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PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS ON SPACES OF NONCONSTANT CURVATURE: I. DARBOUX SPACES DI AND DII C. Grosche1 , G. S. Pogosyan2,3 , A. N. Sissakian2 1 II.

é Theoretische Physik, Universitéat Hamburg, Germany Institute fur 2 Joint Institute for Nuclear Research, Dubna 3 Departamento de Matematicas, CUCEI, Universidad de Guadalajara, Jalisco, Mexico In this paper the Feynman path-integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces DI and DII , respectively. On DI there are three and on DII four such potentials, respectively. We are able to evaluate the path integral in the 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 DI , in particular, show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of at space (constant zero curvature) and the two-dimensional hyperboloid (constant negative curvature) emerge. ‚ ¶·¥¤¸É ¢²¥´´μ° ¸É ÉÓ¥ Ë¥°´³ ´μ¢¸± Ö ɥ̴¨±  ¨´É¥£·¨·μ¢ ´¨Ö ¶μ ¶ÊÉÖ³ ¶·¨³¥´Ö¥É¸Ö ¤²Ö ¸Ê¶¥·¨´É¥£·¨·Ê¥³ÒÌ ¶μÉ¥´Í¨ ²μ¢ ´  ¤¢Ê³¥·´ÒÌ ¶·μ¸É· ´¸É¢ Ì ¶¥·¥³¥´´μ° ±·¨¢¨§´Ò: Ôɨ ¶·μ¸É· ´¸É¢  Ö¢²ÖÕÉ¸Ö ¶·μ¸É· ´¸É¢ ³¨ „ ·¡Ê DI ¨ DII ¸μμÉ¢¥É¸É¢¥´´μ.   DI ¸ÊÐ¥¸É¢ÊÕÉ É·¨,   ´  DII ŠΥÉÒ·¥ É ±¨Ì ¶μÉ¥´Í¨ ² . ŒÒ ³μ¦¥³ ¢ÒΨ¸²¨ÉÓ ¨´É¥£· ² ¶μ ¶ÊÉÖ³ ¢ ¸ ³μ° ¢Ò¤¥²¥´´μ° ¸¨¸É¥³¥ ±μμ·¤¨´ É, ¶·¨¢μ¤ÖÐ¥° ± ¢Ò· ¦¥´¨Ö³ ¤²Ö ËÊ´±Í¨° ƒ·¨´ , ¢μ²´μ¢Ò³ ËÊ´±Í¨Ö³ ¤¨¸±·¥É´μ£μ ¨ ´¥¶·¥·Ò¢´μ£μ ¸¶¥±É·μ¢,   É ±¦¥ ± ¤¨¸±·¥É´μ³Ê Ô´¥·£¥É¨Î¥¸±μ³Ê ¸¶¥±É·Ê. ‚ ´¥±μÉμ·ÒÌ ¸²ÊΠÖÌ, μ¤´ ±μ, ¤¨¸±·¥É´Ò° ¸¶¥±É· ´¥²Ó§Ö ¢ÒΨ¸²¨ÉÓ Ö¢´μ, É ± ± ± μ´ ¨²¨ μ¶·¥¤¥²Ö¥É¸Ö É· ´¸Í¥´¤¥´É´Ò³ Ê· ¢´¥´¨¥³, ¢±²ÕΠÕШ³ ¶ · ¡μ²¨Î¥¸±¨¥ ͨ²¨´¤·¨Î¥¸±¨¥ ËÊ´±Í¨¨ (¶·μ¸É· ´¸É¢μ „ ·¡Ê I), ¨²¨ ¶μ²¨´μ³¨ ²Ó´Ò³ Ê· ¢´¥´¨¥³ ¡μ²¥¥ ¢Ò¸μ±μ£μ ¶μ·Ö¤± . ¥Ï¥´¨Ö ´  DI , ¢ Π¸É´μ¸É¨, ¶μ± §Ò¢ ÕÉ, ± ± ¶μÖ¢²ÖÕÉ¸Ö £· ´¨Î´Ò¥ ¸²ÊΠ¨ ¶²μ¸±μ£μ ¶·μ¸É· ´¸É¢  (¶μ¸ÉμÖ´´ Ö ±·¨¢¨§´Ò · ¢´  ´Ê²Õ) ¨ ¤¢Ê³¥·´μ£μ £¨¶¥·¡μ²μ¨¤ .

PACS: 03.65.Db 1. INTRODUCTION General Overview and Recent Work. In the last years, an enormous amount of work has been archived on solving path integrals in quantum mechanics exactly, and on the application of the path-integral method in various branches of mathematical physics; many of them have been compiled in our publication [22]. In [13] one of us has discussed path-integral representations of the free motion in two and three dimensions for Euclidean space, pseudo-Euclidean space, spheres,

588 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

and hyperboloids. In these studies, the goal was to ˇnd all path-integral representations for the coordinate systems [24, 44Ä47] in which the Schréodinger equation (respectively the path integral) allows separation of variables. Paper [22] was aimed explicitly to give the best to our knowledge list of up-to-date explicitly 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 pathintegral 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 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 DI , there is no limiting case, because we have no free parameter in the metric to choose from. In the 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 nonconstant curvature, called Darboux spaces, DI ÄDIV , respectively. These spaces have been introduced by Kalnins et al. [27, 28]. They can be embedded in threedimensional 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 nonconstant 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 DI , for the radial potential b − a/r2 one obtains a metric ∝ (b − a/r2 ), i.e., the Darboux space DII , etc. The case of two dimensions is especially simple, because one obtains always a conformally at space. This method of constructing new spaces was ˇrst 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 ˇnd the corresponding superintegrable potentials. The notion of superintegrable systems was introduced by Winternitz and co-workers in [9, 52], Wojciechowski [53], and was further developed later on also by Evans [6]. Superintegrable potentials have the property of additional constants of motion: The simplest of the cases of the only conserved quantity is the energy, gives usually a chaotic system [22]; in order that a physical system is just integrable it 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 ˇve (maximal superintegrable) functional independent constants of motion. Well-known examples are the Coulomb potential with its LenzÄRunge vector and the harmonic oscillator with its quadrupole moment.

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 589

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 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 than one coordinate system the superintegrability and the existence of additional constants of motion follow, 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 DiracÄCoulomb possesses some of this symmetry by the conservation of the JohnsonÄLippmann operator which reduces in the nonrelativistic limit to the LenzÄRunge vector [37]. In the previous publications [18Ä21] we have studied 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 Schréodinger equation) one can obtain 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 ˇnd superintegrable potentials on the hyperboloid [29]. The connection of superintegrability and the polynomial solutions was studied (e.g., in [30]) in connection 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 were investigated. In this ˇrst paper on superintegrable potentials on Darboux spaces we discuss only the Darboux spaces DI and DII . The superintegrable potentials on the other two Darboux spaces DIII and DIV will be discussed in a forthcoming publication. The paper is organized as follows: In Sec. 2 Section we treat the superintegrable potentials on Darboux space DI . There are three of them, the third consisting of a constant divided by the metric term which makes the potential almost trivial. The common feature of the ˇrst 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 Sec. 3 the superintegrable potentials on DII are discussed. There are three nontrivial and one trivial potentials. For the ˇrst potential we obtain a quadratic equation for the energy levels, and they show an oscillator-like behavior. An

590 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

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 DII is exactly solvable in two coordinate systems. Here, we also ˇnd a quadratic equation for the energy levels. V2 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 eighth order in E which cannot be solved in general. For a special case, however, we ˇnd a Coulomb-like behavior of the energy levels. The fourth potential is a constant times the metric term, and is therefore trivial. As for DI , this potential is included for completeness. Section 4 contains a discussion of our results and an outlook for the remaining two Darboux spaces DIII and DIV . Introducing Darboux Spaces. Kalnins et al. [27, 28] denoted four types of two-dimensional spaces of nonconstant curvature, labeled by DI ÄDIV , which are called Darboux spaces [40]. In terms of the inˇnitesimal 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): ds2 = (x + y)dx dy, (x = u + iv, y = u − iv), (1.1) = 2u(du2 + dv 2 ),   a (II) ds2 = +b dx dy, (x−y)2   1 bu2 − a 1 = (du2 + dv 2 ), x = (v+iu), y = (v−iu) , (1.2) 2 u 2 2   (III) ds2 = a e−(x+y)/2 + b e−x−y dxdy, (I)

(IV) ds2

= e−2u (b + a eu )(du2 + dv 2 ), (x = u − i v, y = u + iv), (1.3)   a e(x−y)/2 + e(y−x)/2 + b = −  2 dx dy, e(x−y)/2 − e(y−x)/2   a+ a− = + (du2 +dv 2 ), (x = u+iv, y = u−iv),(1.4) sin2 u cos2 u

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 these spaces. They found appropriate coordinate systems, and we will consider all of them. In the majority of the cases we will be able to ˇnd a solution, however in some cases this will be impossible due to a quartic anharmonicity of the problem in question.

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 591

2. SUPERINTEGRABLE POTENTIALS ON DARBOUX SPACE DI We start with Darboux space DI and consider the following coordinate systems: ((u, v) system:) (Rotated (r, q) coordinates:)

(Displaced parabolic:)

x = u + iv, y = u − iv (u  a), u = r cos ϑ + q sin ϑ,

(2.1)

v = −r sin ϑ + q cos ϑ (θ ∈ [0, π]), (2.2) 1 u = (ξ 2 − η 2 ) + c, v = ξη 2 (2.3) (ξ ∈ ⺢, η > 0, c > 0).

The inˇnitesimal distance, i.e., the metric is given by ds2 = 2u(du2 + dv 2 ),

(2.4) 2

2

(Rotated (r, q) coordinates:) = 2(r cos ϑ + q sin ϑ)(dr + dq ),

(2.5)

(Displaced parabolic:) = (ξ − η + 2c)(ξ + η )(dξ + dη ). (2.6) 2

2

2

2

2

2

The Gaussian curvature in a space with metric ds2 = g(u, v)(du2 + dv 2 ) is given by (g = det g(u, v))   2 ∂ ∂2 1 + ln g. (2.7) G=− 2g ∂u2 ∂v 2 Equation (2.7) will be used to discuss shortly the curvature properties of the Darboux spaces, including their limiting cases of constant curvature. We ˇnd, e.g., in the (u, v) system for the Gaussian curvature G=

1 . u4

(2.8)

There is no further parameter in the metric, therefore this space is of nonconstant curvature throughout for all u > a with a being some real constant a > 0. However, DI can be embedded in a three-dimensional Euclidean space. It can then be visualized as an inˇnite surface (similar to one sheet of a double-sheeted hyberboloid) with a circular hole at the bottom∗ . The constant a may be 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: L(u, u, ˙ v, v) ˙ = mu(u˙ 2 +v˙ 2 )−V (u, v), H(u, pu , v, pv ) =

∗ Kalnins E.

Private communication.

1 (p2 +p2 )+V (u, v), 4mu u v (2.9)

592 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

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   1  ∂  ∂ + , (2.10) pu = , pv = i ∂u 2u i ∂v and for the quantum Hamiltonian we ˇnd   2 ∂ ∂2 1 2 1 1 1 √ (p2 +p2 ) √ +V (u, v). (2.11) + H =− +V (u, v) = 2m 2u ∂u2 ∂v 2 2m 2u u v 2u We formulate the path integral (ignoring the half-space constraint for the time being): K(u , u , v  , v  ; T ) = lim



N →∞

× exp  u(t )=u

⎧ N  ⎨i ⎩

u(t )=u

j=1

N N −1

Dv(t)2u exp v(t )=v 

2uj duj dvj ×

j=1

⎫ ⎬ = muj (Δ2 uj + Δ2 vj ) − V (uj , vj ) ⎭

 v(t )=v 

Du(t)

=

m 2πi 

⎧ T ⎨i  ⎩

0

(2.12)

⎫  ⎬ mu(u˙ 2 + v˙ 2 ) − V (u, v) dt , ⎭

(2.13) √ where uj = u(tj ), Δuj = uj − uj−1 , = T /N , uj = uj−1 uj . We have displayed the path integral in our product-lattice deˇnition, which will be used throughout this paper [22]. Due to this lattice deˇnition of the path integral, we have no additional 2 potential because the dimension of the space of nonconstant curvature equals 2, c.f. [22]. According to [27, 28] we introduce the following three integrals of motion in DI . They are ⎫ K = pv , ⎪ ⎬ v (p2 + p2 ), X1 = pu pv − 2u u v (2.14) 2 ⎪ ⎭ v 2 2 X2 = pv (vpu − upv ) − 4u (pu + pv ). They satisfy the relation ˜ 0 X2 + X 2 + K 4 = 0. 4H 1

(2.15)

˜ 0 the classical Hamiltonian without the 1/2m factor is (Let us note that by H meant. Keeping this factor is no problem, however, in the present form the

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 593

algebra has a simpler showing.) These operators satisfy the Poisson algebra relations ˜ 0, {K, X1 } = 2H

{K, X2 } = −X1 ,

{X1 , X2 } = 2K 3 .

(2.16)

The quantum analogues are given by ⎫ ⎪ ⎪ ⎪ ⎬

 = ∂v , K 1 = ∂u ∂v − v (∂ 2 + ∂ 2 ), X v 2u u

(2.17)

⎪ 2 ⎪ ⎭ 2 = 1 {∂v , v∂u − u∂v } − v (∂ 2 + ∂ 2 ), ⎪ X u v 2 4u

where {·, ·} is the anticommutator. These operators satisfy the commutation relations  X 1 ] = −2H  0, [K,

 X  2 ] = X1 , [K,

1 , X  2 ] = 2K  3, [X

(2.18)

with the operator relation 2 + X 2 + K  4 = 0. 0X 4H 1

(2.19)

The operators K, X1 , X2 can be used to characterize the separating coordinate systems on DI , as indicated in Table 1. Table 1. Constants of motion in space DI Metric 2u(du2 + dv 2 ) 2

2

2(r cos ϑ + q sin ϑ)(dr + dq ) (ξ − η + 2c)(ξ + η )(dξ + dη ) 2

2

2

2

2

2

Constants of motion

Coordinate system

K2

(u, v) system

X1

(r, q) system

X2

Parabolic

Let us note again that we do omit here factors of i, , and 1/2m for the sake of simplicity. H0 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 used 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   1 ∂2 ∂2 Λ1 = q sin θ 2 − r cos θ 2 = − sin 2θX1 − cos 2θK 2 , q sin θ + r cos θ ∂r ∂q (2.20)

594 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

  2 2 1 4cξ 2 η 2 4 ∂ 4 ∂ + ξ = η + 2 4 4 2 2 ξ −η ∂ξ ∂η ξ − η2   2 ∂ ∂ ∂2 ∂ ∂ v2 ∂2 2cv 2 − 2v + 2(u − c) 2 + . =− + + 2 2 ∂u ∂u ∂v ∂v 2(u − c) ∂u ∂v (u − c) (2.21)

Λ2 =

These two operators describe the general case. Special cases for Λ1 are: • θ = π/4, we have Λ1 = −X1 (symmetric case), • θ = π/2, we have Λ1 = K 2 , and for c = 0 we have Λ2 = −2X2 . Now we consider the following potentials on DI (following [28], an additional fourth potential is according to [4]): ⎡ V1 (u, v) =

λ2 −

⎤ 1 4⎥ ⎦,

1 ⎢m 2 ⎣ ω (4u2 + v 2 ) + κ + 2u 2 2mv 2

  1 m 2 2 2 ω (u + v ) + κ1 + κ2 v , V2 (u, v) = 2u 2 1 2 v02 , V3 (u, v) = 2u 2m  a0 4u − 2iv 1 √ V4 (u, v) = + a1 + a2 u + a3 √ . 2u u − iv u − iv

(2.22)

(2.23) (2.24) (2.25)

Table 2. Separation of variables for the superintegrable potentials on DI Potential V1

V2

V3

Constants of motion m 2 v4 κ2 v 2 2 ω − − 2 4u 2 u 4m m 2 2 2 λ2 − 1/4 2 R2 = K + ω v + 2 m v2 R1 = X2 −

  1 4u2 + v 2 (u, v) system λ2 − 4 uv 2

κ2 (u2 − v 2 ) m v(u2 − v 2 ) κ1 v + + ω2 u u 2 u R2 = K 2 + 2κ2 v + mω 2 v 2 R1 = X1 −

2 v02 2m 2 v02 R2 = X2 − 4m R3 = K R1 = X1 −

v u v2 u

Separating coordinate system

Parabolic (u, v) system (r, q) system (u, v) system (r, q) system Parabolic

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 595

In Table 2 we have summarized some properties of three of these potentials. Actually, V3 can be considered as a special case either of V1 or V2 , respectively. The fourth potential separates, for instance, in parabolic coordinates (c = 0) and then has the (complex) form V4 (ξ, η) = √  a2 4 1 2 2 4 3/2 3 3 (ξ 2a (ξ + iη) + a (ξ + η ) + − η ) + 2 a (ξ − iη ) . = 4 0 1 3 ξ − η4 2 (2.26) However, this is not tractable and we will not discuss this potential any further. 2.1. The Superintegrable Potential V1 on DI . We start with the potential V1 in DI . V1 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 V1 in the respective coordinate systems ⎡ ⎤ 1 2 − λ 1 ⎢m 2 2 2 4⎥ V1 (u, v) = (2.27) ⎣ ω (4u + v ) + κ + ⎦, 2u 2 2mv 2  1 m 2 6 = ω (ξ + η 6 ) + 2mω 2 (ξ 4 − η 4 ) + 2 2 2u(ξ + η ) 2   2 1 1 2 2 2 2 2 λ +(2mω c + κ)(ξ + η ) +  + 2 . 2m ξ 2 η

(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 DI , which can be put into the form (including already the proper energy term) 1 α VE (u, v) = − √ + E, u u2 + v 2

(2.29)

which yields after a space-time transformation, with unshifted (c = 0) parabolic coordinates VE (u, v) → −2α(ξ 2 − η 2 ) + E(ξ 4 − η 4 ), (2.30) 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¯

596 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

in the sense of Turbiner [50] and Ushveridze [51]. In fact, sextic oscillators with a centrifugal barrier and quartic hyperbolic and trigonometric ones can be considered, and they are very similar in their structure as, for instance, in (2.28). One can ˇnd particular solutions, provided the parameters in the quasi-exactly solvable potentials fulˇll special conditions. Furthermore, well-deˇned expressions for the wave functions and for the energy spectrum can indeed be found if only quadratic, sextic, and particular centrifugal terms are present. The wave 4 functions then have the form of Ψ(x) ∝ P (x4 ) × e−αx , with a polynomial P . However, quasi-exactly solvable potentials have the feature that only a ˇnite 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 that one described in (2.28), and we can omit the metric term for simplicity). Writing down the Schré odinger 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 deˇned 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 ˇnds a quantization condition for the energy E. Because the potential is assumed to be separable in Cartesian coordinates we already know the energy levels, En . The second separation constant λ of the system of coupled differential equations in ξ and η can then be expressed in terms of En , i.e., λn = f (En ). 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 ∈ ⺢, 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 corresponds 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

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 597

quasi-exactly solvable potential are determined by the separation constant λn of the coupled system of differential equations. This feature is common to all quasiexactly 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 nonterminating three-term recursion relations for the scattering states. In comparison to the (conuent) hypergeometric functions little is known about expansion and addition theorems (with the exception of Mathieu and spheroidal wave functions in at 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. 2.1.1. Separation of V1 in the (u, v) System. We insert V1 in (2.13) and obtain

K

(V1 )







 u(t )=u



 v(t )=v 

Du(t)

(u , u , v , v ; T ) = u(t )=u

Dv(t)2u× v(t )=v 

⎧ ⎡ ⎞⎤ ⎫ ⎛ 1 2 ⎪ ⎪ ⎨ i T ⎬ − λ 1 ⎜m 2 ⎢ ⎟⎥ 2 2 2 2 4 × exp dt = ⎣mu(u˙ + v˙ ) − ⎠ ⎦ ⎝ ω (4u + v ) + κ + ⎪ ⎪ 2u 2 2mv 2 ⎩ ⎭ 0

=

∞

√ v  v  Ψ(RHO,λ) (v  )Ψ(RHO,λ) (v  )Kn(V1 ) (u , u ; T ), (2.31) n n n=0

with the path integral Kn (T ) given by

Kn(V1 ) (u , u ; T )

  1/4

 )=u u(t

√ Du(t) 2u×

= (4u u )

× exp

⎧ T ⎨i  ⎩

0

u(t )=u

⎫  ⎬   1 En mω 2 u2 + κ − muu˙ 2 − dt , (2.32) ⎭ 2u 2u

with En = 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

598 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

be imposed. However, we continue with the case v > 0. The wave functions for m 2 λ2 − 1/4 the radial harmonic oscillator V (r) = ω 2 − have the form 2 2m r2 %

 λ/2     mω n! 2m mω 2 (λ) mω 2 r r r Ln r . exp −  Γ(n + λ + 1)  2  (2.33) (λ) The Ln (z) are Laguerre polynomials [10]. Ψ(RHO,λ) (r) = n

At 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

  1) G(V n (u , u ; E)

∞ =

ds exp

0

    i E2  (V1 )  − κ − E (u , u ; s ), n s Kn  2mω 2 (2.34)

with the transformed path integral given by Kn(V1 ) (u , u ; s ) = & s   2  ' i m 2 m E 2 u˙ − (2ω) u − Du(s) exp ds . (2.35)  2 2 mω

 u(s )=u

= u(0)=u

0

This path integral of a shifted harmonic oscillator with frequency 2ω can be solved. The corresponding Green function has the form (

1) G(V (E; u , u ; E) u

  1 E m Γ − = D− 12 +E/2ω × 2π3 ω 2 2ω * ) ( * )( 4mω 4mω u ˜> D− 12 +E/2ω − u ˜< . (2.36) ×  

Here, the Dν (z) are parabolic cylinder functions [10] and u ˜ = u − E/2mω 2 . For  the evaluation of the s integration we use the involution formula  G(u , u , v , v ; E) = 2πi 









d EGv (E; v  , v  ; E) Gu (E; u , u ; −E)

(2.37)

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 599

to obtain K (V1 ) (u , u , v  , v  ; T ) = ∞ ∞ dE −iET / √   (RHO,λ)  (RHO,λ)  e vv Ψ (v )Ψ (v )× = 2π n=0 −∞    E2 (V1 )   × Gu E; u , u ; − κ − En . (2.38) 2mω 2 Solution without Boundary Condition. Let us ˇrst solve the potential problem V1 on DI 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 (HO) (˜ u) with u ˜ = u − E/mω 2 . The wave with wave functions given by Ψl functions for the harmonic oscillator (HO) are given by the well-known form in terms of Hermite polynomials  (x) = Ψ(HO) n

mω π

1/4 √

1 2n n!

( Hn

(V1 )

Evaluating the Green function Gu (V1 ) (u , u , v  , v  ; T ) Kdiscr

×

=

∞ ∞

   mω 2 mω x exp − x .  2

(2.39)

we obtain the solution % mω 2   − i Eln T / vv e × 2Eln

n=0 l=0 (HO)  (HO)  (RHO,λ)  (v )Ψ(RHO,λ) (v  )Ψl (˜ u )Ψl (˜ u ), Ψn n

+ Eln = ± mω 3 (2l + 2n + 2 + λ) + 2mω 2 κ.

(2.40) (2.41)

The spectrum is degenerate in n and l, as it is known for superintegrable potentials. However, this ®solution¯ is seriously awed. If we calculate the norm of the wave functions, we see immediately that the norm is proportional to the energy En , which in the negative-sign case is negative, and it follows that the Hilbert space is not properly deˇned. In the positive-sign case the norm would be positive, however, the corresponding conˇguration 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.

600 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

By exploiting the Dirichlet boundary conditions [12] at u = a we therefore get 1 G(x=a) (u , u , v  , v  ; E) =

(V )

& ×

∞

√ v  v  Ψ(RHO,λ) (v  )Ψ(RHO,λ) (v  )× n n n=0

  1) G(V n (u , u ; E) −

Gn 1 (u , a; E)Gn 1 (a, u ; E) (V )

(V )

(V )

Gn 1 (a, a; E)

' . (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  (   mω El,n Dνl,n 2 = 0, (2.43) a−  mω 2   2 El,n 1 1 νl,n = − + − κ − ω(2n + λ + 1) . (2.44) 2 2ω mω 2 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 curly-bracket 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 typical for superintegrable systems. The solution (2.41) has this degeneracy but the boundary conditions are not fulˇlled and the Hilbert space is not properly deˇned either. For 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 El,n follows, i.e., a set El,0 , El,1 , . . . There is no possibility of ˇnding that a level from the set n = 0 is equal to one level of the set n = 1, for example, Ela ,0 = Elb ,1 for some numbers la , lb . Therefore, we ˇnd 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 SturmÄLiouville 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 for the other way round, and the present examples of potentials on Darboux space DI 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 inˇnite and a wave function at the coordinate origin does not make sense. Depending whether the Darboux space DI is embedded in three-dimensional space with deˇnite or

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 601

indeˇnite metric further determines the parameter a, c.f. [28]. For a positivedeˇnite metric, v is an angle with v ∈ [0, 2π), the two-dimensional surface making up DI has a deˇnite boundary and it follows a = 1/2. For a negative-deˇnite metric the boundary turns out to be constrained by a = 0. In fact, it is impossible to extend the surface beyond u < 0, and all values from 0 to ∞ are deˇnitely excluded. We will see that the same property holds for the potential V2 . 2.2. The Superintegrable Potential V2 on DI . Next, we consider the potential V2 on DI . First, we state the potential in the separating coordinate systems. We have   1 m 2 2 2 ω (u + v ) + κ1 + κ2 v , (2.45) V2 (u, v) = 2u 2   1 m 2 2 = ω (r + q 2 ) + κ1 + κ2 (q cos ϑ − r sin ϑ) . (2.46) 2u 2 2.2.1. Separation of V2 in the (u, v) System. We proceed in a similar way as before and obtain

K

(V2 )









 u(t )=u

Du(t)

(u , u , v , v ; T ) =

× exp

⎧ T ⎨i  ⎩

0

 v(t )=v 

u(t )=u

Dv(t)2u× v(t )=v 

⎫  ⎬   1 m 2 2 ω (u + v 2 ) + κ1 + +κ2 v dt = mu(u˙ 2 + v˙ 2 ) − ⎭ 2u 2 =

∞

Ψ(HO) (˜ v  )Ψ(HO) (˜ v  )Kn(V2 ) (u , u ; T ), (2.47) n n

n=0 (HO)

where Ψn are the wave functions of a shifted harmonic oscillator with v˜ = v + κ2 /mω. Note that we have to require v ∈ ⺢, otherwise for v cyclic, the complicated boundary conditions have to be imposed on the solution in v. The remaining path integral in the variable u has the form

Kn(V2 ) (u , u ; T ) = (4u u )1/4

× exp

⎧ T ⎨i  ⎩

0

 )=u u(t

√ Du(t) 2u×

u(t )=u

1 muu˙ 2 − 2u



⎫   ⎬  m 2 2 1 κ22 ω u + κ1 +  n + dt . − ⎭ 2 2 2mω 2 (2.48)

602 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

This gives in the usual way   2) G(V n (u , u ; E) =      ∞ i 1 κ22 = ds exp − s κ1 + ω n + Kn(V2 ) (u , u ; s ), −  2 2mω 2 0

(2.49) with the transformed path integral given by (˜ u = u − 2E/mω 2 )

Kn(V2 ) (u , u ; s ) =

& s   ' i m 2 2 2 (u˙ −ω u )+2Eu ds = Du(s) exp  2

 u(s )=u

u(0)=u

0



=e



2is E/mω  2





u(s )=u

u(0)=u



im Du(s) exp 2

s

 m ˙2 2 2  (u ˜ −ω u ˜ )ds . (2.50) 2

0

Solution without Boundary Condition. This is again a path integral for a shifted harmonic oscillator, and ˇrst we ignore the boundary condition for the wave functions in the variable u for u = 0, say, we obtain the solution: 2 (u , u , v  , v  ; T ) = Kdiscr

(V )

∞ ∞

% mω 2   −iEln T / vv e × 4Eln

n=0 l=0 (˜ v  )Ψ(HO) (˜ v  )Ψ(HO) (˜ u )Ψ(HO) (˜ u ), × Ψ(HO) n n n n

% Eln = ±

  mω 2 k22 . l + n + 1 + κ1 − 2 2mω 2

(2.51) (2.52)

This spectrum exhibits degeneracy, however the norm is again proportional to the energy, which is negative, and therefore the Hilbert space is not properly deˇned. Solution with Boundary Condition. If we now take into account the boundary condition for some u = a such that the wave function vanishes for u = a, we obtain in a similar manner as in the previous subsection: 2 (u , u , v  , v  ; E) = G(x=a)

(V )

∞

√ v  v  Ψ(HO) (˜ v  )Ψ(HO) (˜ v  )× n n n=0

& ×

  2) G(V n (u , u ; E)

Gn 2 (u , a; E)Gn 2 (a, u ; E) (V )

−

(V )

(V )

Gn 2 (a, a; E)

' , (2.53)

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 603 (V )

with the Green function Gn 2 (E) given by (

  1 m E Γ − × 2π3 ω 2 2ω )( * ) ( * 4mω 4mω u ˜> D− 12 +E/2ω − u ˜< , (2.54) × D− 12 +E/2ω  

2) (E; u , u ; E) = G(V u

E=

  1 2E 2 + κ22 /2 − κ − ω n + . 1 mω 2 2

Bound states can be determined by the quantization condition (   2mω 2El,n = 0, a− Dνl,n  mω 2 νl,n

  2  1 2Eln + κ22 /2 1 1 =− + − κ1 − ω n + . 2 ω mω 2 2

(2.55)

(2.56)

(2.57)

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). 2.2.2. Separation of V2 in the (r, q) System. In order to set up the path integral formulation we follow our canonical procedure. The Lagrangian and Hamiltonian are given by, respectively: L(r, r, ˙ q, q) ˙ = m(r cos ϑ + q sin ϑ)(r˙ 2 + q˙2 ) − V (r, q), 1 (p2 + p2q ) + V (r, q). H(r, pr , q, pq ) = 4m(r cos ϑ + q sin ϑ) r

(2.58) (2.59)

The canonical momenta are

  cos ϑ  ∂ + , i ∂r 2(r cos ϑ + q sin ϑ)   sin ϑ  ∂ + = . i ∂q 2(r cos ϑ + q sin ϑ)

pr =

(2.60)

pq

(2.61)

The quantum Hamiltonian has the form   2 1 ∂ ∂2 2 + H =− + V (r, q) = (2.62) 2m 2(r cos ϑ + q sin ϑ) ∂r2 ∂q 2 1 1 1 + (p2r +p2q ) + +V (r, q). (2.63) = 2m 2(r cos ϑ+q sin ϑ) 2(r cos ϑ+q sin ϑ)

604 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

Using the representation (2.46) we write down the path integral for V2 in the rotated (r, q)-coordinate system, and obtain K(r , r , q  , q  ; T ) =

 r(t )=r 

 q(t )=q

Dr(t) r(t )=r 

Dq(t)2(r cos ϑ + q sin ϑ)× q(t )=q

& T  i × exp m(r cos ϑ + q sin ϑ)(r˙ 2 + q˙2 )−  0

 ' 1 m 2 2 2 ω (r + q ) + κ1 + κ2 (q cos ϑ − r sin ϑ) dt . (2.64) − 2u 2 

Performing a space-time transformation in the usual way gives









∞

G(r , r , q , q ; E) =



ds e−is

κ1 /

K(r , r , q  , q  ; s ),

(2.65)

0

with the transformed path integral K(s ) given by

K(r , r , q  , q  ; s ) =

 r(s )=r 

 q(s )=q

Dr(s) r(0)=r 

Dq(s)× q(0)=q

& s   i m 2 r˙ + q˙2 − ω 2 (r2 + q 2 ) + 2E(r cos ϑ + q sin ϑ)− × exp  2 0  '  ) *  i 4E 2 + κ22 − κ1 s × − κ2 (−r sin ϑ + q cos ϑ) ds = exp  2mω 2 ⎧ s ⎫ ⎨ i m  ⎬ m × exp r2 + q˜2 ) ds = r˙ 2 + q˙2 ) − ω 2 (˜ ⎩ ⎭ 2 2 0  ) *    ∞

i 4E 2 + κ22 1  = exp − ω n + Ψ(HO) (˜ q  )Ψ(HO) (˜ q  )× s − κ 1 n n  2mω 2 2 n=0 ⎡ ⎤   r(s )=r  s im × Dr(s) exp ⎣ (r˙ 2 − ω 2 r˜2 )ds⎦ (2.66) 2 r(0)=r 

0

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 605

(˜ r = r − (2E cos ϑ + κ2 sin ϑ)/mω 2 , q˜ = q − (2E sin ϑ − κ2 cos ϑ)/mω 2 ). 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 deˇned only in the half-space u  a. Setting, for instance, in the deˇnition 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 V2 in the (u, v) system, we impose on the Green function in r the boundary condition r  a and obtain in this limiting case for the bound states the quantization condition (   2mω 2El,n = 0, (2.67) Dνl,n a−  mω 2   2  1 2Eln + κ22 /2 1 1 νn = − + − κ1 − ω n + . 2 ω mω 2 2

(2.68)

This is the result of (2.57). The quantization conditions of (2.57) and (2.68) are identical as it should be. 2.3. The Superintegrable Potential V3 on DI . Next, we consider the potential V3 on DI . First, we state the potential in the separating coordinate systems. We have 1 2 v02 , 2u 2m 1 2 v02 = 2 , ξ − η 2 + 2c 2m 1 2 v02 = . 2(r cos ϑ + q sin ϑ) 2m

V3 (u, v) =

(2.69) (2.70) (2.71)

This potential can be considered as a special case either of V1 or V2 , respectively. However, it has an additional conserved quantum number, i.e., K = pv . 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):

K(u , u , v  , v  ; T ) =

 u(t )=u

 v(t )=v 

Du(t) u(t )=u

× exp

⎧ T ⎨i  ⎩

0

Dv(t)2u× v(t )=v 

⎫  ⎬ 2 2  v 1 0 mu(u˙ 2 + v˙ 2 ) − dt = ⎭ 2u 2m

(2.72)

606 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

= (4u u )1/4

 u(t )=u

  ∞

eil(v −v )

l=0

2π

u(t )=u

× exp

√ Du(t) 2u×

⎧ T ⎨i  ⎩

0

⎫  ⎬ 1  2 (l + v02 ) dt . (2.73) muu˙ 2 − ⎭ 2u 2m 2

We observe that the only effect is the change in the quantum number l in comparison to the v0 = 0 case. Using the solution of [15] we get for the corresponding Green function    ∞

˜l2 2  ˜l2 2 1/2 eil(v −v ) 4m   × G(u , u , v , v ; E) = u − u − 2π 3 4mE 4mE l=−∞ ⎡ 







  ⎢ ˜l2 2  ˜2 2  ⎢ ˜ 1/3 u> − l  K × ⎢I˜1/3 u< − − 4mE 4mE ⎣ ⎤  ˜l2 2  ˜     I1/3 a − ˜2 2 ˜2 2 ⎥ 4mE ⎥ ˜ 1/3 u − l  ˜ 1/3 u − l  K K − ⎥ . (2.74)   ˜l2 2 4mE 4mE ⎦ ˜ K1/3 a − 4mE I˜ν (z) denotes

 √  4 −mE 3/2 ˜ z Iν (z) = Iν , 3

˜ ν (z) similarly, and ˜l2 = l2 + v 2 . Due to the relation satisˇed by the Airy with K 0 + 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, we infer that in this case there are no bound states. For E > 0 there is no real bound state solution, either. This concludes the discussion.

3. SUPERINTEGRABLE POTENTIALS ON DARBOUX SPACE DII In this section we consider superintegrable potentials on the Darboux space DII (1.2). The following four coordinate systems separate the Schréodinger equa-

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 607

tion for the free motion: ((u, v)-system:) x =

1 (v + iu), 2

y=

1 (v − iu), 2

(Polar:) u =  cos ϑ, v =  sin ϑ



 > 0, ϑ ∈

(3.1)  π π − , , (3.2) 2 2

1 2 (ξ − η 2 ) (ξ > 0, η > 0), 2 (Elliptic:) u = d cosh ω cos ϕ, v = d sinh ω sin ϕ    π π . ω > 0, ϕ ∈ − , 2 2

(Parabolic:) u = ξη, v =

(3.3)

(3.4)

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 inˇnitesimal distance is given in these four cases (note that the metric gives us the additional requirement u > 0): bu2 − a (du2 + dv 2 ), u2 b2 cos2 ϑ − a (Polar:) = (d2 + 2 dϑ2 ), 2 cos2 ϑ bξ 2 η 2 − a 2 (ξ + η 2 )(dξ 2 + dη 2 ) = (Parabolic:) = ξ 2 η2     a a = bξ 2 − 2 + bη 2 − 2 (dξ 2 + dη 2 ), ξ η ds2 =

(3.5) (3.6)

(3.7)

bd2 cosh2 ω cos2 ϕ − a (cosh2 ω − cos2 ϕ)(dω 2 + dϕ2 ) = cosh2 ω cos2 ϕ   a 2 2 = bd cosh ω + − cosh2 ω   a − bd2 cos2 ϕ + (3.8) (dω 2 + dϕ2 ). cos2 ϕ

(Elliptic:) =

We can see that the case a = −1, b = 0 leads to the case of the Poincar e 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 semicircular-parabolic system and the elliptic case to the ellipticÄparabolic 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 systems which allow separation of variables of the LaplaceÄBeltrami 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)

608 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

and a space of constant negative curvature (the hyperbolic plane). This includes the emerging of coordinate systems in a at space from curved spaces. Table 3. Constants of motion and limiting cases of coordinate systems on DII Λ(2) (a = −1, E2 (a = 0, b = 0) b = 1)

Metric

Constant of motion

DII

bu2 − a (du2 + dv 2 ) u2

K2

(u, v) system

Horicyclic

Cartesian

X2

Polar

Equidistant

Polar

X1

Parabolic

Semicircular parabolic

Parabolic

b2 cos2 ϑ − a (d2 + dϑ2 ) 2 cos2 ϑ bξ 2 η 2 − a 2 (ξ + η 2 )(dξ 2 + dη 2 ) ξ2 η2

bd2 cosh2 ω cos2 ϕ − a × X2 + d2 K 2 Elliptic EllipticÄparabolic Elliptic cosh2 ω cos2 ϕ 2 2 2 2 2 ×(cosh ω − cos ϕ)(dω + d ϕ )

We ˇnd for the Gaussian curvature in the (u, v) system G=

a(a − 3bu2 ) . (a − 2bu2 )3

(3.9)

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 deˇniteness of the metric (1.2). The following constants of motion (see Table 3) are introduced on DII (without potential): K X1 X2

= pv , 2v(p2v − u2 p2u ) + 2upu pv , = bu2 − a (v 2 − u4 )p2v + u2 (1 − v 2 )p2u + 2uvpu pv . = bu2 − a

(3.10) (3.11) (3.12)

They satisfy the Poisson algebra relations ˜ 0 ), {K, X1 } = 2(K 2 − H

{K, X2 } = X1 ,

{X1 , X2 } = 4KX2 (3.13)

and the relation ˜ 0 X2 − 4 H ˜ 2 = 0. X12 − 4K 2 X2 + 4H 0

(3.14)

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 609

The quantum analogues have the form (again with i, , 2m) K = ∂v , 2v X1 = (∂ 2 − u2 ∂u2 ) + 2u∂u ∂v , 2 bu − a v   1 2 4 2 2 2 2 X2 = − u )∂ + u (1 − v )∂ (v v u + bu2 − a 1 + 2uv∂u ∂v + u∂u + v∂v − 4

(3.15) (3.16)

(3.17)

 0 Å the Hamiltonian operator, {, } Å the and satisfy the operator relation (H anticommutator)  2 − 4H  2, X  2 } + 4H  0X  02 + 4K 2 = 0 12 − 2{K X

(3.18)

and the commutation relations  X 1 ] = 2(K 2 − H  0 ), [K,  X 2 ] = X 1 , [X 1 , X 2 ] = 2{K,  X 2 }. [K,

(3.19)

We consider the following potentials on DII : ⎡

⎤ 1 2 k − bu − a ⎢ m 2 2  2 4⎥ 2 V1 (u, v) = ⎣ ω (u + 4v ) + k1 v + ⎦, u2 2 2m u2 2

2

⎡ V2 (u, v) =

⎛ 2 2 ⎜ k1

bu2 − a ⎢ m 2 2 ⎣ ω (u + v 2 ) + ⎝ u2 2 2m ⎡

(3.20)

⎞⎤ 1 1 2 − k − ⎥ 4 + 2 4⎟ ⎠⎦, (3.21) 2 u v2

⎛

1 k12 − 2m bu − a  ⎜ ⎢ 4 √ + V3 (u, v) = ⎣−α + ⎝√ 2 u2 2m u2 + v 2 u + v2 + v 2

+√

V4 (u, v) =

2

k22 − u2

+

⎞⎤

1 4

v2

−v

⎟⎥ ⎠⎦ ,

bu2 − a 2 2 v . u2 2m 0

(3.22)

(3.23)

In Table 4 we have listed the properties of these potentials (the coordinate systems where an explicit path-integral evaluation is possible are underlined).

610 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

Table 4. Separation of variables for the superintegrable potentials on DII Potential

Constants of motion

Separating coordinate system



V1

 u2 + 4v 2 R1 = X1 + mω v u + + bu2 − a 1   k22 − k1 4v 2 v 2 2 4 + u + 2 − 2 bu − a m bu2 − a R2 = K 2 + 2mω 2 v 2 + k1 v 2

(u, v)

2

system

Parabolic

V2 R1 = X2 +    u2 + v 2 m 2 2 2 1 1 u2 ω (u + v 2 ) − k12 − − (k22 − ) 2 + 2 bu − a 2 2m 4 4 v 1 2 m  2 k2 − 4 R2 = K 2 + ω 2 v 2 + 2 2m v 2

(u, v) system

Polar Elliptic

V3 R1 = X1 + Polar     2  4 1  4 2 1 2 2 4 2 −αξ η + 1 + k1 − η +1 − k2 − (ξ + 1) 2m 4 2m 4 + Parabolic (bξ 2 η 2 − a) (ξ 2 + η 2 ) R2 = X2 − Displaced     2 1 2 1 2 4 2 4 elliptic 2 2 α(ξ + η ) + k1 − (ξ − 1) + k2 − (ξ − 1) 2m 4 2m 4 − 4(bξ 2 η 2 − a) V4 R1 = X1 +

v 2 v02 m bu2 − a

(u, v)

2 v02 u2 + v 2 2m bu2 − a R3 = K = pv

system

R2 = X2 +

Polar Parabolic Elliptic

3.1. The Superintegrable Potential V1 on DII . We state the potential V1 in the respective coordinate systems: ⎡

⎤ 1 2 k − 2 bu − a ⎢ m 2 2  2 4⎥ V1 (u, v) = ⎣ ω (u + 4v ) + k1 v + ⎦= u2 2 2m u2 2

2

(3.24)

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 611

⎤ 1 2  k − 1 bu −a 1 ⎢ m 2 6 6 k1 4 4 2 4 1+ 1 ⎥ = ⎦. ⎣ ω (ξ +η )− (ξ − η )− u2 ξ 2 +η 2 2 2 2m ξ 2 η2 ⎡

2

(3.25) In a at 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 semicircular parabolic coordinates, the limiting cases of the (u, v) system and the parabolic coordinates, respectively. 3.1.1. Separation of V1 in the (u, v) System. We start with the (u, v)-coordinate system. We formulate the classical Lagrangian and Hamiltonian, respectively: m bu2 − a 2 (u˙ + v˙ 2 ) − V (u, v), 2 u2 u2 1 H(u, pu , v, pv ) = (p2 + p2v ) + V (u, v). 2m bu2 − a u L(u, u, ˙ v, v) ˙ =

The canonical momenta are   bu 1  ∂ pu = + 2 − , i ∂u bu − a u

pv =

 ∂ . i ∂v

The quantum Hamiltonian has the form   2 ∂2 2 u2 ∂ + H = − + V (u, v) = 2m bu2 − a ∂u2 ∂v 2 u 1 u √ = (p2 + p2v ) √ + V (u, v). 2m bu2 − a u bu2 − a

(3.26) (3.27)

(3.28)

(3.29) (3.30)

Therefore the path integral for V1 in the (u, v) system has the following form:

K

(V1 )









 u(t )=u

 v(t )=v 

Du(t)

(u , u , v , v ; T ) = u(t )=u

Dv(t) v(t )=v 

bu2 − a exp × u2

⎧ ⎤⎫ ⎞ ⎡ 1 ⎪ 2 T ⎪ ⎨ ⎬ 2 k2 − m 2  ⎟ ⎢m ⎜i 4⎥ dt⎠ , (u˙ + v˙ 2 )− f ⎣ ω 2 (u2 + 4v 2 ) + k1 v + ×⎝ ⎦ ⎪ 2f ⎪  ⎩ 2 2m u2 ⎭ ⎛

0

(3.31) and we have abbreviated f = u2 /(bu2 − a). integration according to

First, we separate the v-path

612 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.  v(t )=v 

Dv(t) exp v(t )=v 

⎧ T ⎨ i m ⎩

 v˙ 2 −

2

∞ (

m 2 2 ω v + k1 v 2

⎫ ⎬

 dt

0

⎭

=

  mω  2 2mω 1  2 exp − (˜ v + v˜ ) × = π 2n n!  n=0 )( * )( * 2mω  2mω  −iEn T / v˜ Hn v˜ e , (3.32) × Hn     1 k12 En = ω n + , + 2 8mω 2

(3.33)

with v˜ = v + k1 /4mω, which is the solution for the shifted harmonic oscillator. (HO) Writing for short the wave functions of the shifted harmonic oscillator by Ψn , we thus obtain: K

(V1 )









(u , u , v , v ; T ) =

∞

Ψ(HO) (˜ v  )Ψ(HO) (˜ v  )Kn(V1 ) (u , u ; T ), n n

(3.34)

n=0

Kn(V1 ) (u , u ; T )

,





−1/4

(

 )=u u(t

Du(t)

= f (u )f (u )]

u(t )=u

bu2 − a × u2

⎧ ⎡ ⎞ ⎤ ⎫ ⎛ 1 ⎪ ⎪ 2 ⎬ ⎨ i T m 2 k2 −  ⎢ ⎟ ⎥ ⎜m 2 2 2 4 . (3.35) + E dt × exp ⎣ u˙ − f ⎝ ω u + ⎠ ⎦ n ⎪ ⎪ 2f 2 2m u2 ⎭ ⎩ 0

We obtain in the usual way by means of a space-time transformation   1) G(V n (u , u ; E)

∞ =

Kn(V1 ) (u , u ; s ) exp

0



i (bE − En )s 

 (3.36)

with the transformed path integral given by

Kn(V1 ) (u , u ; s )r =

 u(s )=u

Du(s)× u(0)=u

⎧ ⎡ ⎤ ⎫  1 2 2 ⎪ ⎪ ⎨ i s m ⎬ 2 k2 + 2maE/ −  ⎢ ⎥ 2 2 2 4 = ( u ˙ − ω u ) − × exp ds ⎣ ⎦ ⎪ ⎪ 2 2m u2 ⎩ ⎭ 0

(3.37)

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 613

√     mωu u mω u u mω  2  2  (u = exp − + u ) cot ωs I = λ i sin ωs 2i i sin ωs ∞

 (RHO,λ)  (RHO,λ)  Ψl (u )Ψl (u ) eis ω(2l+λ+1) . (3.38) = l=0

Alternatively we have for the Green function (λ2 = k22 + 2maE/2 ) Γ   1) G(V n (u , u ; E) =

   1 1 (bE − En ) 1+λ− 2 ω √ ×   ω u u Γ(1 + λ)     mω 2 mω 2 × W bE−En , λ u M bE−En , λ u . (3.39) 2ω 2 2ω 2  > 
= k − k − × 2 2

G(V4 ) (u , u , v  , v  ; E) =

(3.131)

632 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

 = 2 π ∞ × 0

∞



dk −∞



eik(v −v ) × 2π

2p sinh πpdp   Kip 2 1 2 p + −E 2m|a| 4

)(

* )( * 2mbE 2mbE   u Kip u , k˜2 − k˜2 − 2 2 (3.132)

(

with

1 2m|a|E − ≡ ip. 4 2 The wave functions and the energy spectrum are read off: * )( √ eikv 2p sinh πp 2mbE (V4 ) 2 ˜ Kip u , Ψpk (u, v) = √ k − π 2 2π λ=

  2 1 2 E= p + . 2m|a| 4

(3.133)

(3.134)

(3.135)

4. SUMMARY AND DISCUSSION In this paper we have discussed superintegrable potentials on spaces of nonconstant curvature. The results are very satisfactory. According to [4, 27, 28], there are three potentials on DI , four potentials on DII , ˇve potentials on DIII , and four potentials on DIV , 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 PéoschlÄ Teller potential, but also path-integral identities derived from path integration on harmonic spaces like the elliptic and spheroidal path-integral representations with their more complicated special functions [13, 17, 22]. This includes also numerous transformation techniques to ˇnd a particular solution based on one of the basic solutions. Various analysis techniques can be applied to ˇnd not only an expression for the Green function but also for the wave functions and the energy spectrum. We also observe a new feature of superintegrable potentials. We learned from our investigation of potential problems on DI that degeneracy for superintegrable

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 633

Table 5. Solutions of the path integration for superintegrable potentials in Darboux spaces Space and potential DI V1 : (u, v) Parabolic V2 : (u, v) (r, q) V3 : (u, v) (r, q) Parabolic DII V1 : (u, v) Parabolic V2 : (u, v) Polar Elliptic V3 : Polar Parabolic Displaced Elliptic V4 : (u, v) Polar Parabolic Elliptic

Solution in terms of the wave functions Hermite polynomials × Parabolic cylinder functions No explicit solution Hermite polynomials × Parabolic cylinder functions Hermite polynomials × Parabolic cylinder functions Product of Airy functions Product of Airy functions Product of Airy functions Hermite polynomial × Whittaker functions∗ No explicit solution Laguerre polynomial × Whittaker functions∗ Gegenbauer polynomial × Whittaker functions∗ No explicit solution Gegenbauer polynomial × Whittaker functions∗ Gegenbauer polynomial × Whittaker functions∗ No explicit solution Product of Bessel functions Bessel functions × Legendre functions Product of Whittaker functions∗ Spheroidal wave functions

∗

The notion Whittaker functions means in all cases for a discrete spectrum Laguerre polynomials and for a continuous spectrum Whittaker functions Wμ,ν (z), respectively Mμ,ν (z).

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 nonequivalence of these two notions comes from the boundary conditions which had to be imposed on DI in order to guarantee a well-deˇned 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 DII , where we could recover the corresponding solutions for the two-dimensional Euclidean space and the two-dimensional hyperboloid. On DI the energy spectra are only determined by a transcendental equation due to the boundary condition for the coordinate u. On DII 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

634 GROSCHE C., POGOSYAN G. S., SISSAKIAN A. N.

energy spectra. The equations for the energy spectra were on DII algebraic equations in the second and fourth orders in the energy. This allows several solutions depending on the speciˇc values of the parameters a and b and possible further boundary conditions. Also semibound states may be possible. In the forthcoming publications we will treat the two remaining Darboux spaces DIII and DIV , respectively. In particular, on DIII there is already a discrete spectrum possible for the free motion, which has the form 2 b (2n + 2l + 1)2 (4.1) 2m a2 yielding for b > 0 an inˇnite number of bound states. This is similar to the motion on the SU (1, 1) hyperboloid, where continuous and discrete spectra exist [2]. On DIII there are ˇve superintegrable potentials and on DIV there are four superintegrable potentials. Let us ˇnally discuss the following issue: Let us consider a three-dimensional generalization of the Darboux space DII with a line element Enl = −

ds2 =

bu2 − a (du2 + dv 2 + dw2 ), u2

(4.2)

and w is the new variable. DII has the property that for a = 0, b = 1, we recover the two-dimensional Euclidean plane, and all four coordinate systems on the twodimensional Euclidean plane are also separable coordinate systems on DII for the Schré odinger, respectively the Helmholtz equation. However, in order to set up a well-deˇned 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 easy to check that all eleven systems for the three-dimensional Euclidean plane which separate the Schré odinger, respectively the Helmholtz equation, also separate the Schréodinger, respectively the Helmholtz equation on D 3d−II . As has been shown in [16], the corresponding quantum motion can be explicitly evaluated by means of the path integral with the energy spectrum E=

2 (p2 + 1). 2m|a|

(4.3)

As is well known, there are four minimally superintegrable potentials in threedimensional Euclidean space and ˇve maximally superintegrable potentials, and it is obvious how to construct maximally superintegrable potentials on D 3d−II . In the forthcoming publication the details will be worked out. Acknowledgements. This work was supported by the HeisenbergÄLandau programme. The authors are grateful to Ernie Kalnins for fruitful and pleasant discussions on superintegrability and separating coordinate systems. C. Grosche would like to

PATH-INTEGRAL APPROACH FOR SUPERINTEGRABLE POTENTIALS 635

thank the organizers of the XII International Conference on Symmetry Methods in Physics, Yerevan, Armenia, July 3Ä8, for the warm hospitality during the stay in Yerevan. G. S. Pogosyan acknowledges the support of the Direcci on General de Asuntos del Personal Acad emico, Universidad Nacional Aut onoma de M exico (DGAPAÄ UNAM) by the grant 102603 Optica Matematica, SEP-CONACYT project 44845 and PROMEP 103.5/05/1705.

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