Quadratic algebra structure in the 5D Kepler system with non-central potentials and Yang–Coulomb monopole interaction

Annals of Physics 380 (2017) 121–134

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Quadratic algebra structure in the 5D Kepler system with non-central potentials and Yang–Coulomb monopole interaction Md Fazlul Hoque a , Ian Marquette a,∗ , Yao-Zhong Zhang a,b a

School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

b

CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

highlights • • • • •

We construct the integrals of the 5D deformed Kepler system with su(2) Yang–Coulomb monopole. We demonstrate the model is superintegrable. We present the symmetry algebra which is a higher rank quadratic algebra. We present an algebraic derivation of the energy spectrum using realizations as deformed oscillator algebras. We show the corresponding Schrödinger equation is multiseparable.

article

abstract

info

Article history: Received 25 October 2016 Accepted 4 March 2017 Available online 18 March 2017 Keywords: Superintegrable Schrödinger equation Quadratic algebras Monopole interaction Oscillator algebras Orthogonal polynomials

We construct the integrals of motion for the 5D deformed Kepler system with non-central potentials in su(2) Yang–Coulomb monopole field. We show that these integrals form a higher rank quadratic algebra Q (3; Lso(4) , T su(2) ) ⊕ so(4), with structure constants involving the Casimir operators of so(4) and su(2) Lie algebras. We realize the quadratic algebra in terms of the deformed oscillator and construct its finite-dimensional unitary representations. This enable us to derive the energy spectrum of the system algebraically. Furthermore we show that the model is multiseparable and allows separation of variables in the hyperspherical and parabolic coordinates. We also show the separability of its 8D dual system (i.e. the 8D singular harmonic oscillator) in the Euler and cylindrical coordinates. © 2017 Elsevier Inc. All rights reserved.

∗

Corresponding author. E-mail addresses: [email protected] (M.F. Hoque), [email protected] (I. Marquette), [email protected] (Y.-Z. Zhang). http://dx.doi.org/10.1016/j.aop.2017.03.003 0003-4916/© 2017 Elsevier Inc. All rights reserved.

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1. Introduction Quantum mechanical systems in Dirac monopole [1] or Yang’s non-abelian su(2) monopole [2] fields have remained a subject of international research interest. In [3] the structural similarity between su(2) gauge system and the 5D Kepler problem was discussed in the scheme of algebraic constraint quantization. Various 5D Kepler systems with su(2) monopole (or Yang–Coulomb monopole) interactions have been investigated by many authors [4–16]. Superintegrability of certain monopole systems has been shown in [17–23]. There exists established connection between 2, 3, 5 and 9-dimensional hydrogen-like atoms and 2, 4, 8 and 16-dimensional harmonic oscillators, respectively [24–29]. In [30] one of the present authors proved the superintegrability of the 5D Kepler system deformed with non-central terms in Yang–Coulomb monopole (YCM) field. This was achieved by relating this system to an 8D singular harmonic oscillator via the Hurwitz transformations [31,32]. An algebraic calculation of the spectrum of the 8D dual system was performed, which enabled the author to deduce the spectrum of the 5D deformed Kepler YCM system via duality between the two models. However, algebraic structure of the 5D deformed Kepler YCM system and a direct derivation of its spectrum via the symmetry algebra have remained an open problem. Such algebraic approach is expected to provide deeper understanding to the degeneracies of the energy spectrum and the connection of the wavefunctions with special functions and orthogonal polynomials. Symmetry algebras generated by integrals of motion of superintegrable systems are in general polynomial algebras with structure constants involving Casimir operators of certain Lie algebras [33–43]. The purpose of this paper is to obtain the algebraic structure in the 5D deformed Kepler YCM system and apply it to give a direct algebraic derivation of the energy spectrum of the model. We construct the integrals of motion and show that they form a higher rank quadratic algebra Q (3; Lso(4) , T su(2) ) ⊕ so(4) with structure constants involving the Casimir operators of so(4) and su(2) Lie algebras. This quadratic algebra structure enables us to give an algebraic derivation of the energy spectrum of the model. Furthermore, we show that the model is separable in the hyperspherical and parabolic coordinates. We also show the separability of its 8D dual system (i.e. the 8D singular harmonic oscillator) in Euler and cylindrical coordinates. 2. Kepler system in Yang–Coulomb monopole field The 5D Kepler–Coulomb system with Yang–Coulomb monopole interaction is defined by the Hamiltonian [4,30]

∂ −ih¯ − h¯ Aaj Ta H = 2 ∂ xj 1



2 +

h¯ 2 2r 2

Tˆ 2 −

c0 r

,

(2.1)

where j = 0, 1, 2, 3, 4 and {Ta , a = 1, 2, 3} are the su(2) gauge group generators which satisfy the commutation relations

[Ta , Tb ] = iϵabc Tc .

(2.2)

Tˆ 2 = T a T a is the Casimir operator of su(2) and Aaj are the components of the monopole potential Aa , which can be written as Aaj =

2i r (r + x0 )

τjka xk .

(2.3)

Here τ a are the 5 × 5 matrices



0 τ 1 = 0 2 0 1

0 0 iσ 1



0 −i σ 1  , 0



0 τ 2 = 0 2 0 1

0 0 −iσ 3



0

iσ  , 0 3



0 τ 3 = 0 2 0 1

0

σ

2

0



0 0 ,

σ2

(2.4)

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which satisfy [τ a , τ b ] = iϵabc τ c and σ i are the Pauli matrices

σ1 =



0 1



1 , 0

σ2 =



−i

0 i



0

,

σ3 =



1 0



0 . −1

(2.5)

The vector potentials Aa are orthogonal to each other, Aa · Ab =

1 r − x0 r 2 r + x0

δab

(2.6)

and to the vector x = (x0 , x1 , x2 , x3 , x4 ). The Kepler–Coulomb model Hamiltonian has the following integrals of motion [2] Ljk = (xj πk − xk πj ) − r 2 h¯ Fjka Ta , Mk =

1 2

(πj Ljk + Ljk πj ) + c0

xk r

,

(2.7)

which satisfy [H , Ljk ] = 0 = [H , Mk ]. Here Fjka = ∂j Aak − ∂k Aaj + ϵabc Abj Ack .

(2.8)

is the Yang–Mills field tensor and πj = −ih¯ ∂∂x − h¯ Aaj Ta which obey the commutation relations j

[πj , xk ] = −ih¯ δjk ,

[πj , πk ] = i h¯ 2 Fjka Ta .

(2.9)

The integrals Ljk and Mk close to the so(6) algebra [44]

[Lij , Lmn ] = ih¯ δim Ljn − ih¯ δjm Lin − ih¯ δin Ljm + ih¯ δjn Lim , [Lij , Mk ] = ih¯ δik Mj − ih¯ δjk Mi ,

[Mi , Mk ] = −2ih¯ HLik .

(2.10)

The so(6) symmetry algebra is very useful in deriving the energy spectrum of the system algebraically. The Casimir operators of so(6) are given by [45,46] Kˆ 1 = Kˆ 3 =

1 2 1 2

Dµν Dµν ,

Kˆ 2 = ϵµνρσ τ λ Dµν Dρσ Dτ λ ,

Dµν Dνρ Dρτ Dτ µ ,

(2.11)

where µ, ν = 0, 1, 2, 3, 4 and

 D=

Ljk (−2H )1/2 Mk

−(−2H )1/2 Mk 0



.

(2.12)

The eigenvalues of the operators Kˆ 1 , Kˆ 2 and Kˆ 3 are [47] K1 = µ1 (µ1 + 4) + µ2 (µ2 + 2) + µ23 , K2 = 48(µ1 + 2)(µ2 + 1)µ3 , K3 = µ21 (µ1 + 4)2 + 6µ1 (µ1 + 4) + µ22 (µ2 + 2)2 + µ43 − 2µ23 ,

(2.13)

respectively, where µ1 , µ2 and µ3 are positive integers or half-integers and µ1 ≥ µ2 ≥ µ3 . One can represent the operators Kˆ 1 , Kˆ 2 , Kˆ 3 in the form [46]

 1/2 ˆK1 = − c0 + 2Tˆ 2 − 4, ˆK2 = 48 − µ0 Tˆ 2 , 2 h¯ 2 H 2 h¯ 2 H Kˆ 3 = K12 + 6K1 − 4K1 Tˆ 2 − 12Tˆ 2 + 6Tˆ 4 .

(2.14)

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Denote by T the eigenvalue of Tˆ 2 . Then, K1 − 2T (T + 1) = µ1 (µ1 + 4),

µ22 (µ2 + 2)2 + µ43 − 2µ23 = 2T 2 (T + 1)2 .

(2.15)

The energy levels of the Yang–Coulomb monopole system are given by c0 T

ϵn = −

2 h¯ 2

n 2

2 , +2

where use has been made of µ1 =

n 2

(2.16)

with n being nonnegative integer.

3. Kepler system with non-central potentials and Yang–Coulomb monopole interaction Let us now consider the 5D Kepler system deformed with non-central potentials in Yang–Coulomb monopole field. The Hamiltonian is given by [2,4,30] H =

1

 −ih¯

2

∂ − h¯ Aaj Ta ∂ xj

2 +

h¯ 2 2r 2

Tˆ 2 −

c0 r

+

c1 r (r + x0 )

+

c2 r (r − x0 )

,

(3.1)

where c1 and c2 are positive real constants. We can construct the integrals of motion of (3.1) by deforming the integrals (2.7) of (2.1), 2rc1 2rc2 A = L˜ 2 + + , (r + x0 ) (r − x0 ) B = Mk +

c1 (r − x0 ) h¯ 2 r (r + x0 )

+

c2 (r + x0 ) h¯ 2 r (r − x0 )

,

(3.2)

where L˜ 2 =



L2jk ,

j, k = 0, 1, 2, 3, 4.

(3.3)

j
It can be checked that these integrals satisfy the commutation relations

[H , A] = 0 = [H , B].

(3.4)

In addition, still some components of the first order integrals of motion Ljk for j, k = 1, 2, 3, 4 of the Hamiltonian (2.1) commute as well as with the Hamiltonian (3.1). So the 5D deformed Kepler YCM system (3.1) is minimally superintegrable as it has 6 algebraically independent integrals of motion including H. Define Lˆ 2 =



L2jk ,

j, k = 1, 2, 3, 4.

(3.5)

j
Lˆ 2 is a Casimir operator of the so(4) Lie algebra and is also a central element of the commutation algebra. The integrals A, B (3.2) have the following differential operator realization:



∂2 ∂ 2r ˆ 2 ∂2 + x x + 4xj T j k 2 ∂ xj ∂ xk ∂ xj r + x0 ∂ xj  2rc1 2rc2 ∂ + 2ir 2 Aaj Ta + + , ∂ xj (r + x0 ) (r − x0 )  ∂2 ∂2 ∂ ∂ 2 B = h¯ x0 2 − xj + i(r − x0 )Aaj Ta −2 ∂ x0 ∂ xj ∂ xj ∂ x0 ∂ xj  (r − x0 ) ˆ 2 c0 x0 c1 (r − x0 ) c2 (r + x0 ) + T + 2 + 2 + 2 . r (r + x0 ) h¯ r h¯ r (r + x0 ) h¯ r (r − x0 ) A = h¯

2

−r 2

(3.6)

These differential expressions are associated with the multiseparability of the Hamiltonian, as will be seen later. However, let us point out that in general systems with monopole interactions are not separable [48].

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4. Algebra structure, unirreps and energy spectrum By direct computation, we can show that the integrals A, B (3.2) and the central elements H, Lˆ 2 , Tˆ 2 close to form the following quadratic algebra Q (3; Lso(4) , T su(2) ),

[A, B] = C ,

(4.1)

[A, C ] = 2{A, B} + 8B − 2c0 (c1 − c2 ) − 4c0 Tˆ ,

(4.2)

[B, C ] = −2B2 + 8HA − 4Lˆ 2 H + (16 − 4c1 − 4c2 )H + 2c02 .

(4.3)

2

The Casimir operator is a cubic combination of the generators, given explicitly by Kˆ = C 2 − 2{A, B2 } − 4B2 − 2{4c0 (c2 − c1 ) − 4c0 Tˆ 2 }B + 8HA2

+ 2{(16 − 8c1 − 8c2 )H − 4Lˆ 2 H + 2c02 }A.

(4.4)

Using the differential realization (3.6) of the integrals A and B, we can show that the Casimir operator (4.4) takes the form, Kˆ = −8H (Tˆ 2 )2 + 16Lˆ 2 H − 8(c1 − c2 )Tˆ 2 H − 2{(c1 − c2 )2

+ 8(2 − c1 − c2 )}H + 4c02 Lˆ 2 + 4c02 (c1 + c2 − 1).

(4.5)

Notice that the first order integrals of motion {Lij , i, j = 1, 2, 3, 4} generate the so(4) algebra

[Lij , Lmn ] = ih¯ (δim Ljn − δjm Lin − δin Ljm + δjn Lim ).

(4.6) so(4)

su(2)

So the full dynamical symmetry algebra of the Hamiltonian (3.1) is a direct sum Q (3; L ,T )⊕ so(4) of the quadratic algebra Q (3; Lso(4) , T su(2) ) and the so(4) Lie algebra, with structure constants involving the Casimir operators Lˆ 2 , Tˆ 2 of so(4), su(2). In order to obtain the energy spectrum of the system, we now construct a realization of the quadratic algebra Q (3; Lso(4) , T su(2) ) in terms of the deformed oscillator algebra of the form [34,49],

[ℵ, bĎ ] = bĎ ,

[ℵ, b] = −b,

bbĎ = Φ (ℵ + 1),

bĎ b = Φ (ℵ).

(4.7)

Here ℵ is the number operator and Φ (x) is well behaved real function satisfying

Φ (0) = 0,

Φ (x) > 0,

∀x > 0.

(4.8)

It is non-trivial to obtain such a realization and to find the structure function Φ (x). After long computations, we get A = (ℵ + u)2 −

9 4

,

B = b(ℵ) + bĎ ρ(ℵ) + ρ(ℵ)b,

(4.9)

where b(ℵ) =

ρ(ℵ) =

c0 (c1 − c2 ) + c0 Tˆ 2

(ℵ + u)2 −

1 4

, 1

3.220 (ℵ + u)(1 + ℵ + u){1 + 2(ℵ + u)2 }

,

(4.10)

and u is a constant to be determined from the constraints on the structure function Φ . Using (4.1)– (4.4), we find

Φ (x; u, H ) = 98304[2c02 + H {1 − 2(x + u)}2 ][4c12 + 4c22 + {1 − 2(x + u)}2 × {4(x + u)(x + u − 1) − 4Lˆ 2 − 3} − 4c1 [2c2 + {1 − 2(x + u)}2

− 4Tˆ 2 ] + 16(Tˆ 2 )2 − 4c2 [{1 − 2(x + u)}2 + 4Tˆ 2 ]].

(4.11)

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A set of appropriate quantum numbers can be defined in same way as in [3,50]. We can use the subalgebra chains so(4) ⊃ so(3) ⊃ so(2) and so(3) ⊃ so(2) for Lˆ 2 and Tˆ 2 respectively. The Casimir 2 operators Jˆ(α) of the related chains can be written as [50] 2 Jˆ(α) =

α 

Jij2 ,

α = 2, 3, 4.

(4.12)

i
We need to use an appropriate Fock space in order to obtain finite-dimensional unitary irreducible representations (unirreps) of Q (3; Lso(4) , T su(2) ). Let |n, E ⟩ ≡ |n, E , l4 , l3 , l2 ⟩ denote the Fock basis 2 states, where l4 , l3 , l2 are quantum numbers of Jˆ(α) , α = 4, 3, 2, respectively. Then the eigenvalues of the Casimir operators Lˆ 2 and Tˆ 2 are h¯ 2 l4 (l4 + 2) and h¯ 2 T (T + 1), respectively. By acting the structure function on the Fock states |n, E ⟩ with ℵ|n, E ⟩ = n|n, E ⟩ and using the eigenvalues of H, Lˆ 2 and Tˆ 2 , we get

      c0 c0 1 1 −√ +√ x+u− Φ ( x ; u, E ) = x + u − 2 2 −2E −2E    1 1 × x + u − (1 + m1 + m2 ) x + u − (1 + m1 − m2 ) 2



2

  1 × x + u − (1 − m1 + m2 ) x + u − (1 − m1 − m2 ) , 1

2

2

(4.13)

where m21 = 1 + 2c1 + h¯ 2 l4 (l4 + 2) + 2 h¯ 2 T (T + 1), m22 = 1 + 2c2 + h¯ 2 l4 (l4 + 2) − 2 h¯ 2 T (T + 1). For the unirreps to be finite dimensional, we impose the following constraints on the structure function (4.13),

Φ (p + 1; u, E ) = 0,

Φ (0; u, E ) = 0,

Φ (x) > 0,

∀x > 0,

(4.14)

where p is a positive integer. These constraints give (p + 1)-dimensional unirreps and their solution gives the energy E and constant u. The energy spectrum is c02 E=−  2 p+1+

 .

m1 +m2 2 2

(4.15)

The total number of degeneracies depends on p + 1 only when the other quantum numbers would be fixed. These quantum numbers do not increase the total number of degeneracies. 5. Separation of variables In this section we show that the Hamiltonian (3.1) is multiseparable and allows separation of variables in the hyperspherical and parabolic coordinates. We also show that the dual system of (3.1) is separable in the Euler and cylindrical coordinates. 5.1. Hyperspherical coordinates We define the hyperspherical coordinates r ∈ [0, ∞), θ ∈ [0, π], α ∈ [0, 2π ), β ∈ [0, π] and

γ ∈ [0, 4π ) in the space R5 by x0 = r cos θ , x1 + ix2 = r sin θ cos x3 + ix4 = r sin θ sin

β 2

β 2

ei(α+γ )/2 , ei(α−γ )/2 .

(5.1)

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In this coordinate system the differential elements of length, volume and the Laplace operator can be expressed [13] as dl2 = dr 2 + r 2 dθ 2 + r4

dV =

∆=

8

r2

sin2 θ (dα 2 + dβ 2 + dγ 2 + 2 cos θ dα dγ ),

4

sin3 θ sin β drdθ dα dβ dγ ,

1 ∂



r4 ∂r

∂ r ∂r 4



∂ + 2 3 ∂θ r sin θ

∂ sin θ ∂θ



1

3

 −

4Lˆ 2 r 2 sin2 θ

,

(5.2)

with Lˆ 2 = L21 + L22 + L23 and

 ∂ ∂ cos α ∂ + sin α − , ∂α ∂β sin β ∂γ   ∂ sin α ∂ ∂ − cos α − , L2 = −i sin α cot β ∂α ∂β sin β ∂γ 

L1 = i cos α cot β

L3 = i

∂ . ∂α

(5.3)

With the help of the identity [5] iAaj

∂ 2 = La , ∂ xj r ( r + x0 )

(5.4)

with L1 = L3 =

i 2 i 2

i

(D41 + D32 ),

L2 =

(D12 + D34 ),

Djk = −xj

2

(D13 + D42 ), ∂ ∂ + xk , ∂ xk ∂ xj

(5.5)

the Schrödinger equation H ψ = E ψ of (3.1) can be written as

 ∆r θ −

Lˆ 2 + c2 h¯ 2 r 2 sin2 θ2

−

Jˆ2 + c1



h¯ 2 r 2 cos2 θ2

ψ+

2  c0  E + ψ = 0, r h¯ 2

(5.6)

where

∆r θ =

1 ∂ r4 ∂r



r4

∂ ∂r

 +

∂ r 2 sin3 θ ∂θ 1



sin3 θ

∂ ∂θ



,

(5.7)

Jˆ2 = Ja Ja with Ja = La + Ta , a = 1, 2, 3. The operators La and Ja satisfy the commutation relations

[La , Lb ] = iϵabc Lc ,

[Ja , Jb ] = iϵabc Jc .

(5.8)

Eq. (5.6) is separable in the hyperspherical coordinates using the eigenfunctions of Lˆ 2 , Tˆ 2 and Jˆ2 with the eigenvalues h¯ 2 L(L + 1), h¯ 2 T (T + 1) and h¯ 2 J (J + 1), respectively. This is seen as follows. We make the separation ansatz [5]

ψ = Φ (r , θ )DJM (α, β, γ , αT , βT , γT ), LTm′ t ′

(5.9)

where

 JM DLTm′ t ′

(α, β, γ , αT , βT , γT ) =

(2L + 1)(2T + 1)  JM CL,m;T ,t 4π 4 M =m+t

× DLmm′ (α, β, γ )DTtt ′ (αT , βT , γT ),

(5.10)

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M.F. Hoque et al. / Annals of Physics 380 (2017) 121–134 JM

and CL,m;T ,t are the Clebsch–Gordon coefficients that arise in angular momentum coupling and appear as the expansion coefficients of total angular momentum eigenstates in uncoupled tensor product j basis [51]; Dmm′ are the su(2) Wigner D-functions of dimension 2j + 1 with j = 0, 1/2, 1, 3/2, 2, . . . and m = −j, −j + 1, . . . , j [52]. Substituting the ansatz into (5.6) leads to the differential equation for the function Φ (r , θ ) L(L + 1) +

 ∆r θ −

c2 h¯ 2

r 2 sin2 θ2

−

J (J + 1) +

c1 h¯ 2

r 2 cos2 θ2

+

2  h¯ 2



E+

c0  r

Φ (r , θ ) = 0.

(5.11)

Setting the function Φ (r , θ ) = R(r )F (θ ), (5.11) is separated into the ordinary differential equations





d2



dθ 2



d2 dr 2

+ 3 cot θ

+

4 d r dr

2 L( L + 1 ) +

d

−

dθ

+

2  h¯ 2

c2



h¯ 2

−

1 − cos θ c0 

E+

r

−

Λ



r2



2 J (J + 1) + 1 + cos θ

c1 h¯ 2



 + Λ F (θ ) = 0,

R(r ) = 0,

(5.12)

(5.13)

where Λ is the separation constant. Solutions of (5.12) and (5.13) in terms of the Jacobi and confluent hypergeometric polynomials [53] are as follows (δ +L,δ1 +J )

F (θ ) = FλJL (θ; δ1 , δ2 )(1 + cos θ )(δ1 +J )/2 (1 − cos θ )(δ2 +L)/2 Pλ−2J −L R(r ) = Rnλ (r : δ1 , δ2 )e

− κr 2

(κ r )λ+

δ1 +δ2 2

1 F1

(cos θ ),

(−n, 4 + 2λ + δ1 + δ2 ; κ r ),

(5.14)

where

   δ1 + δ2 δ1 + δ2 Λ= λ+ λ+ +3 , 2

 δ1 = −1 +

4c1 h¯ 2

 δ2 = −1 + −n = −

2

4c2 h¯ 2

2c0 h¯ κ 2

+

λ ∈ N,

+ (2J + 1)2 − J , + (2L + 1)2 − L,

δ1 + δ2 2

+ λ + 2,

E=

−κ 2 h¯ 2 8

.

(5.15)

Hence the energy spectrum c02

E=−



2 h¯ 2 n + λ + 2 +

δ1 +δ2

2 .

(5.16)

2

This physical spectrum coincides with (4.15) obtained from algebraic derivation by the identification p = n + λ + 1, δ1 = m1 , δ2 = m2 and h¯ = 1. 5.2. Parabolic coordinates The parabolic coordinates are defined by x0 =

1 2

(µ − ν),

x1 + ix2 = x3 + ix4 =

β √ µν cos ei(α+γ )/2 , 2

√

µν sin

β 2

ei(α−γ )/2 ,

(5.17)

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with µ, ν ∈ [0, ∞). The differential elements of length, volume and Laplace operator in this coordinates can be expressed as 2

dl =

µ+ν



d µ2

µ

4

µν

+

du2



ν

+

µν

(dα 2 + dβ 2 + dγ 2 + 2 cos θ dα dγ ),

4

(µ + ν) sin β dµdν dα dβ dγ ,      4 1 ∂ ∂ 1 ∂ ∂ 4 2 µ2 + ν2 − ∆= Lˆ . µ + ν µ ∂µ ∂µ ν ∂ν ∂ν µν dV =

32

(5.18)

The Schrödinger equation H ψ = E ψ of the Hamiltonian (3.1) in this coordinates becomes

 ∆µν −

4(Jˆ2 + c1 ) h¯ 2 µ(µ + ν)

−

4(Lˆ 2 + c2 ) h¯ 2 ν(ν + µ)

+



2

E+

h¯ 2



c0

µ+ν

ψ = 0,

(5.19)

where

∆µν =



4

µ+ν

1 ∂

∂ µ µ ∂µ ∂µ 

2

1 ∂

 +



ν ∂ν

∂ ν ∂ν 2



.

(5.20)

Making the ansatz of the form [9]

ψ = f1 (µ)f2 (ν)DJM (α, β, γ , αT , βT , γT ), LTm′ t ′

(5.21)

in (5.19), the wave functions are separated and lead to the following ordinary differential equations 1 d



µ dµ

µ

2





df1

+

dµ

E 2 h¯ 2

µ−

J (J + 1) +

c1 h¯ 2

µ

   L(L + 1) + E 2 df2 ν + ν− 2 ν dν dν ν 2 h¯ 1 d

+

c2 h¯ 2

+

c0 2 h¯ 2 c0

˜ f1 = 0, + Λ 2

h¯

2 h¯ 2



h¯

(5.22)



˜ f2 = 0, − Λ 2

(5.23)

˜ is the separation constant. Solutions of (5.22) and (5.23) are given by the confluent where Λ hypergeometric polynomials [53], κµ f1 = e− 2 (κµ)δ1 +J F (−n1 , δ1 + J + 2, κµ), κν f2 = e− 2 (κν)δ2 +L F (−n2 , δ2 + L + 2, κν),

(5.24)

where

 δ1 = −1 +

h¯ 2

 δ2 = −1 + −n 1 = −n 2 =

1 2 1 2

4c1

4c2 h¯ 2

+ (2J + 1)2 − J , + (2L + 1)2 − L,

(δ1 + J ) + 1 − (δ2 + L) + 1 +

h¯ 2κ h¯ 2κ

˜ − Λ ˜ − Λ

c0 2κ h¯ 2 c0 2κ h¯

,

, 2

E=

− h¯ 2 κ 2 2

.

(5.25)

Set n1 + n2 =

c0

κ h¯

1

2

− (δ1 + δ2 + J + L) − 2. 2

(5.26)

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M.F. Hoque et al. / Annals of Physics 380 (2017) 121–134

The energy spectrum c02

E=− 2 h¯

 2

2 .

(5.27)

n1 + n2 + 12 (δ1 + δ2 + J + L) + 2

Making the identification p = n1 + n2 + becomes (4.15).

J +L 2

+ 1, δ1 = m1 , δ2 = m2 and h¯ = 1, the energy spectrum

5.3. Euler spherical coordinates The 5D Kepler system with non-central terms and Yang–Coulomb monopole is dual to the 8D singular oscillator via Hurwitz transformation [30]. The symmetry algebra structure and energy spectrum of the 8D singular oscillator have been studied in [30]. In this and next subsections we show the separability of this dual system in the Euler spherical and cylindrical coordinates, and compare our results for the spectrum with those obtained in [30]. The Hamiltonian of the 8D singular oscillator reads 7 h¯ 2  ∂ 2

7 ω2 

λ1

λ2

u2i + 2 + 2 . 2 i=0 ∂ u2i 2 i =0 u0 + u21 + u22 + u23 u4 + u25 + u26 + u27 In the Euler 8D spherical coordinates [14] H =−

θ

u0 + iu1 = u cos

2

θ

u2 + iu3 = u cos

2

θ

u4 + iu5 = u sin

2

θ

+

sin cos

βT 2

βT 2

βK

sin

2

e −i ei ei

(αT −γT ) 2

(αT +γT ) 2

(αK −γK ) 2

βK

,

, ,

(αK +γK )

e −i 2 , 2 where 0 ≤ u < ∞, 0 ≤ θ ≤ π , we have u6 + iu7 = u sin

2

cos

(5.28)

(5.29)

    7  1 ∂ ∂ 4 ∂ 4 4 ∂2 7 ∂ 3 = u + sin θ − 2 Tˆ 2 − Kˆ 2 , (5.30) θ 2 3 2 θ 7 2 2 2 u ∂ u ∂ u ∂θ ∂θ ∂ u u sin θ u cos u sin i i =0 2 2 with

 ∂ ∂2 ∂2 − 2 cos β + , T ∂αT ∂γT ∂αT2 ∂γT2    2 1 ∂ ∂2 ∂2 ˆK 2 = − ∂ + cot βK ∂ + . − 2 cos βK + ∂βK ∂αK ∂γK ∂βK2 ∂γK2 sin2 βK ∂αK2 Tˆ 2 = −



∂2 ∂ 1 + cot βT + 2 ∂βT ∂βT sin2 βT



(5.31)

(5.32)

For the separation of the wavefunctions in the form

ψ = R(u)F (θ )DTtt ′ (αT , βT , γT )DKkk′ (αK , βK , γK ),

(5.33)

Tˆ 2 DTtt ′ = T (T + 1)DTtt ′ ,

(5.34)

with Kˆ 2 DKkk′ = K (K + 1)DKkk′ ,

the Schrödinger equation H Ψ = ϵ Ψ leads to the ordinary differential equations



d2 du2



1

+

7 d

−

u du d

sin3 θ dθ



Γ u2

sin θ 3

+ d dθ

2ϵ h¯ 2

−

 −

ω2 h¯ 2



u2 R(u) = 0,

T (T + 1) + cos2 θ2

λ1 2 h¯ 2

−

(5.35) K (K + 1) + sin2 θ2

λ2 2 h¯ 2

+

Γ 4

 F (θ ) = 0,

(5.36)

M.F. Hoque et al. / Annals of Physics 380 (2017) 121–134

131

where Γ is the separation of constant. The solution of (5.36) in terms of Jacobi polynomials [53] is as follows (δ +K ,δ1 +T )

F (θ ) = FλTK (θ; δ1 , δ2 )(1 + cos θ )(δ1 +T )/2 (1 − cos θ )(δ2 +K )/2 Pλ−2T −K

(cos θ ),

(5.37)

where

   δ1 + δ2 δ1 + δ2 λ+ +3 , Γ =4 λ+ 2

 δ1 = −1 +

2λ1 h¯ 2

 δ2 = −1 +

2λ2 h¯ 2

2

λ ∈ N,

+ (2T + 1)2 − T , + (2K + 1)2 − K .

(5.38)

The solution of (5.35) in terms of the confluent hypergeometric functions [53] R(u) = Rnλ (u; δ1 , δ2 )e−

κ u2

(κ u2 )λ+

2

δ1 +δ2 2

1 F1

(−n, 4 + 2λ + δ1 + δ2 ; κ u2 ),

(5.39)

where

−n=

δ1 + δ2 2

ϵ

+λ+2−

2κ h¯ 2

,

ω2 = κ 2 h¯ 2 .

(5.40)

Hence



ϵ = 2 h¯ κ n + 2

δ1 + δ2



+λ+2 .

2

(5.41)

Using the relations between the parameters of the generalized Yang–Coulomb monopole and the 8D singular oscillator, c0 =

ϵ 4

,

E=

−ω2 8

,

2ci = λi ,

i = 1, 2,

(5.42)

we obtain E=−

c02



2 h¯ 2 n + λ + 2 +

δ1 +δ2

2 ,

(5.43)

2

which coincides with (5.16). 5.4. Cylindrical coordinates Consider the 8D cylindrical coordinates u0 + iu1 = ρ1 sin u2 + iu3 = ρ1 cos u4 + iu5 = ρ2 sin u6 + iu7 = ρ2 cos

βT 2

βT 2

βK 2

βK 2

e −i ei ei

(αT −γT ) 2

(αT +γT ) 2

(αK −γK )

e−i

2

,

, ,

(αK +γK ) 2

,

(5.44)

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M.F. Hoque et al. / Annals of Physics 380 (2017) 121–134

where 0 ≤ ρ1 , ρ2 < ∞. Then we have in this coordinate system,

    7  ∂2 1 ∂ 1 ∂ 4 4 3 ∂ 3 ∂ = 3 ρ1 + 3 ρ2 − 2 Tˆ 2 − 2 Kˆ 2 , 2 ∂ρ ∂ρ ∂ρ ∂ρ ∂ u ρ ρ ρ ρ 1 1 2 2 i 1 2 1 2 i =0   2  2 2 ∂ ∂ ∂ 1 ∂ Tˆ 2 = − + + cot βT 2 − 2 cos βT , 2 2 ∂α ∂γ ∂βT2 ∂α ∂γ sin βT T T T T   2  1 ∂ ∂2 ∂2 ∂ + Kˆ 2 = − + cot β − 2 cos β . K K ∂αK ∂γK ∂βK2 ∂γK2 sin2 βK ∂αK2

(5.45)

(5.46)

(5.47)

Making the ansatz

ψ = f1 f2 DTtt ′ (αT , βT , γT )DKkk′ (αK , βK , γK ) (5.48)  and change of variables xi = a2 ρi2 , a = ωh¯ , the Schrödinger equation H ψ = ϵψ is converted to x1

x2

d2 f1

+2

dx21 d2 f2

+2

dx22

df1 dx1 df2 dx2

 −

λ1

T (T + 1) + x1

 −

K (K + 1) +

λ2 2 h¯ 2

x2 δi +zi

xi where ϵ1 + ϵ2 = ϵ . Setting fi = e− 2 xi

xi

d 2 W ( xi ) dx2i

where αi =

γi 2

−

+ (γi − xi ) ϵi

2h¯ ω

 δi = −1 +

dW (xi ) dxi

2

 ϵ1 + − f1 = 0, 4 2h¯ ω  x2 ϵ2 + − f2 = 0 4 2h¯ ω x1

2 h¯ 2

(5.49)

(5.50)

W (xi ), i = 1, 2, then (5.49) and (5.50) become

− αi W (xi ) = 0,

(5.51)

, γi = δi + zi + 2, z1 = T , z2 = K and

2λi h¯ 2

+ (2zi + 1)2 − zi ,

i = 1, 2.

(5.52)

Solutions of (5.51) in terms of confluent hypergeometric polynomials [53] are given by fni ,γi = (aρi )γi −2 e−

a2 ρ 2 i 2

1 F1

(−ni , γi , a2 ρi2 ),

(5.53)

where ni =

1 ϵi − (δi + zi + 2), 2h¯ ω 2

i = 1, 2.

(5.54)

We thus found the energy spectrum of the 8D singular oscillator



ϵ = 2h¯ ω n1 + n2 +

δ1 + δ2 2

+

T +K 2



+2 .

(5.55)

Using the relations (5.42) between the parameters of the generalized Yang–Coulomb monopole and the 8D singular oscillator, we obtain E=−

c02



2 h¯ 2 n1 + n2 +

δ1 +δ2 2

+

T +K 2

2 , +2

(5.56)

which coincides with (4.15) by making the identification p = n1 + n2 + T +2 K + 1, δ1 = m1 , δ2 = m2 and h¯ = 1.

M.F. Hoque et al. / Annals of Physics 380 (2017) 121–134

133

6. Conclusion One of the results of this paper is the determination of the higher rank quadratic algebra structure Q (3; Lso(4) , T su(2) ) ⊕ so(4) in the 5D deformed Kepler system with non-central terms and Yang–Coulomb monopole interaction. The structure constants of the quadratic algebra Q (3; Lso(4) , T su(2) ) contain Casimir operators of the so(4) and su(2) Lie algebras. The realization of this algebra in terms of deformed oscillator enable us to provide the finite dimensional unitary representations and the degeneracy of the energy spectrum of the model. We also connected these results with method of separation of variables, solution in terms of orthogonal polynomials and the dual under Hurwitz transformation which is a 8D singular oscillator. It would be interesting to extend these results to systems in curved spaces, in particular to the 8D pseudospherical (Higgs) oscillator and its dual system involving monopole [16]. Moreover, higher dimensional superintegrable models with monopole interactions are still relatively unexplored area. One of the open problems is to construct the non-central deformations of the known higherdimensional models [54,55] and their symmetry algebras. Acknowledgments The research of FH was supported by International Postgraduate Research Scholarship and Australian Postgraduate Award. IM was supported by the Australian Research Council through a Discovery Early Career Researcher Award DE 130101067. YZZ was partially supported by the Australian Research Council, Discovery Project DP 140101492. He would like to thank the Institute of Theoretical Physics, Chinese Academy of Sciences, for hospitality and support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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