Equivalence of superintegrable systems in two dimensions

Equivalence of superintegrable systems in two dimensions J. M. Kress 1, 1 School of Mathematics, The University of New South Wales, Sydney 058, Australia. In two dimensions, all nondegenerate superintegrable systems having constants quadratic in the momenta possess a quadratic algebra. In this paper is it shown how the quadratic algebra can be used to classify all such systems into 7 classes that are preserved by coupling constant metamorphosis. I. INTRODUCTION In a recent paper [9] it was shown that all nondegenerate two-dimensional superintegrable systems having constants quadratic in the momenta can be obtained by coupling constant metamorphosis from those on constant curvature spaces. It has also been shown that the Poisson algebras of these systems close quadratically [8]. In this paper the quadratic algebra is used to classify these systems into seven classes on which coupling constant metamorphosis [7] (or the Stäckel transform [1]) acts transitively. A similar classification, also reported at this meeting, has recently been used by Daskaloyannis and Ypsilantis [4] as the basis for calculating the Hamiltonians and associated integrals for these systems. We consider the Hamiltonian of a system with degrees of freedom, H = i,j=1 g ij p i p j + V (x 1, x ), (1) having constants quadratic in the momenta and potential nondegenerate potential V (that is, apart from an additive constant, it is determined by V 1, V and V 11 at a regular point and hence depends on 3 parameters). by The time evolution of a function L of the position x 1, x and the momenta p 1, p is given Electronic address: J.Kress@unsw.edu.au dl dt = {L, H}

where {, } is the Poisson bracket {a, b} = i=1 a b a b x i p i p i x i and so if {H, L} = 0, the function L is called an integral or constant of the motion. Given two constants in involution (i.e. having vanishing Poisson bracket) the system is said to be Liouville integrable and when three or more constants polynomial in the momenta are known, the system is said to be superintegrable. A similar situation exists for quantum systems with constants replaced by differential operators and the Poisson bracket replaced by the operator commutator. In n dimensions, n constants in involution are required for Liouville integrability and a system is said to maximally superintegrable when n 1 polynomial constants are known. The free particle, Coulomb-Kepler system and Harmonic oscillator, (or their quantum counterparts) are well known superintegrable systems. In 1965 Friš et al [6] initiated a search for other superintergable systems and found all such systems in two-dimensional real Euclidean space having three constants quadratic in the momenta. A similar list of superintegrable potentials has been found in real three-dimensional Euclidean space [5] and recently all two-dimensional nondegenerate Hamiltonians of the form (1) have been found [9]. As an example of the type of superintegrable system considered in this paper, consider one of the four systems found by Friš et al [6] given by the Hamiltonian H = p x + p y + α(x + y ) + β x + γ y. Constants of the motion for this system are R 1 = p x + αx + β x R = M + β y x + γ x y where M = xp y yp x, and the Poisson algebra of these constants along with R = {R 1, R } closes to form a quadratic algebra. {R, R 1 } = 8R 1 8HR 1 + 16αR {R, R } = 16R 1 R + 8HR 16(β + γ)r 1 + 16βH

3 The cubic constant R cannot be functionally independent of R 1, R and H and in fact R = 16R 1R + 16HR 1 R 16(β + γ)r 1 16αR + 3βHR 1 16βH + 64αβγ. This cubic expression for R in terms of R 1, R and H contains the complete structure of the quadratic algebra, which can be determined from it by {R, R 1 } = 1 R and {R, R } = 1 R R. R 1 While the existence of a quadratic algebra for this type of system has been noted and used, in the quantum case, to determine the spectrum of bound states [3], it has only recently be shown to be a generic feature of all nondegenerate quadratically superintegrable systems in two dimensions [8]. II. COUPLING CONSTANT METAMORPHOSIS Transformations mapping one integrable system to another have been put to good use in the literature. One such type of transformation, known as coupling constant metamorphosis [7] interchanges a parameter in the potential with the energy. This transformation can be applied to more general systems than those considered here. In the current context it also known as a Stäckel transform [1] and is briefly described below. Consider a Hamiltonian and corresponding constant H = H 0 + αv 0 and L = L 0 + αl 0 such that {H 0, L 0 } = {H, L} = 0. It can be shown that are also in involution, that is H = H 0 V 0 and L = L 0 l 0 H {H, L } = 0. So starting with a superintegrable Hamiltonian, a new conformally related superintegrable Hamiltonian can be constructed. Identities involving integrals associated with H give rise to identities involving integrals associated with H by making the replacements α H, H 0.

4 If we allow the addition of a constant to the Hamiltonian and multiplication by a constant then a transformation that simply interchanges H and α can be constructed. For example H = p x + p y + αx is a flat space superintegrable system with constants K = p y, R 1 = Mp y α 4 y and R = p x p y + α y and Poisson algebra defined by {K, R 1 } = R, {K, R } = α, {R 1, R } = K 3 + HK and R + K 4 HK + αr 1 = 0. Taking V 0 = x gives the transformed Hamiltonian and constants H = p x + p y x K = p y, R 1 = Mp y + y 4x (p x + p y) and R = p x p y y x (p x + p y) with Poisson algebra defined by {K, R 1} = R, {K, R } = 1 H, {R 1, R } = K 3 and R + K 4 H R 1 = 0. In this way, nonflat superintegrable systems can be generated from known quadratically superintegrable systems in two dimensions. III. HAMILTONIANS WITH TWO ADDITIONAL QUADRATIC CONSTANTS Koenigs [14] found all two-dimensional surfaces ds = 4f(x, y)(dx + dy ) admitting at least two rank Killing tensors in addition to the metric. This gives us an equivalent list of corresponding Hamiltonians H = p x + p y f(x, y)

5 admitting at least two additional quadratic constants. For example, those possessing two quadratic constants and one linear constant: D 1 D : H 0 = p x + p y 4x : H 0 = p x + p y 1 + 1 x D 3 : H 0 = p x + p y 4 + x + y D 4 : H 0 = 1 4 p x + p y a+ x + a y These have been rewritten in a rational form so that it is apparent that each of the denominators is in fact a superintegrable potential from those known to exist in flat space. Hence we can obtain each of these Hamiltonians from a flat space superintegrable Hamiltonian by coupling constant metamorphosis. Since each of the demoninators above appears as a term in several nondegenerate superintegrable potenials on Euclidean space, we can generate a non-degenerate superintegrable potentials on these spaces. For example, the potential in each of E, E9 and E3 (see Appendix A or [10]) contains the term x and so dividing throughout by V 0 = x gives three distinct non-degenerate superintegrable potential on D 1. Alternatively we can start with Koenigs list and use it as a basis for finding all quadratically superintegrable systems in two dimensions. This approach was taken in [11, 1]. For example, starting with D 1 we can look for a Hamiltonian of the form H = p x + p y 4x + V (x, y) having two additional constants of the form X i = a i K + b i R 1 + c i R + d i (x, y), i = 1,, and find H 1 H H 3 = p x + p y 4x = p x + p y 4x = p x + p y 4x + α(4x + y ) x + β x + γ xy + δ + α x + βy x + γ(x + y ) + δ x α + x β(x iy) + x iy x x iy + γ x + δ

6 It is easily seen that each of these is essentially one of E, E9 and E3 divided by throughout by x. The first two of these were given in [1] and the third was noted in the appendix A of [11]. Either approach yields all quadratically superintegrable systems in two dimensions [9]. IV. CLASSIFICATION OF THE QUADRATIC ALGEBRA Each three-parameter potential has an associated quadratic algebra characterised by an identity of the form R = a 1 R 3 1 + a R 3 + a 3 R 1R + a 4 R 1 R + b 1 R 1 + b R + b 3 R 1 R + c 1 R 1 + c R + d, where the a i are numbers, the b i, c i and d are respectively linear, quadratic and cubic in H and the parameters. Hence the coupling constant metamorphosis preserves the form of this cubic expression and we can classify the superintegrable systems accordingly. Since there is no preferred basis for the space spanned by R 1 and R and we can add multiples of H and the parameters to R 1 and R and the expression for R can always be reduced to one of the forms in table I. The classes are given labels in this table that reflect the form of the cubic and quadratic parts. (Note that there is no system with vanishing cubic and a perfect square for its quadratic part.) V. GENERATING THE KNOWN SUPERINTEGRABLE SYSTEMS From one known system we can attempt to generate other systems in two-dimensional Euclidean space with the same type of quadratic algebra. For example, starting with system E8, we can ask when the transformed Hamiltonian, H = p x + p y V 0 = p x + p y αz + + γz z + δ z 3 β z a flat space Hamiltonian? This question has been considered in [] and it amounts to solving z z log V 0 = 0,

7 TABLE I: Forms for cubic and quadratic terms of R. The coefficient f(α i, H) is a linear function of the Hamiltonian and the parameters in the potential. Form of cubic in R 1 and R label systems from [10] R1 3 + f(α i, H)R [3,] E S1 R1 3 + f(α i, H)R 1 R [3,11] E9 E10 R1 3 + 0 [3,0] E15 R1 R + f(α i, H)R [1,] E1 E16 S S4 R1 R + 0 [1,0] E7 E8 E17 E19 R 1 R (R 1 + R ) + f(α i, H)R 1 R [111,11] S7 S8 S9 0 + f(α i, H)R 1 R [0,11] E3 E11 E0 which in the current example has solutions α = γ = 0 or β = δ = 0. Having solved this equation, we would now like to know how many distinct systems are generated in this way. Since the quadratic algebra for a given system is unchanged by coordinate transformations, it is more convenient to compare algebras than any given coordinate representation. In the case of two-dimensional Euclidean space, we have a complete list of the possible systems and their algebras and hence it simply remains to check that each of these is generated. In this way, it can be shown that, including the original system, that the four distinct system in this class can be generated by coupling constant metamorphosis. Note that there is no need to separately account for the translations and rotations of the systems, since two equivalent systems will be recognised as such by matching their quadratic algebras. We also can determine the required scaling transformation that will transform one superintegrable potential to another by examining the quadratic algebras. For example, suppose we wish to determine the scaling required to transform S9 into S8 we can focus on the coefficient of R 1 in the expression for R of system S9 (see appendix A) and attempt to transform one into the other. The following sequence of parameter reassignments and coupling constant metamorphosis has the desired effect.

8 1 (α γ)(α + β + γ H) 56 1 (α γ)(α γ H) (H H + β + γ) 56 1 ( γ)( γ H) (γ γ + α, β β + α) 56 1 γ(γ + α) (α H) 56 1 γα (α 4α γ) 64 1 64 (4β + α ) (α β + iα, β β iα) The step in which α is swapped with H is the coupling constant metamorphosis. At this point in the process, we can examine the part of the potential for which α is the parameter and hence determine the required V 0. H = T + α x + β y + γ z + α + β + γ (H H + β + γ) H + β + γ = T + β y + γ z + α + β + γ H = T + β y + γ z + α β γ (γ γ + α, β β + α) H = T + α x + β + α + γ + α + α (β + α) (γ + α) y z ( 1 = T + α x + 1 y + 1 ) z 1 + β y + γ z β γ We see from this that V 0 = 1 x + 1 y + 1 z 1 will transform S9 into S8. The fact that coupling constant metamorphosis modifies the Possion algebra of a superintegrable system in a simple way allows us to classify the nondegenerate quadratically superintegrable systems in two dimensions into seven classes. All such superintegrable sys-

9 tems can then be generated from seven representative systems. Six of these representatives can be taken from E,C, [3,] α(4x + y ) + βx + γ y + δ, () [3,11] α(x iy) + β (x + iy 3 ) (x iy) + γ (x + y 1 ) (x iy)3 + δ, (3) [3,0] h(x iy), (4) [1,] α(x + y ) + β x + γ y + δ, (5) [1,0] α(x + y β γ(x + iy) ) + + (x iy) (x iy) + δ, 3 (6) [0,11] α(x + y ) + βx + γy + δ, (7) and one occurs on S,C, [111,11] α x + β y + γ z + δ. (8) APPENDIX A: SUPERINTEGRABLE POTENTIALS ON E,C AND S,C. This appendix gives a list of nondegenerate superintegrable potentials on E,C and S,C grouped according to the classification given in table I. The constants R 1 and R have been chosen for ease of comparison with the forms in table I and the labels E1 to E0 and S1 to S9 for the systems are taken from [10]. For those on Euclidean space, z = x + iy, z = x iy and for those on the two-sphere x, y, z are related by x + y + z = 1. Class [3,] E: V = α(4x + y ) + βx + γ y R = R1 3 + αr (1 i)β 4 R 1R + 1 ( H + 6αγ ) (1 + i)β R 1 + 48 48 HR i 864 H3 + iαγ 4 H + iβ γ 56 S1: V = α (x iy) + βz (x iy) + γ(1 4z ) 3 (x iy) 4

R = R 3 1 + γr + β 4 R 1R + 1 48 ( ) 6γH α R 1 + αβ ( ) αγ 48 R 4 + β H α3 56 864 10 Class [3,11] E9: V = α z + β(z + z) + γ(z + 3 z) z R = R 3 1 + βr 1 R + 1/3 4 + 1 16 H3 αγ 1 H α β 8 ( 1αγ H ) R 1 + /3 6 E10: V = α z + β (z 3 ) z + γ (z z 1 ) z3 ( βh + 3γ ) R R = R 3 1 + γr 1 R + 1/3 4 γ 16 H αβ 4 H + α3 16 ( 6βH α ) R 1 + /3 1 (αγ + 3β )R Class [3,0] E15: V = h(x iy) R = 4R 3 1 Class [1,] E1: V = α(x + y ) + β x + γ y R = R1R + αr i(β γ) HR 1 + 1 ( H + 8α(β + γ) ) R 64 64 + 1 ( (β + γ)h + α(β γ) ) 51 E16: V = 1 x + y (α + β x + x + y + ) γ x x + y R = R 1R + HR + + α(β + γ) R 1 + 1 ( ) 4(β γ)h α R 16 16 (β + γ) (β γ)α H 64 64

11 S: V = α z + β γ(x + iy) + (x iy) (x iy) α 3 R = R 1R + γr β 64 HR 1 + 1 64 ( 8γH + 16αγ β ) R + γ 56 H β 51 H αβ 56 S4: V = α (x iy) + βz x + y + γ (x iy) x + y R = R1R + αr iβγ 16 R 1 + 1 ( ) 4αH γ R γ 16 64 H β α 64 Class [1,0] E7: V = α z z 1 + βz z 1 ( z + z 1 ) + γz z R = R 1R + ( H α ) R 1 + 64γ(4β γ)r + 8(β γ)h 3αβH + 8(β c γ)α E8: V = αz + + γz z z 3 β z R = R 1R + βhr 1 + 64αγR + 4αH 4β γ E17: V = α + β z z z + γ z z z R = R1R + iαγ R 1 + βhr c 4 H βα 4 E19: V = α z ( z )( z + ) + β z( z + ) + γ z( z ) R = R 1R + ( H α ) R + i (β γ ) R 1 (β + γ ) H αβγ

1 Class [111,11] S7: V = αx y + z + βy z y + z + γ z + γ R = R 1 R (R 1 + R ) + 1 8 HR 1R + iαβ 64 R 1 + 1 ( H 4γH + (α + iβ) + 4γ ) R 56 iβ(α + iβ) + H + γ(β + α ) 104 104 S8: V = iαx y + z β (x + iy z) (x + iy)(z iy) + iγ (x + iy + z) (x + iy)(z + iy) R = R 1 R (R 1 + R ) + 1 4 HR 1R 4β + α 64 R 1 4γ + α R α 64 56 H + αβγ 64 S9: V = α x + β y + γ z + α + β + γ R = R 1 R (R 1 + R ) + 1 16 HR 1R (α γ)(α + β + γ H) (α β)(α + γ + β H) + R 1 + R 56 56 α 4096 H + (3α(β + γ) + α + βγ) (β + γ)(α + γ)(α + β) H 4096 048 Class [0,11] E3 : V = α(x + y ) + βx + γy R = αr 1 R + E11: V = αz + βz z + γ z (β iγ) R 1 + 3 (β + iγ) R α 3 64 H β + γ 18 H R = αr 1 R + H R 1 + β 8 R βγ 4 H αγ 8

13 E0: V = 1 ) (α + β x x + y + y + x + γ x + y x R = HR 1 R + (β iγ) R 1 + (β + iγ) R α 16 4 H α(β + γ ) 4 Note that E3, while it is a translation of the harmonic oscillator potential, must be included for coupling constant metamorphosis to act transitively on this class. The transformations within each class can be demonstrated explicitly and the required V 0 linking pairs of potentials are given below. [3,] E S1 V 0 = 1 z [3,11] E9 E10 V 0 = z [1,] E1 E16 V 0 = x + y E1 S V 0 = 1 x E1 S4 V 0 = 1 x + 1 y [1,0] E8 E7 V 0 = 1 z + 1 E8 E17 V 0 = x + y = z z E8 E19 V 0 = z z 3 + z z [111,11] S9 S8 V 0 = 1 x + 1 y + 1 z 1 S9 S7 V 0 = 1 x + 1 y [0,11] E11 E3 V 0 = 1 z E11 E0 V 0 = x [1] C. P. Boyer, E. G. Kalnins, and W. Miller. SIAM J. Math. Anal. 17 (1986) 778. [] Peter Collas. Metamorphosis from the viewpoint of differential geometry. Phys. Lett. A 135 (1989) 151.

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