Group-theoretical approach to a non-central extension of the Kepler-Coulomb problem
aa r X i v : . [ m a t h - ph ] M a y Group-theoretical approach to a non-centralextension of the Kepler-Coulomb problem
G. A. Kerimov and A. Ventura , Physics Department, Trakya University, 22030 Edirne, Turkey ENEA, Centro Ricerche Ezio Clementel, Bologna, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Italy
November 17, 2018
Abstract
Bound and scattering states of a non-central extension of the three-dimensional Kepler-Coulomb Hamiltonian are worked out analyticallywithin the framework of the potential groups of the problem, SO (7) forbound states and SO (6 ,
1) for scattering states. In the latter case, the S matrix is calculated by the method of intertwining operators. In classical mechanics, the reduced Kepler problem has been known for morethan two centuries [1] to admit seven integrals of motion. These are the totalangular momentum, the Laplace-Runge-Lenz (or Hermann-Bernoulli-Laplace )vector and the total energy. Since there are two relationships between them( see, for example, [2] ) only five of integrals of motion are independent. Ingeneral, a closed system with N degrees of freedom can have at most 2 N − N integrals of motion (including theHamiltonian ) that are independent and in involution ( i.e. Poisson bracketsof any two integrals are zero). The system is called superintegrable if thereexist q , 1 ≤ q ≤ N −
1, additional independent integrals of motion. The cases q = 1 and q = N − SO (4) in the subspace of negative energies and the Lie algebra of SO (3 ,
1) in the subspace of positive energies. It was realized that the ‘acciden-tal’ degeneracies, i.e. degeneracies not connected with geometrical SO (3) sym-metries of the Hamiltonian, are due to the invariance group SO (4). Moreover,the separation of variables in parabolic coordinates was related to Laplace-Runge-Lenz vector[11]. Later on, Zwanziger [12] showed that the algebra of SO (3 ,
1) may be used to calculate the Coulomb phase-shifts. Ever since, in-variance algebras have been determined for many quantum mechanical systems.The best known of these systems are the oscillator [13] and the MICZ-Keplersystem[14, 15]. This is a situation in which the Hamiltonian H of the systembelongs to the centre of the enveloping algebra of some group G , i.e. H = f ( C ) , (1)where C is the Casimir operator of the invariance group G . For example, inthe Coulomb bound-state problem, H = − γ / C + 1), where C is a Casimiroperator of SO (4).But it could happen that the Hamiltonian H ν can be related to the Casimiroperator C as H ν = f ( C ) | H ν , (2)where H ν a subspace occurring in the subgroup reduction and | H ν denotes therestriction to H ν . In this case the group G describes the same energy statesof a family of Hamiltonians H ν with different potential strength. (This is whythe present group G designated potential group [16].) Such an approach wasproposed by Ghirardi [17], who worked it out in detail for the Scarf potential[18]. It is similar to the approach of Olshanetsky and Perelomov [19, 20], wherequantum integrable systems are related to radial part of the Laplace operatoron homogeneous spaces ( i.e. to radial part of Casimir operator of second order ) of Lie groups.Ref. [21] proposed a method that permits purely algebraic calculations of S -matrices for the systems whose Hamiltonians are related to the Casimir opera-tors C of some Lie group G as (1) or (2). Namely, the S -matrices for the systemsunder consideration are associated with intertwining operators A between Weylequivalent representations U χ and U ∼ χ of G as S = A (3)or S = A | H ν (4)2espectively. ( The representations U and U ∼ χ have the same Casimir eigenvalues.Such representations are called Weyl equivalent.) At this stage we note that theoperator A is said to intertwine the representations U χ and U ∼ χ of the group G if the relation AU χ ( g ) = U ∼ χ ( g ) A for all g ∈ G (5)or AdU χ ( b ) = dU ∼ χ ( b ) A for all b ∈ g (6)holds, where dU χ and dU e χ are the corresponding representations of the algebra g of G . Equations (5) and (6) have much restriction power, determining theintertwining operator up to a constant.The potential group approach has been proven to be useful in variety prob-lems in one dimension. Recently, it has been used to describe some potentials[22, 23, 24, 25] classified in [6] . In Ref. [23] it has been shown that the superpo-sition of the Coulomb potential with one barrier term [6] could be related to thepotential group SO (5). Scattering amplitudes for such system are worked outin detail in Ref.[24] by using an intertwining operator [21] between two Weyl-equivalent unitary irreducible representations of the SO (5 ,
1) potential group.Subject of the present work will be the simultaneous description of boundand scattering states of a quantum mechanical system with Hamiltonian H = − ∇ − γr + s − / x + s − / y + s − / z (7)written in units ~ = m = 1, where s i = 0 , , , . . . . We show that H = − γ C + ) (cid:12)(cid:12) H s s s (8)where C is a Casimir operator of SO (7) (for bound states) or SO (6 ,
1) (forscattering states).This system was proved to be minimally superintegrable [6], since four in-tegrals of motions were explicitly derived, as a consequence of the separabilityof the related Schr˝odinger equation in two coordinate systems. But in Ref. [26]it has been shown that the classical counterpart of Hamiltonian (7) is maxi-mally superintegrable, i.e. it admits five independent integrals of motion, in-cluding the Hamiltonian: four of them derive from separability of the relatedHamilton-Jacobi equation in different coordinate systems, but the fifth integral,first discussed in Ref.[26], is not connected with separability. Moreover, thislast integral is quartic in the momenta, while the other three are quadratic, andhas been rederived in Ref.[27] as an example of application of a more generaltechnique.
Let us start the discussion with the fact that the generators of UIR of SO (7)( or SO (6 , M µν = − M νµ ( µ, ν =3 , , . . . ,
7) which obey the commutation relations[ M µν , M σλ ] = i ( g µσ M νλ + g νλ M µσ − g µλ M νσ − g νσ M µλ ) (9)where g µν = (+ , + , . . . , + , +) for SO (7) (10) g µν = (+ , + , . . . , + , − ) for SO (6 , C = 12 X µ,ν =1 M νµ M µν , (11)It is well-known that the most degenerate representation of algebra so (7)( so (6 ,
1) ) can be realized in the Hilbert space spanned by negative-energy(positive-energy) states corresponding to fixed eigenvalue of the Coulomb Hamil-tonian H Coul in six dimensions H Coul = 12 p − γ √ x , γ > x = ( x , x , . . . , x ) ∈ R , p j = − i ∂∂x j , ( j = 1 , ..., x = P i =1 x i x i , p = P i =1 p i p i . (We are using units with M = ~ = 1.) However, in order to beable to write the relation (2) we introduce the following realization M ij = λ ( x ) ◦ ( x i p j − x j p i ) ◦ λ − ( x ) (13) M i = − M i = | h | − λ ( x ) ◦ (cid:20) x i p − p i ( x · p ) + i p i − γx i x (cid:21) ◦ λ − ( x ) , ( i, j = 1 , ..., λ ( x ) = (cid:2)(cid:0) x + x (cid:1) (cid:0) x + x (cid:1) (cid:0) x + x (cid:1)(cid:3) / (15)and h = λ ( x ) ◦ (cid:18) p − γ √ x (cid:19) ◦ λ − ( x ) . (16)The generators (13-14 ) act in the eigenspace of h equipped with the scalarproduct ( φ , φ ) = Z R φ ∗ ( x ) φ ( x ) dµ ( x ) , x ∈ R (17)where dµ ( x ) = λ − ( x ) dx dx · · · dx . H Coul in six dimen-sions. The unitary mapping W which realizes the equivalence is given by W : Ψ Coul → Φ = λ ( x ) Ψ Coul (18)The operators (13-14) provide most degenerate representations of SO (7) if h is negative definite and of SO (6 ,
1) if h is positive definite. More precisely,they define the most degenerate (symmetric) UIR of SO (7) specified by theinteger number j = 0 , , . . . (when h is negative definite) and the most degenerateprincipal series representations of SO (6 ,
1) labelled by the complex number j = − + iρ, ρ > h is positive definite). If we compute the second-order Casimir operator (11), it becomes C = − − γ h (19)Let us consider the reduction corresponding to the group chain G ⊃ SO (6) ⊃ SO (4) × SO (2) ⊃ SO (2) × SO (2) × SO (2) , where G is SO (6 ,
1) or SO (7).Then, the basis functions can be characterized by the Casimir operators of thechain of groups C | j ; lM i = j ( j + 5) | j ; lM i (20) C SO (6) | j ; lM i = l ( l + 4) | j ; lM i C SO (4) | j ; lM i = m ( m + 2) | j ; lM i C SO (2) | j ; lM i = s | j ; lM i C SO (2) | j ; lM i = s | j ; lM i C SO (2) | j ; lM i = s | j ; lM i where M is a collective index ( m, s , s , s ) and C SO (6) = 12 X i,j =1 M ij , C SO (4) = 12 X i,j =1 M ij , C SO (2) = M , C SO (2) = M , C SO (2) = M (21)According to this, we introduce in place of x , x , . . . , x the variables r, θ, ϕ, α , α , α via x i = rn i with n = sin θ sin ϕ sin α , n = sin θ sin ϕ cos α n = sin θ cos ϕ sin α , n = sin θ cos ϕ cos α n = cos θ sin α , n = cos θ cos α (22)where 0 ≤ r < ∞ , ≤ θ, ϕ ≤ π and 0 ≤ α , α , α ≤ π . If we compute theoperator γ / (cid:0) C + (cid:1) for this parametrization, it becomes γ C + = ∂ ∂r + 2 r ∂∂r + 1 r (cid:18) θ ∂∂θ sin θ ∂∂θ + 1sin θ ∂ ∂ϕ (cid:19) (23)+ 1 r sin θ sin ϕ (cid:18)
14 + ∂ ∂α (cid:19) + 1 r sin θ cos ϕ (cid:18)
14 + ∂ ∂α (cid:19) + 1 r cos θ (cid:18)
14 + ∂ ∂α (cid:19) H s s s be a subspace spanned by | j ; lM i with fixed s , s and s . Thus,the operator (23) restricted to this subspace becomes a differential operator in r , θ and ϕ ; it turns out that γ C + (cid:12)(cid:12)(cid:12)(cid:12) H s s s = ∂ ∂r + 2 r ∂∂r + 1 r (cid:18) θ ∂∂θ sin θ ∂∂θ + 1sin θ ∂ ∂ϕ (cid:19) (24)+ 1 / − s r sin θ sin ϕ + 1 / − s r sin θ cos ϕ + 1 / − s r cos θ with s i = 0 , ± , ± . . . , where we have used that C SO (2) i = − ∂ ∂α i , i = 1 , , H = − ∇ − γr + s − / r sin θ sin ϕ (25)+ s − / r sin θ cos ϕ + s − / r cos θ can be described in terms of the potential groups SO (7) and SO (6 ,
1) since H = − γ C + (cid:12)(cid:12)(cid:12)(cid:12) H s s s , as mentioned in Section 1 (formula (8)). (Due to the symmetry s i → − s i inthe Hamiltonian (25), without loss of generality, we may assume that s , s and s are non-negative integers.) Note that the SO (2) subgroups are related topotential strength.At this point, it is worthwhile pointing out that Hamiltonian (25) does notcontain the pure Coulomb potential as a particular case, within the frameworkof the SO (7) and SO (6 ,
1) symmetries considered in the present work. Inorder to restore it, it is necessary to resort to larger symmetry groups, forexample, SO (10) and SO (9 ,
1) and use the decomposition chain G ⊃ SO (9) ⊃ SO (6) × SO (3) ⊃ SO (3) × SO (3) × SO (3) , where now the SO (3) subgroupsare related to potential strength.Here again, use is made of polar coordinates x = (sin θ sin ϕe , sin θ cos ϕe , cos θe )where x ∈ R , e i = (sin α i sin β i , sin α i cos β i , cos α i ) , i = 1 , ,
3. Then, a proce-dure similar to that described above would lead to the Hamiltonian H = − ∇ − γr + l ( l + 1)2 r sin θ sin ϕ + l ( l + 1)2 r sin θ cos ϕ + l ( l + 1)2 r cos θ l i ( i = 1 , ,
3) are integer and are allowed to take the null value, thusrestoring the pure Coulomb potential.Finally, we note that the operators I = L + s − sin θ sin ϕ + s − sin θ cos ϕ + s − cos θ (26) I = L z + s − sin ϕ + s − cos ϕ where L and L z are the square of “angular momentum” and of its projectionon the third axis, commute with the Hamiltonian. These integrals of motionare related to the Casimir operators of SO (6) and its SO (4) subgroup in thesense that I = C SO (6) (cid:12)(cid:12)(cid:12) H s s s , I = C SO (4) (cid:12)(cid:12)(cid:12) H s s s . (27)where C SO (6) = − (cid:18) θ ∂∂θ sin θ ∂∂θ + 1sin θ ∂ ∂ϕ (cid:19) − θ sin ϕ (cid:18)
14 + ∂ ∂α (cid:19) − θ cos ϕ (cid:18)
14 + ∂ ∂α (cid:19) − θ (cid:18)
14 + ∂ ∂α (cid:19) and C SO (4) = − ∂ ∂ϕ − ϕ (cid:18)
14 + ∂ ∂α (cid:19) − ϕ (cid:18)
14 + ∂ ∂α (cid:19) The bound state spectrum can now be easily obtained if we note that the eigen-value of the Casimir operator C of the potential groups SO (7) is j ( j + 5). Wethen find E = − γ (cid:0) j + (cid:1) (28)where j = s + s + s + 2 k + 2 k + n, ( k , k , n = 0 , , , . . . ). It is easy tocheck that states (28) have degeneracy d ( d +1)2 , where d = (cid:2) j − s − s − s (cid:3) + 1, and[ q ] is the largest integer less than or equal to q .We give for reference the expression of the bound-state wave functions ψ ( x ) = R jl ( r ) Y lM ( θ, ϕ ) , (29)where R jl ( r ) is the radial part of the wave function, while Y lM ( θ, ϕ ) is theangular part of it : R jl ( r ) = cu l + e − u L l +4 n ( u ) , u = 2 γr/ (cid:18) j + 52 (cid:19) (30)7ith n = j − l ( n = 0 , , , . . . ), c = (cid:18) γj + (cid:19) " Γ ( j − l + 1)2 (cid:0) j + (cid:1) Γ ( j + l + 5) (31)and Y lM ( θ, ϕ ) = χ sin m +1 θ cos s + θ sin s + ϕ cos s + ϕ × P ( m +1 ,s ) k (cos 2 θ ) P ( s ,s ) k (cos 2 ϕ ) (32)with 2 k = l − m − s , k = m − s − s ( k , k = 0 , , , , . . . ) and χ = " Γ (cid:0) ( l + m + s + 4) (cid:1) Γ (cid:0) ( l − m − s + 2) (cid:1) Γ (cid:0) ( m + s + s + 2) (cid:1) Γ (cid:0) ( l + m − s + 4) (cid:1) Γ (cid:0) ( l − m + s + 2) (cid:1) Γ (cid:0) ( m + s − s + 2) (cid:1) × " Γ (cid:0) ( m − s − s + 2) (cid:1) Γ (cid:0) ( m − s + s + 2) (cid:1) (2 l + 4) (2 m + 2) (33)Here, L αn and P ( α,β ) n are Laguerre and Jacobi polynomials, respectively.It is worth noting that (see Appendix ) the Y lM functions are related to 5-dimensional spherical harmonics Y lM ( n ) (see Section 10.5 of [28]) in polyspher-ical coordinates, while R jl is related to the radial part of the 6-dimensionalCoulomb wave function [29] as R jl ( r ) = r R Couljl ( r ) Once the group structure of the problem has been recognized, the associated S matrix can be computed by using Eqs. (3-6). This requires knowledge of ma-trices h l ′ M ′ | A | lM i that intertwine Weyl-equivalent representations of SO (6 , SO (6 , ⊃ SO (6) ⊃ SO (4) × SO (2) ⊃ SO (2) × SO (2) × SO (2) reduction. One has (see Appendix) h l ′ M ′ | A | lM i = A l δ ll ′ δ MM ′ , (34)where A l = Γ (cid:0) + iρ + l (cid:1) Γ (cid:0) − iρ + l (cid:1) . (35)According to this, we have S ( θ, ϕ ; θ ′ , ϕ ′ ) = X lM A l Y lM ( θ, ϕ ) Y ∗ lM ( θ ′ , ϕ ′ ) . (36)Thus, the scattering amplitude, f ( θ, ϕ ; θ ′ , ϕ ′ ), is defined by f ( θ, ϕ ; θ ′ , ϕ ′ ) = 2 πip X lM ( A l − Y lM ( θ, ϕ ) Y ∗ lM ( θ ′ , ϕ ′ ) . (37)8ince X lM Y lM ( θ, ϕ ) Y ∗ lM ( θ ′ , ϕ ′ ) = δ (cos θ − cos θ ′ ) δ ( ϕ − ϕ ′ )we can omit unity in the brackets of formula (37) when θ = θ ′ , ϕ = ϕ ′ , leaving f ( θ, ϕ ; θ ′ , ϕ ′ ) = 2 πip X lM A l Y lM ( θ, ϕ ) Y ∗ lM ( θ ′ , ϕ ′ ) . (38)Moreover, formulas (49),(52) and (48) imply the following integral represen-tation of the scattering amplitude f ( θ, ϕ ; θ ′ , ϕ ′ ) = 2 πip η √ b π Z π Z π Z (1 − a sin θ sin θ ′ − cos θ cos θ ′ cos α ) − − iρ × exp ( − is α − is α − is α ) dα dα dα (39)where a = sin ϕ sin ϕ ′ cos α + cos ϕ cos ϕ ′ cos α (40)and b = sin θ sin θ ′ sin ϕ sin ϕ ′ (41) We have shown in the present work, based on the potential group approach,how a non-central extension of the Coulomb Hamiltonian, considered in theliterature as an example of maximal superintegrability, can be worked out in afully analytic way, with bound states described by most degenerate represen-tations of SO (7) and scattering states by most degenerate representations of SO (6 , SO (10) and SO (9 , U (4) symmetry has been discussed inRef.[22], while a non-central extension of the null potential with E (4) symme-try has been worked out in Ref.[25]. Other cases of physical interest with morecomplicated symmetries will be considered for future work. A Here we calculate the matrix elements of A which intertwine Weyl-equivalentrepresentations of SO (6 ,
1) or so (6 ,
1) in the bases corresponding to SO (6 , ⊃ O (6) ⊃ SO (4) × SO (2) ⊃ SO (2) × SO (2) × SO (2) reduction. We find itexpedient to use, for this purpose, equation (6).We shall start with the fact that the most degenerate principal series repre-sentations of SO (6 ,
1) can be realized on L (cid:0) S (cid:1) (see Section 9.2.1 of [28]) U j ( g ) f ( n ) = ( ω g ) j f ( n g ) , n ∈ S (42)where ω g = X i =1 g − i n i + g , ( n g ) k = P i =1 g − ki n i + g k P i =1 g − i n i + g The operator A defined by( Af ) ( n ) = Z K ( n, n ′ ) f ( n ′ ) dn ′ (43)intertwines representations j and − − j , if K (cid:0) n g , n ′ g (cid:1) = ( ω g ) j (cid:0) ω ′ g (cid:1) j K ( n, n ′ ) . (44)The kernel, K , is uniquely determined by Eq. (44) up to a constant and isgiven by K ( n, n ′ ) = η (1 − n · n ′ ) − − j . (45)with η = 2 − + iρ Γ (cid:0) + iρ (cid:1) π Γ ( − iρ ) (46)With this factor the operator A becomes unitary for j = − + iρ (see equation(50) ).Taking into account the fact that 5-dimensional spherical harmonics Y lM ofdegree l [28] forms a bases in L (cid:0) S (cid:1) , corresponding to above reduction, wehave the following integral representation for the matrix elements of A h l ′ M ′ | A | lM i = Z K ( n, n ′ ) Y ∗ l ′ M ′ ( n ′ ) Y lM ( n ) dndn ′ . (47)where dn = sin θ cos θ sin ϕ cos ϕdθdϕdα dα dα for n as in (22) and Y lM ( n ) = Y lM ( θ, ϕ ) Y j =1 √ π e is j α j . (48)By using the expansion η (1 − n · n ′ ) − − iρ = 12 π ∞ X v =0 ( ν + 2) Γ (cid:0) + iρ + ν (cid:1) Γ (cid:0) − iρ + ν (cid:1) C v ( n · n ′ ) , (49)we have h l ′ M ′ | A | lM i = A l δ ll ′ δ MM ′ (50)10ith A l = Γ (cid:0) + iρ + l (cid:1) Γ (cid:0) − iρ + l (cid:1) (51)In arriving at equation (50) we have used the addition formula C ν ( n · n ′ ) = 2 π ν + 2 X M Y νM ( n ) Y ∗ νM ( n ′ ) (52) References [1] Goldstein H 1975 Am. J. Phys. , 354.[5] Winternitz P, Smorodinsky YaA,, Mandrosov V, Uhlir M and Fris I 1967Sov. J. Nucl. Phys. , 444[6] Makarov AA, Smorodinsky YaA, Valiev K and Winternitz P 1967 NuovoCimento A , 875.[9] Pauli W 1926 Z. Phys. Representation of Lie Groups and SpecialFunctions
Vol.2, Kluwer Academic Publishers, Dordrecht, 1993.[29] Nieto M M 1979 Am. J. Phys.47