. However in our applications, we need X b;m on the lines m þ b À 1 2 ¼ n 2 Z, where K b;m and X b;m are linearly independent.

A.2 The Laguerre cases
Let us now consider two special cases, namely when À 1 2 À m þ b :¼ n 2 N and when À 1 2 À m À b :¼ n 2 N. In the former case, observe that the Wronskian of I b;m and K b;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 2 C and n 2 N, the Laguerre polynomials by the formulas L ðpÞ n ðzÞ ¼ In the special case À 1 2 À m À b :¼ n 2 N a similar analysis with p ¼ 2m leads to

A.3 The degenerate case
In this section we consider the special case m 2 1 2 Z, which will be called the degenerate case, see Fig. 1. In this situation the Wronskian of I b;m and I b;Àm vanishes, see (A.4). More precisely, for any p 2 N one has the identity or equivalently, Based on this equality and by a limiting procedure, one can provide an expression for the functions K b; p 2 (see [7, Thm. 2.2]), namely where w is the digamma function defined by wðzÞ ¼ C 0 ðzÞ CðzÞ . Note that the equality (or definition) ðaÞ j ¼ CðaþjÞ CðaÞ has also been used for arbitrary j 2 Z. For our applications the most important functions correspond to m ¼ 1 2 and m ¼ 0: ðA:18Þ ðA:19Þ Let us still provide the expression for the function X b; 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 Á Àj ðj À 1Þ!z Àj ðp À jÞ! : In particular, the expansions for m ¼ 1 2 and m ¼ 0 will be useful: ðA:21Þ 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 2 Z, b 2 Z þ 1 2 , or m 2 Z þ 1 2 , b 2 Z. This situation will be called the doubly degenerate case. We will again set m ¼ p 2 with p 2 Z. Note that for ðm; bÞ in (A.22) we have the identity which is a special case of (A.12). In this case we also have WðK b;m ; X b;m ; xÞ ¼ ðÀ1Þ mþbþ 1 2 : ðA:24Þ Hence K b;m and X b;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 Fig. 2). Region which follows for example from (A.23). By setting n 1 :¼ b À m À 1 2 2 N and n 2 ¼ Àb À m À 1 2 2 N, then K b;m ¼ K1þp Region I þ . b þ m 2 N þ 1 2 ; Àb þ m 2 N þ 1 2 : First note that ðm; bÞ 2 I À if and only if ðÀm; bÞ 2 I þ . By setting n 1 :¼ b þ m À 1 2 2 N and n 2 :¼ Àb þ m À 1 2 2 N, one has b ¼ n 1 Àn 2 2 , m ¼ n 1 þn 2 þ1 2 , and the equality (A.23) can be rewritten as Thus I b;m is proportional to X b;m and corresponds to the exploding Laguerre case. The second solution is K b;m . It decays exponentially and has a logarithmic singularity at zero, therefore we call this function the decaying logarithmic solution. Thus I b;m is proportional to K b;m and corresponds to the decaying Laguerre case. The second solution is X b;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 Fig. 2.
Lemma A.1 For i 2 f1; 2g, suppose that v i 2 DðL max b;a Þ satisfies L b;a v i ¼ k i v i for some k i 2 C. Then, for all a; b 20; 1½, where W is the Wronskian introduced in (2.1).

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  Proposition A.4 Let l; g [ 0 with l\ AE Im À bÞ and g\ AE Im À bÞ.

B.1 The modified Bessel equation
The modified (or hyperbolic type) Bessel equation for dimension 1 is up to a trivial rescaling, a special case of the Whittaker equation with b ¼ 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 be For the Wronskian we have WðI m ; I Àm ; zÞ ¼ À sinðpmÞ: The function K m can be introduced for m 6 2 Z by K m ðzÞ ¼ 1 sinðpmÞ À À I m ðzÞ þ I Àm ðzÞ Á : For m 2 Z the definition is extended by continuity. Note that the relation K m ðzÞ ¼ K Àm ðzÞ holds, and that WðK m ; I m ; zÞ ¼ 1: 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 ðzÞ :¼ 1 2 e Àipð 1 2 þmÞ K m À e ip z Á þ e ipð 1 2 þmÞ K m À e Àip z Á : Then the following relations hold: Àm È ðm À 1Þða 2mþ2 À b 2mþ2 Þ þ ðm þ 1Þa 2 b 2 ðb 2mÀ2 À a 2mÀ2 Þ É sinðpmÞðb 2 À a 2 Þ 3 :

ðB:6Þ
In addition, the following limiting cases hold: lnðbÞ À lnðaÞ Á ; Proof Assume first that À1\ReðmÞ\0. By using twice the recurrence relations of Sect. B.2 one gets Then, we infer that where we have used (B.5) with m þ 1 instead of m, and the fact that o a ða À 1 2 Àm Þ ¼ À mþ 1 2 a a À 1 2 Àm . Clearly, a similar relation holds for a replaced by b. By computing the derivatives, one gets the expressions provided in the statement. This proves (B.6) for À1\ReðmÞ\0. We then extend the equality to jReðmÞj\1 by analytic continuation. Finally, the limiting cases are obtained by taking the limit m ! 0 in the first case, the limit b ! a in the second case, and from this result the limit m ! 0. Note that the same result is obtained if we take the limits in the reverse order. h

B.4 The degenerate case
For m 2 Z the following relation holds: I Àm ðzÞ ¼ I m ðzÞ: Note that unlike for the Whittaker equation, in both regions I À and I þ the functions I m , I Àm 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 2 N and m ¼ AEð 1 2 þ nÞ one has K AEð 1 2 þnÞ ðzÞ ¼ ðÀ1Þ n n!ð2zÞ Àn e Àz L ðÀ1À2nÞ