C N -Smorodinsky-Winternitz system in a constant magnetic field
aa r X i v : . [ h e p - t h ] J a n C N -Smorodinsky-Winternitz system in a constant magnetic field Hovhannes Shmavonyan ∗ Yerevan Physics Institute, 2 Alikhanian Brothers St., Yerevan 0036 Armenia
We propose the superintegrable generalization of Smorodinsky-Winternitz system on the N -dimensional complex Euclidian space which is specified by the presence of constant magnetic field.We find out that in addition to 2 N Liouville integrals the system has additional functionally inde-pendent constants of motion, and compute their symmetry algebra. We perform the Kustaanheimo-Stiefel transformation of C - Smorodinsky-Winternitz system to the (three-dimensional) generalizedMICZ-Kepler problem and find the symmetry algebra of the latter one. We observe that constantmagnetic field appearing in the initial system has no qualitative impact on the resulting system. I. INTRODUCTION N -dimensional mechanical system will be called integrable if it has N mutually commuting and functionally inde-pendent constants of motion. In addition to these constants of motion the system can have additional ones. In thatcase we will say that the system is superintegrable . Particularly if N -dimensional mechanical system has 2 N − maximally superintegrable . The one-dimensional singularoscillator is a textbook example of a system which is exactly solvable both on classical and quantum levels.The sumof its N copies, i.e. N -dimensional singular isotropic oscillator is, obviously, exactly solvable as well. It is given bythe Hamiltonian H = N X i =1 I i , with I i = p i g i x i + ω x i , { p i , x j } = δ ij , { p i , p j } = { x i , x j } = 0 (1)It is not obvious that in addition to Liouville Integrals I i this system possesses supplementary series of constantsof motion, and is respectively, maximally superintegrable , i.e. possesses 2 N − interacting with constantmagnetic field . It is defined on the N -dimensional complex Euclidian space parameterized by the coordinates z a bythe Hamiltonian H = N X a =1 (cid:18) π a ¯ π a + g a z a ¯ z a + ω z a ¯ z a (cid:19) , with { π a , z b } = δ ab , { π a , ¯ π b } = ıBδ ab (2)The (complex) momenta π a have nonzero Poisson brackets due to the presence of magnetic field with magnitude B [24, 25]. We will refer this model as C N -Smorodinsky-Winternitz system. For sure, in the absence of magnetic field ∗ Electronic address: [email protected] this model could be easily reduced to the conventional Smorodinsky-Winternitz model, but the presence of magneticfield could have nontrivial impact which will be studied in this paper. So, our main goal is to investigate the wholesymmetry algebra of this system . Notice that this is not only for academic interest: the matter is that C -Smorodinsky-Winternitz system is a popular model for the qualitative study of the so-called quantum ring [26–28], and the studyof its behaviour in external magnetic field is quite a natural task. Respectively, C N -Smorodinsky-Winternitz could beviewed as an ensemble of N quantum rings interacting with external magnetic field. So investigation of its symmetryalgebra is of the physical importance.It is well-known for many years that the energy surface of two-/three-/five-dimensional Coulomb systemcould be transformed to those of two-/four-/eight-dimensional oscillator by the use of so-called Levi-Civita-Bohlin/Kustaanheimo-Stiefel/Hurwitz transformation [29–31]. More generally, reducing the two-/four-/eight-dimensional oscillator (-like) models by the action of Z /U (1) /SU (2) group action, we can get the two-/three-/five-dimensional Coulomb-like systems specified by the presence of ( Z /Dirac/Yang)monopole [32]. Since C -Smorodinsky-Winternitz system is manifestly invariant with respect to U (1) group action, we can perform its Kustaanheimo-Stiefeltransformation, in order to obtain three-dimensional Coulomb-like system. It was done about ten years ago [39],but in the absence of magnetic field in initial system. Repeating this transformation for the system with constantmagnetic field we get unexpected result: it has no qualitative impact in the resulting system, which was referred in[40] as ”generalized MICZ-Kepler system”[36–38]. In addition, we obtain, in this way, the explicit expression of itssymmetry generators and their symmetry algebra, which as far as we know was not constructed before.We already mentioned that both oscillator and Smorodinsky-Winternitz system admit superintegrable generaliza-tions to the spheres. On the other hand the isotropic oscillator on C N admits the superintegrable generalizationon the complex projective space, moreover, the inclusion of constant magnetic field preserves all symmetries of thatsystem [42, 43]. It will be shown that introduction of a constant magnetic field doesn’t change these properties of the C N -Smorodinsky-Winternitz system. Thus, presented model could be viewed as a first step towards the constructionof the analog of Smorodinsky-Winternitz system on CP N .The paper is organized as follows.In the Section 2 we review the main properties of the conventional ( R N -)Smorodinsky-Winternitz system, presentingexplicit expressions of its symmetry generators, as well as wavefunctions and Energy spectrum. We also presentsymmetry algebra in a very simple, and seemingly new form via redefinition of symmetry generators.In the Section 3 we present C N -Smorodinsky-Winternitz system in a constant magnetic field, find the explicitexpressions of its constants of motion. We compute their algebra and find that it is independent from the magnitudeof constant magnetic field. Then we quantize a system and obtain wavefunctions and energy spectrum. We noticethat the C N -Smorodinsky-Winternitz system has the same degree of degeneracy as R N - one, due to the lost part ofadditional symmetry.In the Section 4 we perform Kustaanheimo-Stieffel transformation of the C -Smorodinsky-Winternitz system inconstant magnetic field and obtain, in this way, the so-called “generalized MICZ-Kepler system”. We find thatconstant magnetic field appearing in the initial system, does not lead to any changes in the resulting one.In the Section 5 we discuss the obtained results and possibilites of furthur generalizations. Possible extensionsof discussed system include supersymmetrization and quaternionic generalization as well as generalization of thesesystems in curved background.
II. SMORODINSKY-WINTERNITZ SYSTEM ON R N Smorodinsky-Winternitz system is defined as a sum of N copies of one-dimensional singular oscillators (1), each ofthem defined by generators I i which obviously form its Liouville integrals { I i , I j } = 0. About fifty years ago it wasnoticed that this system possesses additional set of constants of motion given by the expressions [1] I ij = L ij L ji − g i x j x i − g j x i x j , { I ij , H } = 0 , (3)where L ij are the generators of SO ( N ) algebra, L ij = p i x j − p j x i : { L ij , L kl } = δ ik L jl + δ jl L ik − δ il L jk − δ jk L il . (4)The generators I ij provides additional N − { I i , I jk } = δ ij S ik − δ ik S ij , { I ij , I kl } = δ jk T ijl + δ ik T jkl − δ jl T ikl − δ il T ijk (5)where S ij = − I i I j I ij + I i g j − I j g i + ω I ij − g i g j ω ) , T ijk = − I ij I jk I ik + g k I ij + g j I ik + g i I jk − g i g j g k ) . (6)The generators S ij and T ijk are of the sixth-order in momenta and antisymmetric over i, j, k indices. The abovesymmetry algebra could be written in a compact form if we introduce the notation M ij = I ij , M i = I i , M ii = g i , M = ω , R ijk = T ijk , R ij = S ij . (7)Then one can introduce capital letters which will take values from 0 to N . It is worth to mention that M IJ issymmetric, whereas R IJK is antisymmetric with respect to all indices. In this terms the whole symmetry algebra ofSmorodinsky-Winternitz system reads { M IJ , M KL } = δ JK R IJL + δ IK R JKL − δ JL R IKL − δ IL R IJK (8)where R IJK = − M IJ M JK M IK + M IJ M KK + M IK M JJ + M KL M II − M II M JJ M KK ) (9)One important fact should be mentioned, although in this algebra on the right side we have sum of many terms(square roots), only one term always survives, since in case of three indices are equal, the result is automatically0. Consequently in this algebra we always have one square root on the right hand side. Quantum-mechanically themaximal superintegrability is reflected in the dependence of its energy spectrum on the single,“principal” quantumnumber only. Having in mind that in Cartesian coordinates the system decouples to the set of one-dimensionalsingular oscillators, we can immediately extract the expressions for its wavefunctions and spectrum from the standardtextbooks on quantum mechanics, e.g. [44], E n | ω = ~ ω (cid:16) n + 1 + N X i =1 r
14 + g i ~ (cid:17) , Ψ = N Y i =1 ψ ( x i , n i ) , n = N X i =1 n i (10)where ψ ( x i , n i ) = F (cid:16) − n i , r
14 + g i ~ , ωx i ~ (cid:17)(cid:16) ωx i ~ (cid:17) √ g i / ~ e − ωx i ~ (11)Here F is the hypergeometric function. With these expressions at hands we are ready to study Smorodinsky-Winternitzsystem on complex Euclidean space in the presence of constant magnetic field. III. C N -SMORODINSKY-WINTERNITZ SYSTEM Now let us study 2 N -dimensional analog of Smorodinsky-Winternitz system interacting with constant magneticfield. It is defined by (2) and could be viewed as an analog of Smorodinsky-Winternitz system on complex Euclidianspace (cid:16) C N , ds = P Na =1 dz a d ¯ z a (cid:17) . Thus, we will refer it as C N -Smorodinsky-Winternitz system. The analog of SW-system which respects the inclusion of constant magnetic field is defined as follows, H = X a I a , I a = π a ¯ π a + g a z a ¯ z a + ω z a ¯ z a , (12)where z a , π a are complex (phase space) variables with the following non-zero Poisson bracket relations { π a , z b } = δ ab , { ¯ π a , ¯ z b } = δ ab , { π a , ¯ π b } = ıBδ ab . (13)For sure, it can be interpreted as a sum of N two-dimensional singular oscillators interacting with constant magneticfield perpendicular to the plane. It is obvious that in addition to N commuting constants of motion I a this systemhas another set of N constants of motion defining manifest ( U (1)) N symmetries of the system L a ¯ a = ı ( π a z a − ¯ π a ¯ z a ) − Bz a ¯ z a : { L a ¯ a , H} = 0 (14)and supplementary, non-obvious, set of constants of motion defined in complete analogy with those of conventionalSmorodinsky-Winternitz system: I ab = L a ¯ b L b ¯ a + (cid:16) g a z b ¯ z b z a ¯ z a + g b z a ¯ z a z b ¯ z b (cid:17) , { I ab , H} = 0 , a = b (15)with L a ¯ b being generators of SU ( N ) algebra L a ¯ b = ı ( π a z b − ¯ π b ¯ z a ) − B ¯ z a z b : { L a ¯ b , L c ¯ d } = iδ a ¯ d L c ¯ b − iδ c ¯ b L c ¯ d . (16)These symmetry generators, and I a obviously commute with L a ¯ a due to manifest U (1) N symmetry { L a ¯ a , I b } = { L a ¯ a , I bc } = { L a ¯ a , L b ¯ b } = { I a , I b } = 0 (17)The rest Poisson brackets between them are highly nontrivial { I a , I bc } = δ ab S ac − δ ac S ab , { I ab , I cd } = δ bc T abd + δ ac T bcd − δ bd T acd − δ ad T abc , (18)where S ab = 4 I ab I a I b − ( L a ¯ a I b + L b ¯ b I a ) − g a I b − g b I a − ω I ab ( I ab − L a ¯ a L b ¯ b ) + 4 ω g b L a ¯ a + 4 g a ω L b ¯ b + 16 g a g b ω − B ( I ab − L a ¯ a L b ¯ b )( L a ¯ a I b + L b ¯ b I a ) − B ( I ab − L a ¯ a L b ¯ b ) + 4 B ( g b I a L a ¯ a + g a I b L b ¯ b ) + 4 B g a g b (19) T abc = 2( I ab − L a ¯ a L b ¯ b )( I bc − L b ¯ b L c ¯ c )( I ac − L a ¯ a L c ¯ c ) + 2 I ab I ac I bc + L a ¯ a L b ¯ b L c ¯ c − ( I bc L a ¯ a + I ab L c ¯ c + I ac L b ¯ b ) − g c I ab ( I ab − L a ¯ a L b ¯ b )+ g a I bc ( I bc − L b ¯ b L c ¯ c )+ g b I ac ( I ac − L a ¯ a L c ¯ c ))+ 4 g b g c L a ¯ a + 4 g a g c L b ¯ b + 4 g a g b L c ¯ c + 16 g a g b g c (20)To write the symmetry algebra in a simpler form we can redefine the generators M aa = L a ¯ a + 4 g a , M ab = I ab − L a ¯ a L b ¯ b , M a = I a − B L a ¯ a , M = 4 ω + B . (21)Since L a ¯ a commute with all other generators Poisson brackets of M will exactly coincide with the Poisson bracketsof I ab and I a . Similarly the R tensor is defined as in the real case. So the algebra will have the following form { M ab , M cd } = δ bc T abd + δ ac T bcd − δ bd T acd − δ ad T abc , { M a , M ab } = δ ab S ac − δ ac S ab . (22)where S ab = 4 M ab M a M b + (cid:16) ω + B (cid:17) ( M aa M bb − M ab ) − M b M aa − M a M bb (23) T abc = 4 M ab M bc M ac − M ab M cc − M ac M bb − M bc M aa + 14 M aa M bb M cc (24)Needless to say that L a ¯ a commute with all the other constants of motion. Finally the full symmetry algebra thenreads { M AB , M CD } = δ BC R ABD + δ AC R BCD − δ BD R ACD − δ AD R ABC (25)where R ABC = 4 M AB M BC M AC − M AB M CC − M AC M BB − M BC M AA + 14 M AA M BB M CC (26)Again capital letters take values from 0 to N . In the complex case R ABC and M AB are again respectively antisymmetricand symmetric as in the real case. Up to multiplication by a constant this has the same form as the symmetry algebrafor the real case.Let us briefly discuss the number of conserved quantities. We have N real functionally independent constants ofmotion ( I a ). Moreover let us mention that I ab is also real, and although it has N ( N − / N −
1. In addition to this, the complex system has N real conservedquantities ( L a ¯ a ). So the total number of constants of motion is 3 N − N = 1 the system is integrable. For N = 2 the system is superintegrable, but it hasonly one additional constant of motion. In this case the system is called minimally superintegrable . Quantization
Quantization will be done using the fact that C N -Smorodinsky-Winternitz system is a sum of two dimensionalsingular oscillators. This allows to write the wave function as a product of N wave functions and total energy of thesystem as a sum of the energies of its subsystems. So the initial problem reduces to two-dimensional one.ˆ I a Ψ a ( z a , ¯ z a ) = E a Ψ a ( z a , ¯ z a ) , ˆ H Ψ tot = E tot Ψ tot , Ψ tot = N Y a =1 Ψ a ( z a , ¯ z a ) , E tot = N X a E a . (27)After this reduction, complex indices can be temporarily dropped. Now it is obvious to introduce the momentaoperators and commutation relations, which will have the following form in the presence of constant magnetic field.ˆ π = − ı ( ~ ∂ + B z ) , ˆ¯ π = − ı ( ~ ¯ ∂ − B z ) [ π, ¯ π ] = ~ B, [ π, z ] = − ı ~ (28)Schr¨odinger equation can be written down h − ~ ∂ ¯ ∂ + (cid:16) ω + B (cid:17) z ¯ z − ~ B z ¯ ∂ − ∂z ) + g z ¯ z i Ψ( z, ¯ z ) = E Ψ( z, ¯ z ) . (29)Even in this two-dimensional system additional separation of variables can be done if one writes this system in a polarcoordinates using the fact that z = r √ e iφ . h ∂ ∂r + 1 r ∂∂r + 2 ~ (cid:16) E + ~ r ∂ ∂φ − g r − (cid:16) ω + B (cid:17) r + ıB ~ ∂∂φ (cid:17)i Ψ( r, φ ) = 0 . (30)Further separation of variables can be done and one can use the fact that L is a constant of motion.Ψ( r, φ ) = R ( r )Φ( φ ) , ˆ L Φ = ~ m Φ . (31)Using the explicit form of the U (1) generator, normalized solution can be writtenˆ L = − ı ~ ∂∂φ , Φ( φ ) = 1 √ π e ımφ . (32)This result allows to write the equation (30) in the following form h d dr + 1 r ddr + 2 ~ (cid:16) E − ~ m r − g r − (cid:16) ω + B (cid:17) r − B ~ m (cid:17)i R ( r ) = 0 . (33)Solution of this kind of Schr¨odinger equation can be written down. The final result for the wave functions of two-dimensional system and the energy spectrum are as follows ψ ( z, ¯ z, n, m ) = C n,m √ π ( p z/ ¯ z ) m F (cid:16) − n, r m + 4 g ~ + 1 , q ω + B ~ z ¯ z (cid:17)(cid:16) q ω + B ~ z ¯ z (cid:17) / q m + g ~ e − √ ω B ~ z ¯ z (34) E = ~ r ω + B (cid:16) n + 1 + r m + 4 g ~ (cid:17) + B ~ m C N can be recovered. The total wave function is a product of the wavefunctions and the totalenergy is the sum of the energies of two-dimensional subsystemsΨ( z, ¯ z ) = N Y a =1 ψ ( z a , ¯ z a , n a , m a ) (36) E tot = N X a =1 E n a ,m a = ~ r ω + B (cid:16) n + N + N X a =1 r m a + 4 g a ~ (cid:17) + B ~ N X a =1 m a , (37) n = N X a =1 n a , n = 0 , , ... m a = 0 , ± , ± , ... (38)In contrast to the real case the energy spectrum of the C N -Smorodinky-Winternitz system depends on N + 1 quantumnumbers, namely n and m a . IV. KUSTAANHEIMO-STIEFEL TRANSFORMATION
There is a well-known procedure reducing two-/four-/eight-dimensional oscillator to the two-/three-/five-dimensional Coulomb system. It is related with the Hopf maps and assumes the reduction by Z − /U (1) − /SU (2) − group action. In general case it results in the Coulomb like systems specified by the presence of Z -/Dirac-/Yang-monopole[32–35]. Since C N -Smorodinsky-Winternitz system has manifest U (1) invariance, we could apply its re-spective reduction procedure related with first Hopf map S /S = S , which is known as Kustaanheimo-Stieffeltransformation, for the particular case of N = 2. Such a reduction was performed decade ago [39] and was found tobe resulted in the so-called “generalized MICZ-Kepler problem” suggested by Mardoyan a bit earlier [40, 41]. How-ever the initial system was considered, it was not specified by the presence of constant magnetic field, furthermore,the symmetry algebra of the reduced system was not obtained there. Hence, it is at least deductive to performKustaanheimo-Stiefel transformation to the C -Smorodinsky-Winternitz system with constant magnetic field in orderto find its impact (appearing in the initial system) in the resulting one. Furthermore, it is natural way to find theconstants of motion of the “generalized MICZ-Kepler system” and construct their algebra.So, let us perform the reduction of C -Smorodinsky-Winternitz system by the U (1)-group action given by thegenerator J = L + L = ı ( zπ − ¯ z ¯ π ) − Bz ¯ z (39)For this purpose we have to choose six independent functions of initial phase space variables which commute withthat generators, q k = zσ k ¯ z, p k = zσ k π + ¯ πσ k ¯ z z ¯ z , k = 1 , , σ k are standard 2 × U (1)- gener-ator J = 2 s . As a result, we get the reduced Poisson brackets { q k , q l } = 0 , { p k , q l } = δ kl , { p k , p l } = sǫ klm q m | q | (41)Expressing the Hamiltonian via q i , p i , J and fixing the value of the latter one, we get H SW = 2 | q | h p s | q | + Bs | q | + 12 (cid:16) B ω (cid:17) + g | q | ( | q | + q ) + g | q | ( | q | − q ) i (42)So, we reduced the C -Smorodinsky-Winternitz Hamiltonian to the three-dimensional system. To get the Coulomb-like system we fix the energy surface or reduced Hamiltonian, H SW − E SW = 0 and divide it on 2 | q | . This yields theequation H gMICZ − E = 0 , with E ≡ − ω + B /
42 (43)and H gMICZ = p s | q | + g | q | ( | q | + q ) + g | q | ( | q | − q ) − γ | q | with γ ≡ E SW − Bs . (44)The latter expression defines the Hamiltonian of “generalized MICZ-Kepler problem”. Hence, we transformed theenergy surface of the reduced C -Smorodinsky-Winternitz Hamiltonian to those of (three-dimensional) “GeneralizedMICZ-Kepler system”. Additionally it has an inverse square potential and this system has an interaction with a Diracmonopole magnetic field which affects the symplectic structure. Surprisingly, the reduced system contains interaction with Dirac monopole field only, i.e. the constant magneticfield in the original system does not contribute in the reduced one. All dependence on B is hidden in s and γ , whichare fixed, so the reduced system does not depend on B explicitly. Now this reduction can be done for constants of motion. Before doing that it is convenient to present the initialgenerators of u (2) algebra given by (16) in the form J = i ( zπ − ¯ z ¯ π ) − Bz ¯ z, J k = i zσ k π − ¯ πσ k ¯ z ) − Bzσ k ¯ z { J , J i } = 0 , { J i , J j } = ε ijk J k . (45)After reduction we get J = 2 s . After the reduction, the rest su (2) generators result in the generators of the so (3) rotations of three-dimensional Euclidian space with the Dirac monopole placed in the beginning of Cartesiancoordinate frame, J k = ǫ klm p l q m − s q k | q | (46)Then the symmetry generators for the “generalized MICZ-Kepler system” can be written down, I = I − I B L − L ) = p J − p J + x γr + g ( r − x ) r ( r + x ) − g ( r + x ) r ( r − x ) (47) L = 12 ( L − L ) = J = p q − q p − sq | q | , J = I = J + J + g ( r − q ) r + q + g ( r + q ) r − q . (48)It is important to notice that I is a generalization of the z -component of the Runge-Lenz vector.The relation of the initial system and the reduced one will allow to find the symmetry algebra of the final systemusing the previously obtained result for the complex Smorodinsky-Winternitz system. First of all the constants ofmotion in the initial system will also commute with the reduced Hamiltonian. {H gMICZ , I} = {H gMICZ , J } = {H gMICZ , L} = 0 (49)Moreover, since in the initial system L a ¯ a generators commute with all the other constants of motion one can write. {L , J } = {L , I} = 0 (50)There is only one non-trivial commutator {I , J } = S (51) S here coincides with S of C -Smorodinsky-Winternitz system and can be written using the generators of thereduced system. S = 2 H gMICZ h (cid:16) J + 12 (cid:16) L − s (cid:17)(cid:17) − (cid:16) g + ( L + s ) (cid:17)(cid:16) g + ( L − s ) (cid:17)i − (cid:16) g + ( L + s ) (cid:17)(cid:16) I + γ (cid:17) − (cid:16) g + ( L − s ) (cid:17)(cid:16) I − γ (cid:17) − (cid:16) J + 12 ( L − s ) (cid:17)(cid:16) I − γ (cid:17)(cid:16) I + γ (cid:17) (52)There is a crucial fact that should be mentioned. Although the initial system had an interaction with magnetic field,after reduction we don’t have any dependence on B both in symplectic structure and in generators of the symmetryalgebra, at least in classical level. In other words, the reduced system does not feel the magnetic field of the initialsystem. V. DISCUSSION AND OUTLOOK
In this paper we formulated the analog of the Smorodinksy-Winternitz system interacting with a constant magneticfield on the N -dimensional complex Euclidian space C N . We found out it has 3 N − C N -Smorodinsky-Winternitz energy spectrum depends on N + 1 quantum numbers. Then weperformed Kustaanheimo-Stiefel transformation of the C -Smorodinsky-Winternitz system and reduced it to the so-called ”generalized MICZ-Kepler problem”. We obtained the symmetry algebra of the latter system using the resultobtained for the initial ones. Moreover, we have shown that the presence of constant magnetic field in the initialproblem does not affect the reduced system.There are several generalizations one can perform for this system. Straightforward task is the construction of aquaternionic ( H N -) analog of this system. While complex structure allows to introduce constant magnetic field withoutviolating the superintegrability, quaternionic structure should allow to introduce interaction with SU (2) instanton.It seems that one can also introduce the superintegrable analogs of the C N -/ H N -Smorodinsky-Winternitz systemson the complex/quaternionic projective space CP N / HP N , having in mind the existence of such generalization for the C N -/( H N -) oscillator [42, 45]. We expect that the inclusion of a constant magnetic/instanton field does not cause anyqualitative changes for this system. These generalizations will be discussed later on. Acknowledgments
I am grateful to Armen Nersessian for his impact on this work, which includes suggestion of the problem andintroduction to subject, numerous discussions and help in preparation of manuscript. Thanks to his great supportand encouragement this work is finally completed. I am also thankful to Lusine Goroyan and Ruben Hasratyan forediting.This work was done within ICTP Affiliated Center programs AF-04 and Regional Doctoral Program on Theoreticaland Experimental Particle Physics sponsored by Volkswagen Foundation. I also acknowledge partial financial supportwithin research grant from the ANSEF-Armenian National Science and Education Fund based in New York, USA. [1] I. Fris, V. Mandrosov, Ya. A. Smorodinsky, M. Uhlir, and P. Winternitz,
On higher symmetries in quantum mechanics ,Phys. Lett. (1965) 354;[2] P. Winternitz, Ya. A. Smorodinsky, M. Uhlir, I. Fris, Symmetry groups in classical and quantum mechanics , Soviet J.Nuclear Phys. (1967), 444;[3] A. A. Makarov, Ya. A. Smorodinsky, Kh. Valiev, and P. Winternitz, A systematic search for non-relativistic system withdynamical symmetries , Nuovo Cim. A (1967) 1061.[4] N. W. Evans Super-integrability of the Winternitz system
Phys. Lett. A, (1990)483-486[5] N. W. Evans,
Superintegrability in classical mechanics ,Phys. Rev. A (1990) 5666-5676;[6] N. W. Evans, Group theory of the Smorodinsky-Winternitz system ,J. Math. Phys. (1991) 3369-3375[7] E. G. Kalnins, G. C. Williams, W. Miller Jr. and G. S. Pogosyan, Superintegrability in three-dimensional Euclidean space ,J. Math.Phys. (1999)708-725[8] C.Grosche,G. S. Pogosyan, A. N.Sissakian, Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- andThree Dimensional Euclidean Space
Fortschritte der Physik (1995) 453-521[9] W. Miller Jr, S. Post , P. Winternitz Classical and quantum superintegrability with applications.
Journal of Physics A:Mathematical and Theoretical, (2013) 423001[10] Heinzl, T., Ilderton A. Superintegrable relativistic systems in spacetime-dependent background fields.
Journal of PhysicsA: Mathematical and Theoretical (2017) 345204[11] M. F. Hoque, I. Marquette and Y. Z. Zhang, Recurrence approach and higher rank cubic algebras for the N -dimensionalsuperintegrable systems, J. Phys. A (2016) no.12, 125201 [arXiv:1511.03331 [math-ph]];[12] M. F. Hoque, I. Marquette and Y. Z. Zhang, Quadratic algebra structure and spectrum of a new superintegrable system inN-dimension,
J. Phys. A (2015) no.18, 185201.[13] M. F. Hoque, Superintegrable systems, polynomial algebra structures and exact derivations of spectra arXiv:1802.08410[math-ph][14] M.A. Olshanetsky and A.M. Perelomov,
Classical integrable finite dimensional systems related to Lie algebras
Phys. Rept.71 (1981) 313[15] P. W. Higgs,
Dynamical Symmetries in a Spherical Geometry. 1
J. Phys. A (1979) 309.[16] H. I. Leemon, Dynamical Symmetries in a Spherical Geometry. 2,
J. Phys. A (1979) 489.[17] C.Grosche,G. S. Pogosyan, A. N.Sissakian, Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- andThree Dimensional Euclidean Sphere
Fortschritte der Physik (1995) 523-563[18] J. Harnad and O. Yermolayeva, Superintegrability, Lax matrices and separation of variables,
CRM Proc. Lect. Notes (2004) 65 [nlin/0303009 [nlin.SI]].[19] A. Galajinsky, A. Nersessian and A. Saghatelian, Superintegrable models related to near horizon extremal Myers-Perryblack hole in arbitrary dimension,
JHEP (2013) 002 [arXiv:1303.4901 [hep-th]].[20] E. Rosochatius, ¨Uber die Bewegung eines Punktes , Doctoral dissertation, University of Gttingen, 1877.[21] T. Hakobyan, O. Lechtenfeld and A. Nersessian,
Superintegrability of generalized Calogero models with oscillator or Coulombpotential,
Phys. Rev. D (2014) no.10, 101701 [arXiv:1409.8288 [hep-th]].[22] F. Correa, T. Hakobyan, O. Lechtenfeld and A. Nersessian, Spherical Calogero model with oscillator/Coulomb potential:quantum case,
Phys. Rev. D (2016) no.12, 125009 [arXiv:1604.00027 [hep-th]].[23] F. Correa, T. Hakobyan, O. Lechtenfeld and A. Nersessian, Spherical Calogero model with oscillator/Coulomb potential:classical case,
Phys. Rev. D (2016) no.12, 125008 [arXiv:1604.00026 [hep-th]].[24] A.Nersessian Elements of (super-)Hamiltonian Formalism
Lect.Notes Phys.698 (2006) 139-188[25] A. Marchesiello , L. Snobl
An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher OrderIntegrals
SIGMA (2018), 092[26] T. Chakraborty, P. Pietilinen, Electron-electron interaction and the persistent current in a quantum ring
Physical ReviewB (1994)8460-8468;[27] W.-C.Tan, J.C.Inkson, Electron states in a two-dimensional ring- an exactly soluble model
Semicond. Sci. Technology, (1996) 1635-1641;[28] J. Simonin, C. R. Proetto, Z. Barticevic, G. Fuster, Single-particle electronic spectra of quantum rings:A comparative study
Phys. Rev. B (2004), 205305 [29] K. Bohlin, Bull. Astr., (1911), 144[30] P. Kustaanheimo, E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization
J. Reine Angew. Math. (1965) 204[31] A. Hurwitz ber die Komposition der quadratischen Formen von beliebig vielen Variablen
Nachr. K. Gesellschaft Wis-senschaft. Gttingen (1898) 309[32] A. Nersessian, V. Ter-Antonian and M. M. Tsulaia,
A Note on quantum Bohlin transformation,
Mod. Phys. Lett. A (1996) 1605 [hep-th/9604197].[33] A. Nersessian and V. Ter-Antonian, ’Charge dyon’ system as the reduced oscillator, Mod. Phys. Lett. A (1994) 2431[hep-th/9406130];[34] A. Nersessian and V. Ter-Antonian, Quantum oscillator and a bound system of two dyons,
Mod. Phys. Lett. A (1995)2633 [hep-th/9508137];[35] L. G. Mardoian, A. N. Sisakian and V. M. Ter-Antonian, Phys. Atom. Nucl. (1998) 1746 [hep-th/9712235].[36] D. Zwanziger Exactly Soluble Nonrelativistic Model of Particles with Both Electric and Magnetic Charges
Phys. Rev., ,1480, (1968)[37] H. McIntosh, A. Cisneros
Degeneracy in the Presence of a Magnetic Monopole
J. Math. Phys., , 896, (1970).[38] B. Cordani The Kepler problem Springer (2003)[39] L. G. Mardoyan and M. G. Petrosyan,
4D singular oscillator and generalized MIC-Kepler system,
Phys. Atom. Nucl. (2007) 572[quant-ph/0604127].[40] L. Mardoyan, The Generalized MIC-Kepler system,
J. Math. Phys. (2003) 4981 [quant-ph/0306168];[41] L. Mardoyan, Spheroidal analysis of the generalized MIC-Kepler system
Phys. Atom. Nucl. (2005) 1746 [quant-ph/0310143].[42] S. Bellucci and A. Nersessian, (Super)oscillator on CP**N and constant magnetic field , Phys. Rev. D (2003) 065013[hep-th/0211070];[43] S. Bellucci, A. Nersessian and A. Yeranyan Quantum oscillator on CP**n in a constant magnetic field
Phys.Rev. D70(2004) 085013 [hep-th/0406184][44] L.D. Landau, L.M. Lifshiz
Quantum Mechanics ( Volume 3 of A Course of Theoretical Physics )
Pergamon Press 1965[45] S. Bellucci, S. Krivonos, A. Nersessian and V. Yeghikyan,
Isospin particle systems on quaternionic projective spaces,
Phys.Rev. D87