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Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII

In this paper the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of non-constant curvature: these spaces are Darboux spaces D_I and D_II, respectively. On D_I there are three and on D_II four such potentials, respectively. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D_I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge.


General Overview and Recent Work
In the last years an enormous amount of work has been archived in solving path integrals in quantum mechanics exactly, and to the application of the path integral method in various branches of mathematical physics; many of this has been compiled in our publication [22]. In [13] one of us have discussed path integral representations of the free motion in two and three dimensions for Euclidean space, Pseudo-Euclidean space, spheres and hyperboloids. In these studies it was the goal to find all path integral representations for the coordinate systems [24,44]- [47] in which the Schrödinger equation respectively the path integral allows separation of variables. [22] was devoted to give a best to our knowledge list of up-to-date explictly known path integral solutions.
In the present work we extend our studies of superintegrable potentials to spaces of nonconstant curvature, i.e. Darboux spaces, by means of the path integral method. In the following sections we discuss two Darboux spaces: we set up the Lagrangian, the Hamiltonian, the quantum operator, and formulate and solve (if this is possible) the corresponding path integral. We also discuss some of the limiting cases of the Darboux-spaces, i.e. where we obtain a space of constant (zero or negative) curvature. In the case of D I there is no limiting case, because we have no free parameter in the metric to choose from.
In a recent publication one of us [15] has applied the path integral technique [7,22,38,49] to the quantum motion on two-dimensional spaces of non-constant curvature, called Darboux spaces, D I -D IV , respectively. These spaces have been introduced by Kalnins et al. [27,28]. They can be embedded in three-dimensional spaces which can be either of Euclidean or Minkowskian type, respectively. Then the Darboux spaces consist of surfaces, which are also called surfaces of revolution [4]. In two dimensions Darboux spaces of non-constant curvature can be constructed as follows. One takes for instance two-dimensional Euclidean space and takes for the metric a superintegrable potential in its simplest form in radial coordinates. For the Coulomb potential 1/r one obtains a metric ∝ r, which gives the Darboux space D I , for the radial potential b − a/r 2 one obtains a metric ∝ (b − a/r 2 ), i.e. the Darboux space D II , etc. The case of two-dimensions is especially simple, because one obtains always a conformally flat space. This method to construct new spaces was first discussed by Koenigs [40].

Superintegrable Potentials
The intention of [27,28] was however, not only to construct new spaces, and to study their properties, but another equally important motivation was to find the corresponding superintegrable potentials. The notion of superintegrable systems was introduced by Winternitz and co-workers in [9,52], Wojciechowski [53], and was developed further later on also by Evans [6]. Superintegrable potentials have the property that one finds additional constants of motion: The simples case of the case of the only conserved quantity is the energy gives usually a chaotic system [22]; in order that a physical system is just integrable requires d constants of motion, where 2d denotes the number of degrees of freedom. In two dimensions one obtains in total three functional independent constants of motion and in three dimensions one has four (minimal superintegrable) and five (maximal superintegrable) functional independent constants of motion. Well-known examples are the Coulomb potential with its Lenz-Runge vector and the harmonic oscillator with its quadrupole moment.
Moreover, the existence of an additional conserved quantity in (maximally) superintegrable potentials leads in classical mechanics to the fact that the orbits of a particle in such a potential are closed: Kepler ellipses are stable and do not "rotate". In quantum mechanics it follows that the spectrum is usually degenerate. A perturbation of the pure Newtonian potential causes the Kepler ellipses to rotate (Mercury's or the Moon's perihelion rotation), and in quantum mechanics degeneration is lost, respectively.
Another feature of superintegrable potentials is that the corresponding equations in classical and quantum mechanics separate in more than one coordinate system. (However, whereas from the separability in more that one coordinate system the superintegrability and the existence of additional constants of motion follows, a system with additional constants of motion may not be easily separable.) It turns out that the Coulomb potential in three dimensions separates in spherical, conical, parabolic and prolate-spheroidal coordinates [42]. Even the relativistic Dirac-Coulomb possesses some of this symmetry by the conservation of the Johnson-Lippmann operator which reduces in the non-relativistic limit to the Lenz-Runge vector [37].
In previous publications [18]- [21] we have studies superintegrable potentials in two and three dimensions in Euclidean space, on spheres and on hyperboloids. We restricted ourselves to real spaces and omitted their corresponding complex extensions [25,31,33,34]. Let us also note that by integrating out ignorable coordinates (i.e. variables which have plane waves, respectively circular waves as solutions of the Schrödinger equation) one can obtain from a higher dimensional more complicated space interacting systems on spaces with constant curvature: the interaction has the form of a superintegrable potential. One example is the hermitian hyperbolic space [3,14] where one can find superintegrable potentials on the hyperboloid [29]. The connection with superintegrability and the polynomial solutions was studied e.g. in [30], in the connections with contractions of Lie algebras e.g. in [23,32,48], where the various limiting cases from spaces of positive or negative constant curvature to zero curvature was investigated.
In this first paper on super integrable potentials on Darboux spaces we discuss only the Darboux spaces D I and D II . The superintegrable potentials on the other two Darboux spaces D III and D IV will be discussed in a forthcoming publication.
The paper is organized as follows: In the next Section we treat the superintegrable potentials on Darboux space D I . There are three of them, the third consisting of a constant divided by the metric term which makes the potential almost trivial. The common features of the first two potentials is that the energy eigenvalues are determined by a transcendental equation involving parabolic cylinder functions. For the third (trivial) potential no bound states can be found.
In the third section the superintegrable potentials on D II are discussed. There are three non-trivial potentials and one trivial. For the first potential we obtain a quadratic equation for the energy-levels, and they show an oscillator-like behavior. An exact solution can be found only in the (u, v)-system. This is very similar to the Holt potential in two-dimensional Euclidean space.
The second superintegrable potential on D II is exactly solvable in two coordinate systems. Here, we also find a quadratic equation for the energy levels. V 2 is similar to the singular oscillator in two-dimensional Euclidean space.
The third superintegrable potential has a relation to the Coulomb potential in two-dimensional Euclidean space. The energy levels are determined by an equation of eight order in E which cannot be solved in general. For a special case, however, we find a Coulomb-like behavior of the energy-levels.
The fourth potential is a constant times the metric term, and is therefore trivial. As for D I this potential is included for completeness.
The fourth Section contains a discussion of our results and an outlook for the remaining two Darboux spaces D III and D IV .

Introducing Darboux Spaces
Kalnins et al. [27,28] denoted four types of two-dimensional spaces of non-constant curvature, labeled by D I -D IV , which are called Darboux spaces [40]. In terms of the infinitesimal distance they are described by (the coordinates (u, v) will be called the (u, v)-system; the (x, y)-system in turn can be called light-cone coordinates): a and b are additional (real) parameters (a ± = (a ± 2b)/4). Kalnins et al. [27,28] studied not only the solution of the free motion, but also emphasized on the superintegrable systems in theses spaces. They found appropriate coordinate systems, and we will consider all of them. In the majority of the cases we will be able to find a solution, however in some cases this will not be possible due to a quartic anharmonicity of the problem in question.

Superintegrable Potentials on Darboux Space D I
We start with Darboux Space D I and consider the following coordinate systems The infinitesimal distance, i.e., the metric is given by (Rotated (r, q)-Coordinates:) = 2(r cos ϑ + q sin ϑ)(dr 2 + dq 2 ) , (2.5) (Displaced parabolic:) = (ξ 2 − η 2 + 2c)(ξ 2 + η 2 )(dξ 2 + dη 2 ) . (2.6) The Gaussian curvature in a space with metric ds 2 = g(u, v)(du 2 + dv 2 ) is given by (g = det g(u, v)) Equation (2.7) will be used to discuss shortly the curvature properties of the Darboux spaces, including their limiting cases of constant curvature. We find e.g. in the (u, v)-system for the Gaussian curvature There is no further parameter in the metric, therefore this space is of non-constant curvature throughout for all u > a with a some real constant a > 0. However, D I can be embedded in a three-dimensional Euclidean space. It can then be visualized as an infinite surface (similar to one sheet of a double-sheeted hyberboloid) with a circular hole at the bottom 1 . The constant a maybe taken as a = 1 2 . In order to set up the path integral formulation we follow our canonical procedure as presented in [22]. The free Lagrangian and Hamiltonian are given by, respectively: and we must require u > a for some a > 0, and ϕ ∈ [0, 2π] can be considered as a cyclic variable [28]. The canonical momenta are and for the quantum Hamiltonian we find We formulate the path integral (ignoring the half-space constraint for the time being): We have displayed the path integral in our product-lattice definition, which will be used throughout this paper [22]. Due to this lattice definition of the path integral, we have no additional 2 -potential because the dimension of the space of non-constant curvature equals 2, c.f. [22].
According to [27,28] we introduce the following three integrals of motion in D I . They are (2.14) 1 E.Kalnins, private communication.
(2.15) (Let us note that byH 0 the classical Hamiltonian without the 1/2m-factor is meant. Keeping this factor is no problem, however, in the present form the algebra has a simpler showing). These operators satisfy the Poisson algebra relations The quantum analogues are given by where {·, ·} is the anti-commutator. These operators satisfy the commutation relations with the operator relation The operators K, X 1 , X 2 can be used to characterize the separating coordinate systems on D I , as indicated in Table 1.
Let us note again that we do omit here factors of i, and 1/2m for the sake of simplicity. H 0 therefore is the quantum Hamiltonian without the usual − 2 /2m. However, in the tables with the constants of motion, these factors are meant to be included. In the remaining Darboux spaces this notation as long as the algebra is concerned will be for the sake of simplicity in the same way.
For the operators which characterize separation of variables in the (r, q)-systems and parabolic coordinates, respectively, we introduce . (2.21) These two operators describe the general case. Special cases for Λ 1 are: • θ = π/4, we have Λ 1 = −X 1 (symmetric case), • θ = π/2, we have Λ 1 = K 2 , We now consider the following potentials on D I (following [28], an additional fourth potential is according to [4]): In Table 2 we have summarized some properties of three of these potentials. Actually, V 3 can be considered as a special case either of V 1 or V 2 , respectively. The fourth potential separates for instance in parabolic coordinates (c = 0), and then has the (complex) form However, this is not tractable and we will not discuss this potential any further.

2.1
The Superintegrable Potential V 1 on D I .
We start with the potential V 1 in D I . V 1 is separable in the (u, v)-system and in parabolic coordinates. However, only in the (u, v)-system a closed solution can be found. We state for V 1 in the respective coordinate systems (2.28) The separation procedure in the space-time transformation gives additional terms according to −E[(ξ 4 − η 4 ) + 2c(ξ 2 + η 2 )] in the respective Lagrangian. Although symmetric in ξ and η the involvement of quartic and sextic terms make any further evaluation impossible in parabolic coordinates.
The same observations are valid in the case of a Coulomb-like potential on D I , which can be put into the form (including already the proper energy term) which yields after a space-time transformation, with unshifted (c = 0) parabolic coordinates and is not tractable either. In particular, the metric term 2u spoils any further investigation. There exist some attempts in the literature to treat such potential systems, and these studies go with the name "quasi-exactly solvable potentials" in the sense of Turbiner [50] and Ushveridze [51]. In fact, sextic oscillators with a centrifugal barrier and quartic hyperbolic and trigonometric can be considered, and they are very similar in their structure as for instance in (2.28). One can find particular solutions, provided the parameters in the quasi-exactly solvable potentials fulfill special conditions. Furthermore, well-defined expressions for the wave-functions and for the energy-spectrum can indeed be found if only quadratic, sextic, and a particular centrifugal term are present. The wave-functions then have the form of Ψ(x) ∝ P (x 4 ) × e −αx 4 , with a polynomial P . However, quasi-exactly solvable potentials have the feature that only a finite number of bound states can be calculated (usually the ground state and some excited states). Another important observation is due to [35,41]: The authors found quasi-exactly solvable potentials that emerge from dimensional reduction from two-and three-dimensional complex homogeneous spaces. The sextic potential in the Hamiltonian (2.28) is exactly of that type. This observation now opens an interpretation of two-dimensional systems with higher anharmonic terms. Let us assume that we have a two-dimensional superintegrable potential system. This system has additional constants of motion, respectively observables, and there are in total three of them (including the energy). Let us assume further that we choose an example which is separable in at least two coordinate systems, say in Cartesian and parabolic coordinates (i.e., a system which is similar to the one described in (2.28) and we can omit the metric term for simplicity).
Writing down the Schrödinger equation of potentials like this, one obtains a coupled system of differential equations in ξ and η, respectively, which are functionally identical. Their difference is that they are defined on another domain in the complex plane [35]. If one looks now for bound state solutions, i.e., solutions which can be written in terms of polynomials and which are therefore square-integrable, one finds a quantization condition for the energy E. Because the potential is assumed to be separable in Cartesian coordinates we already know the energy levels, E n . The second separation constant λ of the system of coupled differential equations in ξ and η can then be expressed in terms of E n , i.e. λ n = f (E n ). The wave-functions of the bound state solutions are determined by three-term recursion relations, terminating to give polynomials. However, they cannot be solved to give explicit formulas for the polynomials. Now we can return to the quasi-exactly solvable potentials. We take one of the two coupled differential equations and rename the variable ξ → x ∈ IR, say. This one-dimensional quasi-exactly solvable potential "remembers" its origin from a two-dimensional superintegrable potential: The subset of wave-functions which can be explicitly found correspond to the case where one of the coupling constants corresponds in a simple way with the energy-levels of the superintegrable potential labeled by n, and the emerging energy-levels of the quasi-exactly solvable potential are determined by the separation constant λ n of the coupled system of differential equations. This feature is common to all quasi-exactly solvable potentials, and even more, one is able to construct quasi-exactly solvable potentials from superintegrable potentials in two, three, etc. dimensions. They are of power-like behavior, or powers of trigonometric, hyperbolic, and elliptic functions.
However, there does not exist a theory of the corresponding wave-functions, which are determined by terminating three-term recursion relations for the bound states and non-terminating three-term recursion relations for the scattering states. In comparison to the (confluent) hypergeometric functions little is known about expansion and addition theorems (with the exception of Mathieu and spheroidal wave-functions in flat space [43]). In some few cases, an interbasis expansion is known to switch from, say, Hermite polynomials to these new wave-functions [35]. Summarizing, we are not able to treat systems with the structure of (2.28), and similar with powers of trigonometric and hyperbolic functions any further.
We insert V 1 in (2.13) and obtain with the path integral K n (T ) given by with E n = ω(2n + λ + 1) and we have inserted the path integral solution for the radial harmonic oscillator (RHO) with parameter λ and the variable v > 0. If v is more restricted, say v is an angular variable, additional boundary conditions must be imposed. However, we continue with the case v > 0. The wave-functions for the radial harmonic oscillator The L (λ) n (z) are Laguerre polynomials [10]. In the next step we perform a space-time transformation in (2.32) by eliminating the term 2u in the metric. This gives in the usual way with the transformed path integral given by This path integral of a shifted harmonic oscillator with frequency 2ω can be solved. The corresponding Green function has the form (2.36) Here, the D ν (z) are parabolic cylinder functions [10] For the evaluation of the s ′′ -integration we use the involution formula to obtain

Solution without Boundary Condition
Let us first solve the potential problem V 1 on D I without any boundary condition on the variables. In this case the path integral in the variable u is just a path integral for a shifted harmonic oscillator with wave-functions given by Ψ The wave-functions for the harmonic oscillator (HO) are given by the well-known form in terms of Hermite-polynomials Evaluating the Green function G we obtain the solution: The spectrum is degenerate in n and l, as it is known for superintegrable potentials. However, this "solution" is seriously flawed. If we calculate the norm of the wave-functions, we see immediately that the norm is proportional to the energy E n , which in the negative-sign case is negative, and it follows that the Hilbert space is not properly defined. In the positive-sign case the norm would be positive, however, the corresponding configuration space cannot be extended to u → −∞, and this does not make sense either.

Solution with Boundary Condition
Due to the coordinate singularity for u = 0 we must impose some boundary condition. The simplest way to incorporate such a boundary condition is to require that the wave-functions vanish at u = 0, or generally the motion in the variable u takes place only in the half-space u > a. By exploiting the Dirichlet boundary-conditions [12] at u = a we therefore get . (

2.42)
This Green function cannot be evaluated further. However, we can determine bound states by the poles of (2.42) and obtain the quantization condition (2.44) According to [28] the asymptotic behavior of the energy-eigenvalues is in accordance with (2.41) for high-level states. The wave-functions can be obtained by taking the residuum of the curlybracket expression in (2.42). Our last quantization condition, however, rises a problem. It is not obvious for us how to determine the degeneracy of the energy-values which is usually typically for superintegrable systems. The solution (2.41) has this degeneracy but the boundary conditions are not fulfilled and the Hilbert space is not properly defined either. For the solution (2.44) it is just the other way round. In the original paper [28] this issue was not addressed any further.
We can see from the quantization condition (2.44) that for each value of the number n a set of energy levels E l,n follows, i.e. a set E l,0 , E l,1 , . . .. There is no possibility to find that a level from the set n = 0 is equal to one level of the set n = 1, for example E la,0 = E l b ,1 for some numbers l a , l b . Therefore we find that the degeneracy of the energy levels is lost. The usual lore in the study of superintegrable systems is that the statements that a potential is superintegrable and that the spectrum of such a potential is degenerate are equivalent. Indeed, from the Sturm-Liouville theory for differential equations, i.e. in our case the quantum Hamiltonian, it follows that degeneracy implies superintegrability, i.e. additional constants of motion. However, this statement is not valid the other way round, and the present examples of potentials on Darboux space D I serve as counter examples for such an attempt.
If we look at (2.42) we see that the "lost" degeneracy is due to the boundary condition for the Green function and the wave-functions, respectively, for some u > a > 0. For u = 0 the curvature of the space becomes infinite and a wave-function at the coordinate origin does not make sense. Depending whether the Darboux space D I is embedded in three-dimensional space with definite or indefinite metric further determines the parameter a, c.f. [28]. For a positive-definite metric, v is an angle with v ∈ [0, 2π), the two dimensional surface making up D I has a definite boundary and it follows a = 1/2. For a negative-definite metric the boundary turns out to be constraint by a = 0. In fact, it is not possible to extend the surface beyond u < 0 and all values from 0 to ∞ are definitely excluded. We will see that the same property holds for the potential V 2 .

The Superintegrable Potential
Next, we consider the potential V 2 on on D I . First, we state the potential in the separating coordinate systems. We have We proceed in a similar way as before and obtain where Ψ (HO) n are the wave-functions of a shifted harmonic oscillator withṽ = v + κ 2 /mω. Note that we have to require v ∈ IR, otherwise for v cyclic complicated boundary conditions have to imposed on the solution in v. The remaining path integral in the variable u has the form This gives in the usual way with the transformed path integral given by (ũ = u − 2E/mω 2 )

Solution without Boundary Condition
This is again a path integral for a shifted harmonic oscillator, and first we ignore the boundary condition for the wave-functions in the variable u for u = 0, say, we obtain the solution: This spectrum exhibits degeneracy, however the norm is again proportional to the energy, which is negative, and therefore the Hilbert space is not properly defined..

Solution with Boundary Condition
If we now take into account the boundary condition for some u = a such that the wave-function vanish for u = a, we obtain in a similar manner as in the previous subsection: with the Green function G n (E) given by Bound states can be determined by the quantization condition Again, degeneracy in the quantum numbers n and l is lost. According to [28] the asymptotic behavior of the energy-eigenvalues (2.57) is in accordance with (2.52). The wave-functions can be obtained by taking the residuum of the curly-bracket expression in (2.55).

Separation of
In order to set up the path integral formulation we follow our canonical procedure. The Lagrangian and Hamiltonian are given by, respectively: The canonical momenta are The quantum Hamiltonian has the form Using the representation (2.46) we write down the path integral for V 2 in the rotated (r, q)coordinate system, and obtain Performing a space-time transformation in the usual way gives with the transformed path integral K(s ′′ ) given by +2E(r cos ϑ + q sin ϑ) − κ 2 (−r sin ϑ + q cos ϑ) ds Here, we have inserted the path integral solution for the shifted harmonic oscillator in the variable q.

Solution with Boundary Condition
For the path integral for the shifted harmonic oscillator in the variable r we now take care that the variable u is defined only in the half-space u ≥ a. Setting for instance in the definition of the (r, q)-system ϑ = 0 yields r = u and q = v. For ϑ = π/2 the roles of r and q are reversed. In the view of the previous paragraph of V 2 in the (u, v)-system we impose of the Green function in r the boundary condition r ≥ a and obtain in this limiting case for the bound states the quantization condition D ν l,n 2mω a − 2E l,n mω 2 = 0 , (2.67) This is the result of (2.57). The quantization conditions of (2.57) and (2.68) are identical as it should be.

The Superintegrable Potential
Next, we consider the potential V 3 on on D I . First, we state the potential in the separating coordinate systems. We have This potential can be considered as a special case either of V 1 or V 2 , respectively. However, it has an additional conserved quantum number, i.e. K = p v . Therefore we will sketch only the solution in the (u, v)-system. Proceeding in the usual way, we obtain for the path integral (assuming v cyclic): We observe that the only effect is change in the quantum number l in comparison to the v 0 = 0 case. Using the solution of [15] we get for the corresponding Green function (2.74) withK ν (z) similarly, andl 2 = l 2 +v 2 0 . Due to the relation of the Airy-function [1, 10] K ±1/3 (ζ) = π 3/z Ai(z), z = (3ζ/2) 2/3 , and the observation that for E < 0 the argument of Ai(z) is always greater than zero, and there are no bound states. For E > 0 there is no real bound state solution, either. This concludes the discussion.
Elliptic-parabolic Elliptic 2d is the interfocal distance in the elliptic system. For convenience we also display in the following the special case of the parameters a = −1 and b = 1 [27]. The infinitesimal distance is given in these four cases (note that the metric gives us the additional requirement u > 0): (Elliptic:) = bd 2 cosh 2 ω cos 2 ϕ − a cosh 2 ω cos 2 ϕ (cosh 2 ω − cos 2 ϕ)(dω 2 + dϕ 2 ) , We can see that the case a = −1, b = 0 leads to the case of the Poincaré upper half-plane u > 0 endowed with the metric (3.5) [13], i.e. the two-dimensional hyperboloid Λ (2) in horicyclic coordinates. The parabolic case corresponds to the semi-circular-parabolic system and the elliptic case to the elliptic-parabolic system on the two-dimensional hyperboloid. On the other hand, the case a = 0, b = 1 just gives the usual two-dimensional Euclidean plane with its four coordinate system which allow separation of variables of the Laplace-Beltrami equation, i.e., the Cartesian, polar, parabolic, and elliptic system. Hence, the Darboux space II contains as special cases a space of constant zero curvature (Euclidean plane) and a space of constant negative curvature (the hyperbolic plane). This includes the emerging of coordinate systems in flat space from curved spaces.
We find for the Gaussian curvature in the (u, v)-system For b = 0 we have G = 1/a which is indeed a space of constant curvature, and the quantity a measures the curvature. In particular, for the unit-two-dimensional hyperboloid we have G = 1/a, with a = −1 as the special case of Λ (2) . In the following we will assume that a < 0 in order to assure the positive definiteness of the metric (1.2).
The following constants of motion are introduced on D II (without potential): They satisfy the Poisson algebra relations and the relation 14) The quantum analogues have the form (again with i, , 2m)

16)
and satisfy the operator relation ( H 0 the Hamiltonian operator, {, } the anti-commutator) 18) and the commutation relations We consider the following potentials on D II : In the Table 4 we have listed the properties of these potentials (the coordinate systems where an explicit path integral evaluation is possible are underlined).
Parabolic Elliptic

The Superintegrable Potential V 1 on D II
We state the potential V 1 in the respective coordinate systems: In flat space, the corresponding potential is known as the Holt-potential [18]. It consists of a radial harmonic oscillator in one variable (here in the variable u), and a harmonic oscillator plus a linear term in the second variable (here in the variable v). There is an analogue of this potential on the two-dimensional hyperboloid [20], which separates in horicyclic and semi-circular parabolic coordinates, the limiting cases of the (u, v)-system and the parabolic coordinates, respectively.

Separation of
We start with the (u, v)-coordinate system. We formulate the classical Lagrangian and Hamiltonian, respectively:

26)
H(u, p u , v, p v ) = 1 2m The canonical momenta are (3.28) The quantum Hamiltonian has the form Therefore the path integral for V 1 in the (u, v)-system has the following form and we have abbreviated f = u 2 /(bu 2 − a). First we separate the v-path integration according to withṽ = v + k 1 /4mω, which is the solution for the shifted harmonic oscillator. Writing for short the wave-functions of the shifted harmonic oscillator by Ψ (HO) n , we thus obtain: We obtain in the usual way by means of a space-time transformation with the transformed path integral given by Alternatively we have for the Green function (λ 2 = k 2 2 + 2maE/ 2 ) The W µ,ν (z) are Whittaker functions [10]. We can either evaluate the s ′′ -integration or analyze the poles of the Green function. The latter give the poles in terms of the poles of the Γ-function yielding the quantization condition for the bound states E ln : which is equivalent to Let us analyze this equation in more detail. We obtain similar equations for the other potentials, and the present case serves as a standard example for those which come later. Let us note that the specific form of the discrete spectrum and the corresponding wave-functions depend on the special choice of the parameters a and b and the special space of revolution one considers. For instance, the plus-respectively the minus-sign in the square-root expression below may be allowed giving positive normed states for some cases, and for others the minus sign may be allowed. Similarly, the radicand of the square-root can be become negative and we may obtain semi-bound states. The quadratic equation in E ln gives (ǫ ln = (2l + n + 3/2)) In the latter (special) case this gives bound states for m 2 ω 4 + 2 ω 2 2 k 2 2 − 2m ω 3 ǫ ln − k 2 1 /4 ≥ 0, i.e., the number of levels is determined by otherwise we may have semi-bound states, that is bound states with energy ℜ(E ln ) and with a decay width ℑ(E ln ). They are located in the continuous spectrum. In particular, we have a ground state Note that if the radicand of the square root equals the upper bound of the energy-levels for the case ab < 1 we get: The spectrum is similar to the spectrum of the Holt potential: Flat Euclidean space corresponds to a = 0, then (3.42) is identical with the result of [18]. Note that different energy spectra emerge depending on the signs of the parameters a and b. For both parameters positive the discrete spectrum cannot be simultaneously located in the continuous spectrum. For b negative, the properties of the space D II must be further analyzed, if a discrete spectrum with negative infinite values is allowed (which is the case for the singlesheeted hyperboloid).
In order to extract the continuous spectrum we consider the dispersion relation [11] I λ (z) = 2 π 2 ∞ 0 dp sinh πp This gives The continuous spectrum has the form 51) and the wave-functions are Note that for k 2 = ± 1 2 , i.e. the radial potential equals zero, we obtain the case from the free motion on D II .
Finally we state the kernel K (V 1 ) (T ) and the Green function G (V 1 ) (E) which have the form The normalization constant N ln emerges form evaluating the residuum of the Green function (3.39) at the energy E ln as given in (3.42).

Separation of V 1 in Parabolic Coordinates on D II
The classical Lagrangian and Hamiltonian are given by The canonical momenta are given by The quantum Hamiltonian has the form: We obtain for the path integral in parabolic coordinates, c.f. (3.25), (1/f (ξ, η) = (bξ 2 η 2 − a)/ξ 2 η 2 ): Performing the space-time transformation yields with the transformed path integral given by (3.63) These path integrals are due to the anharmonic terms in ξ and η not tractable, a well-known fact due to its relation to the Holt-potential.

The Superintegrable Potential
We consider the potential V 2 . The corresponding quantum mechanical problem is separable in the (u, v)-system, in polar and elliptic coordinates. First we state the potential V 2 in the respective coordinate systems: The potential V 2 can be interpreted as a two-dimensional oscillator with radial term similarly as its analogue in flat space. Note that a Higgs-like harmonic oscillator on D II could have a form according to (with the limiting case the Higgs-oscillator on the hyperboloid) with ̺ = e τ 2 , cos ϑ = 1/ cosh τ 1 , τ 1,2 being equidistant coordinates. The corresponding path integral cannot be solved, and V Higgs is not superintegrable in D II either.
We start with the consideration in the (u, v)-system, and the path integral has the form where E n = ω(2n + |k 2 | + 1). Performing a space-time transformation in the usual way yields: with the transformed path integral given by where λ 2 1 = k 2 1 + 2maE/ 2 . This path integral has almost the same form as the path integral (3.37), the only difference being another E n . Thus we can write the solution as follows: and alternatively we have for the Green function We can either evaluate the s ′′ -integration or analyze the poles of the Green function. The latter give the poles in terms of the poles of the Γ-function yielding the quantization condition for the bound states E ln : which is equivalent to The quadratic equation in E ln gives (ǫ ln = (2l + 2n + 2 + |k 2 |)) This gives for the special case bound states for m 2 ω 4 + 2 ω 2 k 2 1 − 2mω 2 ǫ ln ≥ 0, otherwise we can infer for semi-bound states, that is bound states with energy ℜ(E ln ) and with a decay width ℑ(E ln ). They are located in the continuous spectrum. Again, the limiting case of flat space emerges from a = 0, b = 1 E ln = ω(2l + 2n + |k 1 | + |k 2 | + 2) . (3.79) Finally we state the kernel K (V 2 ) (T ) and the Green function G (V 2 ) (E) which have the form The normalization constant N ln emerges form evaluating the residuum of the Green function (3.73) at the energy E ln as given in (3.76). We omit the continuous part of K (V 2 ) due to its similarity to the case of V 1 .

The Superintegrable Potential
We consider the potential V 3 and start by expressing V 3 in the respective coordinate systems. We have In the last case the rotated elliptic coordinates are given by Due to the complicated structure of the path integral in rotated elliptic coordinates no closed solution can be stated. We will omit a path integral discussion of V 3 in these coordinates. The potential V 3 can be interpreted as an analogue of the Coulomb potential. Similarly as in flat space and on the two-dimensional hyperboloid it is separable in three coordinate systems, i.e. in spherical, parabolic, and rotated elliptic coordinates (there are no conical coordinates in D II ).

Separation of V 3 in Polar Coordinates.
We start with the investigation of V 3 in polar coordinates and we immediately switch form the (ρ, ϑ)-system to the (r, ϕ)-system. This gives for the path integral: with 1/f = br 2 − a/r 2 sin 2 ϕ cos 2 ϕ. Proceeding in the usual way by means of a space time transformation gives and the path integral K (V 3 ) (s ′′ ) given by This is the usual radial harmonic oscillator solution, and we have set Λ = 2n + λ 1 + λ 2 + 1, ω 2 = −2Eb/m. The bound states are determined by the quantization condition 2 ω(l + n + 1) or alternatively For a = 0, b = 1 we recover the two-dimensional flat space Coulomb-spectrum. In general this is an equation of eighth order in E, where no closed solution can be stated. However, we can study the special case k 1 = k 2 = 0, which gives the quantization condition (we also take a < 0, b > 0) 2(l + n + 1) − α m 2b This is a quadratic equation in E with solution (only one solution is physical, set N = l + n + 1) showing a Coulomb-behavior. In order to extract the continuous spectrum we use (3.49) and obtain for the entire kernel with the discrete energy spectrum as determined by (3.115), and the normalization constant N ln resulting from the residuum in (3.113). The continuous spectrum is given by (k < denotes the smaller of k 1 , k 2 ) Finally we consider V 3 in parabolic coordinates. The formulation of the path integral for a potential on D II we know already from V 1 . We therefore have (f = b − a/ξ 2 η 2 ) with the transformed path integral K(s ′′ ) given by We have inserted for the discrete spectrum the solution of the radial harmonic oscillator in the usual way. Performing the s ′′ -integration gives the same spectrum as in (3.115), as it should be. The continuous spectrum is extracted in the usual way by means of (3.49) and we obtain: (p 1,2 ≡ b(p 2 + k 2 1,2 )/|a|), with the discrete energy spectrum as determined by (3.115), and the normalization constant N ln resulting from the residuum in (3.124).

The Superintegrable Potential
We consider the potential V 4 in the respective coordinate systems We have displayed the potential in a somewhat more complicated way to demonstrate the effect of the separation procedures. The quantity v 0 enters the formulas in a way that only the respective quantum numbers are altered. We will not go into the details, and consider the potential V 4 only in the (u, v)-system. For the remaining systems we refer to [15]. Let us note that the separability of V 4 in all the four coordinate systems on D II shows that a quantum system of a three-dimensional analogue of D II is also separable in three-dimensional "cylindrical" versions of the (u, v)-system, spherical, parabolic and elliptic coordinates [16]. The additional quantum number associated with the third coordinate can be identified with v 0 . Inserting V 4 into the path integral in the (u, v)-systems yields We observe that the principal effect of the introduction of V 4 consists in a change in the quantum number k which can be formulated ask 2 = k 2 + v 2 0 . We can therefore write down the solution by referring to [15] and get The wave functions and the energy spectrum are read off:

Summary and Discussion
In this paper we have discussed superintegrable potentials on spaces of non-constant curvature. The results are very satisfactory. According to [4,27,28] there are three potentials on D I , four potentials on D II , five potentials on D III , and four potentials on D IV , respectively. We could solve many of the emerging quantum mechanical problems. To give an overview, we summarize our results in Table 5. We list for each space the corresponding potentials including the general form of the solution (if explicitly possible). We omit the trivial potentials here, because they are separable in all corresponding coordinate systems. We were able to solve the various path integral representations, because we have now to our disposal not only the basic path integrals for the harmonic oscillator, the linear oscillator, the radial harmonic oscillator, and the Pöschl-Teller Potential, but also path integral identities derived from path integration on harmonic spaces like the elliptic and spheroidal path integral representations with its more complicated special functions [13,17,22]. This includes also numerous transformation techniques to find a particular solution based on one of the basic solutions. Various analysis techniques can be applied to find not only an expression for the Green function but also for the wave-functions and the energy spectrum.  Spheroidal wave-functions ( * : The notion Whittaker functions means in all cases for a discrete spectrum Laguerre polynomials, and for a continuous spectrum Whittaker functions W µ,ν (z), respectievly M µ,ν (z).) We also observe a new feature of superintegrable potentials. We learned from our investigation of potential problems on D I that degeneracy for superintegrable potentials does not follow automatically. In fact, our (counter-)examples show that the usually accepted opinion that superintegrability and degeneracy of a quantum system are equivalent statements is not true in general. It would be interesting to formulate the precise additional mathematical requirements that these statements are actually true in general. In our case the non-equivalence of these two notions comes from the boundary-conditions which had to imposed on D I in order to guarantee a well-defined Hilbert space.
We found in all cases a discrete and a continuous spectrum for the superintegrable potentials. We also could compare some limiting cases, e.g. for the Darboux space D II , where we could recover the corresponding solutions for the two-dimensional Euclidean space and the twodimensional hyperboloid. On D I the energy spectra are only determined by a transcendental equation due to the boundary condition for the coordinate u. On D II we found analogues of the singular oscillator, the Holt potential and the Coulomb potential in two dimensional Euclidean space. We could recover these limiting cases in the equations for the energy spectra. The equations equations for the energy spectra were on D II algebraic equations in second and fourth order in the energy. This allows several solutions depending on the specific values of the parameters a and b and possible further boundary conditions. Also semi-bound states may be possible.
In a forthcoming publications we will treat the two remaining Darboux spaces D III and D IV , respectively. In particular on D III there is already a discrete spectrum possible for the free motion, and has the form E nl = − 2 2m b a 2 (2n + 2l + 1) 2 . (4.1) yielding for b > 0 an infinite number of bound states. This is similar as the motion on the SU(1, 1) hyperboloid, where a continuous and a discrete spectrum exists [2]. On D III there are five superintegrable potentials and on D IV there are four superintegrable potentials. Let us finally discuss the following issue: Let us consider a three-dimensional generalization of the Darboux space D II with a line element and w is the new variable. D II has the property that for a = 0, b = 1, we recover the two-dimensional Euclidean plane, and all four coordinate systems on the two-dimensional Euclidean plane are also separable coordinate systems on D II for the Schrödinger, respectively the Helmholtz equation. However, in order to set up a well-defined quantum theory a curvature term ( 2 /2m) · (R/8) must be introduced in the quantum Hamiltonian [16,26]. In the present case of (4.2), which we might call three-dimensional Darboux space II, for short D 3d−II , it is easily checked that all eleven systems for the three-dimensional Euclidean plane which separate the Schrödinger, respectively the Helmholtz equation, also separate the Schrödinger, respectively the Helmholtz equation on D 3d−II . As it has been shown in [16], the corresponding quantum motion can be explicitly evaluated by means of the path integral with energy spectrum E = 2 2m|a| (p 2 + 1) .

(4.3)
As it is well known, there are four minimally superintegrable potentials in three-dimensional Euclidean space and five maximally superintegrable potentials, and it is obvious how to construct maximally superintegrable potentials on D 3d−II . In a forthcoming publication the details will be worked out.