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The Runge-Lenz vector for quantum Kepler problem in the space of positive constant curvature and complex parabolic coordinates

By analogy with the Lobachevsky space H_{3}, generalized parabolic coordinates (t_{1},t_{2},\phi) are introduced in Riemannian space model of positive constant curvature S_{3}. In this case parabolic coordinates turn out to be complex valued and obey additional restrictions involving the complex conjugation. In that complex coordinate system, the quantum-mechanical Coulomb problem is stu- died: separation of variables is carried out and the wave solutions in terms of hypergeometric functions are obtained. At separating the variables, two parameters k_{1} and k_{2} are introduced, and an operator B with the eigen values (k_{1}+k_{2}) is found, which is related to third component of the known Runge-Lenz vector in space S_{3} as follows: i B = A _{3} + i \vec{L}^{2}, whereas in the Lobachevsky space as B =A_{3} + \vec{L}^{2}. General aspects of the possibility to employ complex coordinate systems in the real space model S_{3} are discussed.


Introduction
In Euclidean 3-dimension space E 3 there exist 11 coordinate systems [1][2][3], allowing for the complete separation of variables in the Helmholtz equation [ 1 √ g ∂ ∂x α √ g g αβ ∂ ∂x α + const ] Φ(x 1 , x 2 , x 3 ) = 0 ; (1) g αβ (x) stands for the metric tensor of space E 3 specified for curvilinear coordinates (x 1 , x 2 , x 3 ). Solution of the same problem for spaces of constant positive and negative curvature, Riemannian S 3 and Lobachevsky H 3 models was given by Olevsky in [4], see also [5]. It was established that there exist 34 such coordinate systems for hyperbolic space H 3 , whereas in the case of spherical model S 3 the number of those systems is only 6. This result may seem rather unexpected and even intriguing by some reasons. Indeed, there must exist a limiting procedure from both curved space models H 3 and S 3 to flat space: and it is natural to expect the reduction 34 −→ 11, however one can hardly perform the reduction 6 −→ 11. The above asymmetry between H 3 and S 3 may be seen as even more strange if one calls the known relations of these models through the analytical continuation: where R stands for a curvature radius for S 3 and H 3 .
The asymmetry of the models H 3 and S 3 with respect to coordinate systems finds its logical corollary when turning to the study of the quantum mechanical model for a hydrogen atom on the background of a curved space. Firstly, such a model was considered in [6,7,8] where the wave function in spherical coordinates and energy spectrum were established. In particular, an additional degeneracy like in the case of flat space was observed, which presumes existence of a hidden symmetry in the (curved space) problem. In [9][10][11][12] the symmetry operators accounting for such additional degeneracy in Kepler problem on curved space background ware found for both model H 3 and S 3 , and an analog of the conventional Runge-Lenz vector in the flat space was constructed.
Connection between the Runge-Lenz operator A in the quantum Kepler problem and parabolic coordinates in Euclidean space is well known: by solving the Schrödinger equation in these coordinates the eigenfunctions of the third component A 3 arise [13]. Analogous situation exists in the hyperbolic space H 3 but not in in the spherical S 3 [14]. In the Lobachewsky space, among 34 coordinates established by Olevsky one may select one special case, parabolic system of coordinates in H 3 , in which the Schrödinger equation allows the separation of variables and the wave functions arisen turn out to be eigenfunctions of the operator B = A 3 + L 2 . Among six coordinate systems mentioned in [4] an analog of parabolic coordinates is not encountered.
If one looks at 34 and 6 systems in H 3 and S 3 respectively, one can note that all six ones from S 3 have their counterparts in H 3 . The main purpose of the present paper consists is the search of some counterparts of remaining 34 − 6 = 28 systems. It turns out that such 28 systems in S 3 can be constructed, but they should be complex-valued; to preserve real nature of the geometrical space one must impose additional restrictions including complex conjugation.
In particular, the complex analog for parabolic coordinates in space of the positive curvature S 3 can be introduced and used in studying the quantum mechanical Kepler problem in this space.
2 Complex parabolic coordinates in real space S 3 Let us start with the following fact: from the the metric in Lobachevsky space by means of the formal change χ → iχ , sinh χ −→ i sin χ one can obtain the corresponding metric of the Riemannian space This simple observation on H 3 − S 3 connection leads us to interesting consequences. Indeed, let us compare, for instance, wave functions and spectra for hydrogen atom in spaces of negative and constant curvature [14]: quantity (Mh 2 /R 2 provides us with natural unit for energy, e is a dimensionless parameter characterizing intensity of the Coulomb interaction. One may readily note that these two solutions turn into each other by the following formal substitutions: This example indicates that the relation between H 3 and S 3 reflected by substitution χ → iχ is meaningful. In the context of the described above situation with coordinate systems in H 3 and S 3 , let us make use of this correspondence (χ −→ i χ) as follows: Let in the Lobachevsky space H 3 be chosen a coordinate system (ρ 1 , ρ 2 , ρ 3 ) (one of those 34 found by Olevsky), then as a first step one has to establish connection of such a system with spherical one: and a second step is to introduce a corresponding coordinate system in the space S 3 through the formal change χ −→ i χ: With help of this prescription one can determine 34 coordinate systems in space S 3 in comparison to six ones given in [4]. It turns out that 28 new (added) coordinate systems are complex-valued and therefore additional restrictions should be imposed which involve complex conjugation. All these extra coordinate systems permit the full separation of variables in the Helmholtz equation on the sphere S 3 . Below, only one example of such coordinates, analog of the parabolic ones in space H 3 , will be examined in detail and applied to the study of the quantum-mechanical Kepler problem on the sphere S 3 .

Complex parabolic coordinates in S and the hydrogen atom
In [4] Olevsky had given the following coordinate system (the case XXV ) in Lobachevsky space: where (ρ 1 , ρ 2 , ρ 3 ) are connected with the four "Cartesian" (dimensionless) coordinates (x 0 , x 1 , x 2 , x 3 ) by the formulas With a = +1, b = 0 and notations x 2 Combining two last relations one obtains whence having in mind σ 2 + U V = −1, one gets and further As a result, for U, V, σ we arrive at (take notice that the Lobachevsky model is realized on the upper sheet of hyperboloid x 0 > +1, and therefore Thus, explicit formulas relating ρ 1 , ρ 2 , ρ 3 with Cartesian coordinates ( 0 , x l look as and the inverse formulas are Now, instead of the introduced ρ 1 , ρ 2 , ρ 3 one can define other coordinates which behave simply in the limit R → ∞ (the curvature vanishes). Such a limiting procedure for spherical coordinates of the hyperbolic space H 3 with metric going over into spherical ones of the flat space E 3 dl 2 = [ dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ) ] , looks as follows: Eliminating x 0 through q l : 1 + x 2 , q l = tanh χ n l , n l = (sin θ cos φ, sin θ sin φ, cos θ) , q = q 2 1 + q 2 2 + q 2 3 = tan χ ; we can readily see that when R → ∞ the coordinates q l will reduce to lim R→∞ (Rq l ) = lim R→∞ (R tanh χ n l ) = rn l .
So, to have coordinates with the known and understandable behavior in the limit R → ∞ we define coordinates t 1 , t 2 , φ: in the limit of the flat space they provide us with the known parabolic coordinates (ξ, η, φ) (see in [13]): The metric (12) in coordinates (t 1 , t 2 , φ) takes the form Now, with the help of the rules (10)-(11) one has to define corresponding parabolic coordinates t 1 , t 2 on the sphere S 3 . To this end, coordinates (t 1 , t 2 ) in H 3 must be expressed in term of spherical ones (χ, θ): From (21) we get defining relations for corresponding coordinates in S 3 Take special notice that (t 1 and t 2 ) in (22) are complex-valued expressed through two real (χ, θ). The inverse formulas are readily found: and therefore So defined parametrization of S 3 by coordinates t 1 , t 2 can be additionally detailed by the formulas: From (24) one can derive the relationship between t 1 and t 2 : its existence may evidently be referred to the real nature of the space S 3 . The values of θ = 0 and θ = π are peculiar: In the following, so defined coordinates (t 1 , t 2 , φ) are called parabolic coordinates on the sphere S 3 . In the limit of the flat space, they reduce to the ordinary parabolic coordinates [...] in accordance with Now, we transform the metric of the space S 3 in spherical coordinates to complex parabolic t 1 , t 2 , φ. As a first step, with the help of we get sin 2 χ sin 2 θ dφ 2 = t 1 t 2 dφ 2 .
As a second step, we have Finally, taking into account relations and we get Therefore, for the metric in parabolic coordinates in S 3 we have arrived at the form Solutions of eqs. (35) and (36) can be searched for with the help of substitution Below, all calculations will be done for the case (35); at any point, transition to H 3 space is accomplished by the formal changes ǫ → −ǫ, −ie → e. It suffices to consider in detail only the first equation for f 1 (t 1 ) (index 1 is omitted below) Both terms proportional to t −1 and (1 − t) −1 may be eliminated from the equation by adding the requirements: then eq. (38) results in That is, S(t) turns out to be a hypergeometric function S(t) = F (α, β, γ; t) whose parameters are determined by which implies Thus, the quantum mechanical Kepler problem in complex parabolic coordinates t 1 , t 2 , φ) has been solved in hypergeometric functions. The separation constants k 1 and k 2 are connected by the relation (34): Some additional study is required to obtain the physical wave solutions of the Schrödinger equation.

The Runge-Lenz vector in S 3 and complex parabolic coordinates
At separating the variables in Schrödinger equation two constants were introduced k 1 k 2 ; the problem is to find an operator that is diagonalized on wave functions (31) with eigenvalues (k 1 + k 2 ):B Taking into account (36), one can obtain for the operatorB the following representation: or after substituting the expression for H from (30) note the identity − ie t 1 + t 2 − 2t 1 t 2 t 1 − t 2 = −ie cos θ = −ie q 3 q .
In H 3 and S 3 the quantum mechanical Runge-Lenz operator is constructed in term of momentum and orbital momentum by the formula [11,12.] where P = (P i ), upper sign corresponds to the model H 3 , lower corresponds to S 3 model; operators and L P are measured in unitsh andh/ρ respectively. In correspondence with symmetry of space models, the components of P , L obey the commutation relations of Lie algebras so(3.1) and so(4): [L a , L b ] = i ǫ abc L c , [L a , P b ] = i ǫ abc P c , [P a , P b ] = ±i ǫ abc L c .
As in the above expression forB, specific term −ie q 3 /q is presented, (in the model H 3 we see the term e q 3 /q , it is natural to look for certain relationship betweenB and A 3 -they look as follows (all details are omitted here) in H 3 ,B = ( A + L 2 ) , in S 3 , iB = ( A + i L 2 ) .