Tools for Verifying Classical and Quantum Superintegrability
aa r X i v : . [ m a t h - ph ] A ug Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2010), 066, 23 pages Tools for Verifying Classicaland Quantum Superintegrability
Ernest G. KALNINS † , Jonathan M. KRESS ‡ and Willard MILLER Jr. §† Department of Mathematics, University of Waikato, Hamilton, New Zealand
E-mail: [email protected]
URL: ‡ School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
E-mail: [email protected]
URL: http://web.maths.unsw.edu.au/~jonathan/ § School of Mathematics, University of Minnesota, Minneapolis, Minnesota,55455, USA
E-mail: [email protected]
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Received June 04, 2010, in final form August 06, 2010; Published online August 18, 2010doi:10.3842/SIGMA.2010.066
Abstract.
Recently many new classes of integrable systems in n dimensions occurring inclassical and quantum mechanics have been shown to admit a functionally independent setof 2 n − n = 2 but cases where n > k in the potential. Key words: superintegrability; hidden algebras; quadratic algebras
We define an n -dimensional classical superintegrable system to be an integrable Hamiltoniansystem that not only possesses n mutually Poisson – commuting constants of the motion, but inaddition, the Hamiltonian Poisson-commutes with 2 n − n independent mutually commutingdifferential operators, and that commutes with a set of 2 n − H = P ni,j =1 g ij p i p j + V andcorresponding quantum systems H = ∆ n + ˜ V . These systems, including the classical Kepler andanisotropic oscillator problems and the quantum anisotropic oscillator and hydrogen atom havegreat historical importance, due to their remarkable properties, [1, 2]. The order of a classicalsuperintegrable system is the maximum order of the generating constants of the motion (withthe Hamiltonian excluded) as a polynomial in the momenta, and the maximum order of the E.G. Kalnins, J.M. Kress and W. Miller Jr.quantum symmetries as differential operators. Systems of 2nd order have been well studiedand there is now a structure and classification theory [3, 4, 5, 6, 7, 8]. For 3rd and higherorder superintegrable systems much less is known. In particular there have been relatively fewexamples and there is almost no structure theory. However, within the last three years there hasbeen a dramatic increase in discovery of new families of possible higher order superintegrableclassical and quantum systems [9, 10, 11, 12, 13, 14, 15, 16]. The authors and collaborators havedeveloped methods for verifying superintegrability of these proposed systems, [17, 18, 19, 20].In the cited papers the emphasis was on particular systems of special importance and recentinterest. Here, however, the emphasis is on the methods themselves.In Section 2 we review a method for constructing classical constants of the motion of allorders for n -dimensional Hamiltonians that admit a separation of variables in some orthogonalcoordinate system. Then we apply it to the case n = 2 to show how to derive superintegrablesystems for all values of a rational parameter k in the potential. Many of our examples are new.In Section 3 we review our method for establishing a canonical form for quantum symmetry oper-ators of all orders for 2-dimensional Schr¨odinger operators such that the Schr¨odinger eigenvalueequation admits a separation of variables in some orthogonal coordinate system. Then we applythis method to establish the quantum superintegrability of the caged anisotropic oscillator for allfrequencies that are rationally related, and for a St¨ackel transformed version of this system onthe 2-sheet hyperboloid. We give a second proof for the caged oscillator in all dimensions n thatrelies on recurrence relations for associated Laguerre polynomials. We also apply the canonicalequations to establish the quantum superintegrability of a deformed Kepler–Coulomb systemfor all rational values of a parameter k in the potential. There are far more verified superintegrable Hamiltonian systems in classical mechanics than wasthe case 3 years ago. The principal method for constructing and verifying these new systemsrequires that the system is already integrable, in particular, that it admits a separation ofvariables in some coordinate system. For a Hamiltonian system in 2 n -dimensional phase spacethe separation gives us n second order constants of the motion in involution. In this paper wefirst review a general procedure, essentially the construction of action angle variables, whichyields an additional n − n − n new constants of the motion that arepolynomial in the momenta, so that the system is superintegrable. We will show how this can bedone in many cases. We start first with the construction of action angle variables for Hamiltoniansystems in n dimensions, and later specialize to the case n = 2 to verify superintegrability.Consider a classical system in n variables on a complex Riemannian manifold that admitsseparation of variables in orthogonal separable coordinates x , . . . , x n . Then there is an n × n St¨ackel matrix S = ( S ij ( x i ))such that Φ = det S = 0 and the Hamiltonian is H = L = n X i =1 T i (cid:0) p i + v i ( x i ) (cid:1) = n X i =1 T i p i + V ( x , . . . , x n ) , where V = P ni =1 T i v i ( x i ), T is the matrix inverse to S : n X j =1 T ij S jk = n X j =1 S ij T jk = δ ik , ≤ i, k ≤ n, (1)ools for Verifying Classical and Quantum Superintegrability 3and δ ik is the Kronecker delta. Here, we must require Π ni =1 T i = 0. We define the quadraticconstants of the motion L k , k = 1 , . . . , n by L k = n X i =1 T ki (cid:0) p i + v i (cid:1) , k = 1 , . . . , n, or p i + v i = n X j =1 S ij L j , ≤ i ≤ n. (2)As is well known, {L j , L k } = 0 , ≤ j, k ≤ n. Here, {A ( x , p ) , B ( x , p ) } = n X i =1 ( ∂ i A ∂ p i B − ∂ p i A ∂ i B ) . Furthermore, by differentiating identity (1) with respect to x h we obtain ∂ h T iℓ = − n X j =1 T ih S ′ hj T jℓ , ≤ h, i, ℓ ≤ n, where S ′ hj = ∂ h S hj .Now we define nonzero functions M kj ( x j , p j , L , . . . , L n ) on the manifold by the requirement { M kj , L ℓ } = T ℓj S jk , ≤ k, j, ℓ ≤ n. It is straightforward to check that these conditions are equivalent to the differential equations2 p j ∂ j M kj + − v ′ j + n X q =1 S ′ jq L q ! ∂ p j M kj = S jk , ≤ j, k ≤ n. We can use the equalities (2) to consider M kj either as a function of x j alone, so that ddx j M kj = S jk / p j or as a function of p j alone. In this paper we will take the former point of view.Now define functions˜ L q = n X j =1 M qj , ≤ q ≤ n. Then we have { ˜ L q , L ℓ } = n X j =1 T ℓj S jq = δ ℓq . (3)This shows that the 2 n − H = L , L , . . . , L n , ˜ L , . . . , ˜ L n , are constants of the motion and, due to relations (3), they are functionally independent. E.G. Kalnins, J.M. Kress and W. Miller Jr.Now let’s consider how this construction works in n = 2 dimensions. By replacing eachseparable coordinate by a suitable function of itself and the constants of the motion by suitablelinear combinations of themselves, if necessary, e.g. [21], we can always assume that the St¨ackelmatrix and its inverse are of the form S = (cid:18) f f − (cid:19) , T = 1 f + f (cid:18) f − f (cid:19) , where f j is a function of the variable x j alone. The constants of the motion L = H and L are given to us via variable separation. We want to compute a new constant of the motion ˜ L functionally independent of L , L . Setting M = M, M = − N , we see that2 p ddx M = 1 , p ddx N = 1 , (4)from which we can determine M , N . Then ˜ L = M − N is the constant of the motion that weseek.The treatment of subgroup separable superintegrable systems in n dimensions, [18], is alsoa special case of the above construction. Suppose the St¨ackel matrix takes the form S = − f , · · · · · ·
00 1 − f · · ·
00 0 . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . · · · − f n − · · · where f i = f i ( x i ), i = 1 , . . . , n . The inverse matrix in this case is T = f f f · · · · · · f f · · · f n − f f f · · · f · · · f n − . . . . . . . . . . . . . . . . . . · · · f n − · · · , which leads exactly to the construction found in [18]. n = 2 As we have seen, for n = 2 and separable coordinates u = x , u = y we have H = L = 1 f ( x ) + f ( y ) (cid:0) p x + p y + v ( x ) + v ( y ) (cid:1) , L = f ( y ) f ( x ) + f ( y ) (cid:0) p x + v ( x ) (cid:1) − f ( x ) f ( x ) + f ( y ) (cid:0) p y + v ( y ) (cid:1) . We will present strategies for determining functions f , f , v , v such that there exists a 3rdconstant of the motion, polynomial in the momenta. The constant of the motion ˜ L = M − N constructed by solving equations (4) is usually not a polynomial in the momenta, hence notdirectly useful in verifying superintegrability. We describe a procedure for obtaining a polynomialconstant from M − N , based on the observation that the integrals M = 12 Z dx √ f H + L − v , N = 12 Z dx √ f H − L − v , ools for Verifying Classical and Quantum Superintegrability 5can often be expressed in terms of multiples of the inverse hyperbolic sine or cosine (or theordinary inverse sine or cosine), and the hyperbolic sine and cosine satisfy addition formulas.Thus we will search for functions f j , v j such that M and N possess this property. Thereis a larger class of prototypes for this construction, namely the second order superintegrablesystems. These have already been classified for 2-dimensional constant curvature spaces [4] and,due to the fact that every superintegrable system on a Riemannian or pseudo-Riemannian spacein two dimensions is St¨ackel equivalent to a constant curvature superintegrable system [22, 23],this list includes all cases. We will typically start our construction with one of these secondorder systems and add parameters to get a family of higher order superintegrable systems.A basic observation is that to get inverse trig functions for the integrals M , N we can choosethe potential functions f ( z ), v ( z ) from the list Z ( z ) = Az + Bz + C, Z ( z ) = A + B sin pz cos py ,Z ( z ) = A cos pz + B sin pz , Z ( z ) = Ae ipz + Be ipz ,Z ( z ) = Az + Bz , Z ( z ) = Az + Bz , sometimes restricting the parameters to special cases. Many of these cases actually occur inthe lists [4] so we are guaranteed that it will be possible to construct at least one second orderpolynomial constant of the motion from such a selection. However, some of the cases lead onlyto higher order constants. Cartesian type systems
For simplicity, we begin with Cartesian coordinates in flat space. The systems of this type arerelated to oscillators and are associated with functions Z j for j = 1 , ,
6. We give two examples. [E1] . For our first construction we give yet another verification that the extended caged har-monic oscillator is classically superintegrable. We modify the second order superintegrablesystem [E1], [4], by looking at the potential V = ω x + ω y + βx + γy , with L = p x + ω x + βx . (For ω = ω this is just [E1].) It corresponds to the choice ofa function of the form Z for each of v , v . Evaluating the integrals, we obtain the solutions M ( x, p x ) = i ω A , sinh A = i (2 ω x − L ) p L − ω β ,N ( y, p y ) = − i ω B , sinh B = i (2 ω y − H + L ) p ( H − L ) − ω γ , to within arbitrary additive constants. We choose these constants so that M , N are proportionalto inverse hyperbolic sines. Then, due to the formula cosh u − sinh u = 1, we can use theidentities (2) to compute cosh A and cosh B :cosh A = 2 ω xp x p L − ω β , cosh B = 2 ω yp y p ( H − L ) − ω γ . Now suppose that ω /ω = k is rational, i.e. k = pq where p , q are relatively prime integers.Then ω = pω , ω = qω andsinh( − ipqω [ M − N ]) = sinh( q A + p B ) , cosh( − ipqω [ M − N ]) = cosh( q A + p B ) , E.G. Kalnins, J.M. Kress and W. Miller Jr.are both constants of the motion. Each of these will lead to a polynomial constant of the motion.Indeed, we can use the relations(cosh x ± sinh x ) n = cosh nx ± sinh nx, cosh( x + y ) = cosh x cosh y + sinh x sinh y, sinh( x + y ) = cosh x sinh y + sinh x cosh y, cosh nx = [ n/ X j =0 (cid:18) n j (cid:19) sinh j x cosh n − j x, sinh nx = sinh x [( n =1) / X j =1 (cid:18) n j − (cid:19) sinh j − x cosh n − j − x. recursively to express each constant as a polynomial in cosh A , sinh A , cosh B , sinh B . Then,writing each constant as a single fraction with denominator of the form(( H − L ) − ω γ ) n / ( L − ω β ) n / we see that the numerator is a polynomial constant of the motion. Note that, by construction,both sinh( − ipqω [ M − N ]) and cosh( − ipqω [ M − N ]) will have nonzero Poisson brackets with L ,hence they are each functionally independent of H , L . Since each of our polynomial constantsof the motion differs from these by a factor that is a function of H , L alone, each polynomialconstant of the motion is also functionally independent of H , L . Similar remarks apply to allof our examples. [E2] . There are several proofs of superintegrability for this system, but we add another. Here, wemake the choice v = Z , v = Z , corresponding to the second order superintegrable system [E2]in [4]. The potential is V = ω x + ω y + αx + βy , where L = p x + ω x + αx and the system is second order superintegrable for the case ω = ω .Applying our method we obtain M ( x, p x ) = i ω A , sinh A = i ( ω x + α ) p ω L + α , cosh A = 2 ω p x p ω L + α ,N ( y, p y ) = i ω B , sinh B = i (2 ω y − H + L ) p ( H − L ) − ω β , cosh B = 2 ω yp y p ( H − L ) − ω β . Thus if ω / ω is rational we obtain a constant of the motion which is polynomial in themomenta. Polar type systems
Next we look at flat space systems that separate in polar coordinates. The Hamiltonian is ofthe form H = p r + 1 r (cid:0) p θ + f ( r ) + g ( θ ) (cid:1) = e − R (cid:0) p R + p θ + v ( R ) + v ( θ ) (cid:1) ,x = R = ln r, y = θ, f = e R , f = 0 . ools for Verifying Classical and Quantum Superintegrability 7Cases for which the whole process works can now be evaluated. Possible choices of f and g are(1) f ( r ) = αr , g ( θ ) = Z ( kθ ) , (2) f ( r ) = αr , g ( θ ) = Z ( kθ ) , (3) f ( r ) = αr , g ( θ ) = Z ( kθ ) , (4) f ( r ) = αr , g ( θ ) = Z ( kθ ) , (5) f ( r ) = αr , g ( θ ) = Z ( kθ ) , (6) f ( r ) = αr , g ( θ ) = Z ( kθ ) . In each case if p is rational there is an extra constant of the motion that is polynomial in thecanonical momenta. [E1] . Case (1) is system [E1] for k = 1. For general k this is the TTW system [12], which wehave shown to be supperintegrable for k rational. Case (2).
This case is not quadratic superintegrable, but as shown in [19] it is superintegrablefor all rational k . [E8] . For our next example we take Case (3) where z = x + iy : V = αz ¯ z + β z k − ¯ z k +1 + γ z k/ − ¯ z k/ . for arbitrary k . (If k = 2 this is the nondegenerate superintegrable system [E8] listed in [4].) Inpolar coordinates, with variables r = e R and z = e R + iθ . Then we have H = e − R (cid:0) p R + p θ + 4 αe R + βe ikθ + γe ikθ (cid:1) , −L = p θ + βe ikθ + γe ikθ , H = e − R (cid:0) p R − L + 4 αe R (cid:1) . Our method yields N ( θ, p θ ) = ik √−L B , sinh B = i (2 L e − ikθ + γ ) p − β L + γ , cosh B = 2 ip θ √−L p − β L + γ e − ikθ ,M ( R, p R ) = i √−L A , sinh A = ( − L e − R − H ) √− α L − H , cosh A = 2 ip R √−L √− α L − H e − R . This system is superintegrable for all rational k . [E17] . Taking Case (4) we have, for z = x + iy : V = α √ z ¯ z + β ¯ z k − z k +1 + γ ¯ z k/ − z k/ . (For k = 1 this is the superintegrable system [E17] in [4].) Then we have H = e − R (cid:0) p R + p θ + αe R + βe − ikθ + γe − ikθ (cid:1) , (5) −L = p θ + βe − ikθ + γe − ikθ , H = e − R (cid:0) p R − L + αe R (cid:1) . Applying our procedure we find the functions N ( θ, p θ ) = 1 k √L B , sinh B = (2 L e ikθ + γ ) p β L − γ , cosh B = − ip θ √L p β L − γ e ikθ ,M ( R, p R ) = 1 √L A , sinh A = ( − L e − R + α ) √ HL − α , cosh A = 2 √L p R √ HL − α e − R . This demonstrates superintegrability for all rational k . [E16] . Case (5) corresponds to [E16] for k = 1, and our method shows that it is superintegrablefor all rational k . Case (6).
This case is not quadratic superintegrable, but it is superintegrable for all rational k . E.G. Kalnins, J.M. Kress and W. Miller Jr. Spherical type systems
These are systems that separate in spherical type coordinates on the complex 2-sphere. TheHamiltonian is of the form H = cosh ψ (cid:0) p ψ + p ϕ + v ( ϕ ) + v ( ψ ) (cid:1) ,x = ϕ, y = ψ, f = 0 , f = 1cosh ψ . Embedded in complex Euclidean 3-space with Cartesian coordinates, such 2-sphere systems canbe written in the form H = J + J + J + V ( s ) , where J = s p s − s p s , J = s p s − s p s , J = s p s − s p s , s + s + s = 1 . [S9] . Here, we have the case v ( ϕ ) = Z and v ( ψ ) is a special case of Z . V = αs + βs + γs . It is convenient to choose spherical coordinates s = cos ϕ sinh ψ , s = sin ϕ cosh ψ, s = tanh ψ. In terms of these coordinates the Hamiltonian has the form H = cosh ψ (cid:20) p ψ + p ϕ + α cos ϕ + β sin ϕ + γ sinh ψ (cid:21) , L = p ϕ + α cos ϕ + β sin ϕ , H = cosh ψ (cid:20) p ψ + L + γ sinh ψ (cid:21) , system [S9] in [4]. We extend this Hamiltonian via the replacement ϕ → kϕ and proceed withour method. Thus H = cosh ψ (cid:20) p ψ + p ϕ + α cos kϕ + β sin kϕ + γ sinh ψ (cid:21) , L = p ϕ + α cos kϕ + β sin kϕ with H expressed as above. The functions that determine the extra constant are M ( ϕ, p ϕ ) = i k √L A , sinh A = i ( L cos(2 kϕ ) − α + β ) p ( L − α − β ) − αβ , cosh A = sin(2 kϕ ) p ϕ p ( L − α − β ) − αβ ,N ( ψ, p ψ ) = i √L B , sinh B = i ( L cosh(2 ψ ) + γ − H ) p ( H + L − γ ) + 4 L γ , cosh B = i sinh(2 ψ ) p ψ p ( H + L − γ ) + 4 L γ . Thus this system is superintegrable for all rational k .ools for Verifying Classical and Quantum Superintegrability 9 [S7] . This system corresponds to v ( φ ) = Z and v ( ψ ) a variant of Z . The system is secondorder superintegrable: V = αs p s + s + βs s p s + s + γs . Choosing the coordinates ψ and ϕ we find H = cosh ψ (cid:18) p ψ + p ϕ + α sinh ψ cosh ψ + β cos ϕ sin ϕ + γ sin ϕ (cid:19) , L = p ϕ + β cos ϕ sin ϕ + γ sin ϕ , H = cosh ψ (cid:18) p ψ + L + α sinh ψ cosh ψ (cid:19) . We make the transformation ϕ → kϕ and obtain H = cosh ψ (cid:18) p ψ + p ϕ + α sinh ψ cosh ψ + β cos kϕ sin kϕ + γ sin kϕ (cid:19) , L = p ϕ + β cos kϕ sin kϕ + γ sin kϕ , with H as before. The functions that determine the extra constants are M ( ϕ, p ϕ ) = 1 √L k A , sinh A = i ( L cos( kϕ ) + β ) p β + 4 L − L γ , cosh A = 2 √L sin( kϕ ) p ϕ p β + 4 L − L γ ,N ( ψ, p ψ ) = i √L B , sinh B = 2 L sinh ψ + α p − α + 4 L − L H , cosh B = 2 i cosh ψp ψ p − α + 4 L − L H . Thus this system is superintegrable for all rational k . [S4] . Here v ( ϕ ) = Z and v ( ψ ) is a variant of Z . This is another system on the sphere thatis second order superintegrable and separates in polar coordinates: V = α ( s − is ) + βs p s + s + γ ( s − is ) p s + s . In terms of angular coordinates ψ , ϕ the Hamiltonian is H = cosh ψ (cid:18) p ψ + p ϕ + αe ikϕ + γ ikϕ + β sinh ψ cosh ψ (cid:19) . After the substitution ϕ → kϕ we have L = p ϕ + αe ikϕ + γ ikϕ , H = cosh ψ (cid:18) p ψ + L + β sinh ψ cosh ψ (cid:19) . The functions that determine the extra constants are M ( ϕ, p ϕ ) = i √L k A , sinh A = i (2 L e − ikϕ − γ ) p L α + γ , cosh A = 2 i √L p ϕ p L α + γ e − ikϕ ,N ( ψ, p ψ ) = i √L B , sinh B = L sinh ψ − β p L + 4 L − β , cosh B = 2 i cosh ψp ψ p L + 4 L − β , so this systems is also superintegrable for all rational k .0 E.G. Kalnins, J.M. Kress and W. Miller Jr. [S2] . Here v ( ϕ ) = Z and v ( ψ ) is a variant of Z : V = αs + β ( s − is ) + γ ( s + is )( s − is ) . which is is second order superintegrable. After the substitution ϕ → kϕ we obtain the system H = cosh ψ (cid:18) p ψ + p ϕ + α sinh ψ + βe ikϕ + γe ikϕ (cid:19) , (6) L = p ϕ + βe ikϕ + γe ikϕ , H = cosh ψ (cid:18) p ψ + L + α sinh ψ (cid:19) . Applying our procedure we find M ( ϕ, p ϕ ) = i √L k A , sinh A = (2 L e − ikϕ − β ) p L γ − β , cosh A = 2 √L p ϕ p L γ − β e − ikϕ ,N ( ψ, p ψ ) = i √L B , sinh B = i ( L cosh(2 ψ ) − α + H ) p ( L − α − H ) − α H , cosh B = √L sinh(2 ψ ) p ψ p ( L − α − H ) − α H . Thus the system is superintegrable for all rational k . In terms of horospherical coordinates on the complex sphere we can construct the Hamiltonian H = y (cid:0) p x + p y + ω x + ω y + α + βx (cid:1) . If ω /ω is rational then this system is superintegrable. However, there is no need to go intomuch detail for the construction because a St¨ackel transform, essentially multiplication by 1 /y ,takes this system to the flat space system generalizing [E2] and with the same symmetry algebra.There is a second Hamiltonian which separates on the complex 2 sphere H = y (cid:16) p x + p y + αx + β + ω x + ω y (cid:17) . It is superintegrable for ω /ω rational, as follows from the St¨ackel transform 1 /y from thesphere to flat space. If ω = ω then we obtain the system [S2]. We note that each of thepotentials associated with horospherical coordinates can quite generally be written in termsof s , s , s coordinates using the relations y = − is − is , x = − s s − is , s + s + s = 1 . These are systems of the form H = 1 Z j ( x ) − Z k ( y ) (cid:0) p x − p y + ˆ Z j ( x ) − ˆ Z k ( y ) (cid:1) , where j , k can independently take the values 2, 3, 4 and Z j , ˆ Z j depend on distinct parameters.Superintegrability is possible because of the integrals Z dx √ C + Be ikx + Ae ikx = 12 k √ C arcsin (cid:18) Ce − ikx + B √ AC − B (cid:19) , ools for Verifying Classical and Quantum Superintegrability 11 Z dx q A + B sin kx cos kx + C = − k √ C arcsin (cid:18) − C sin kx + B √ C + 4 CA + B (cid:19) , Z dx q A cos kx + B sin kx + C = − k √ C arcsin C cos 2 kx + A − B p ( A + B + C ) − AB ! . Consider an example of this last type of system: H = 1 A ′ e ikx + B ′ e ikx − a ′ e iqy − b ′ e iqy (cid:0) p x − p y + Ae ikx + Be ikx − ae iqy − be iqy (cid:1) . The equation H = E admits a separation constant p x + ( A − EA ′ ) e ikx + ( B − EB ′ ) e ikx = p y + ( a − Ea ′ ) e iqy + ( b − Eb ′ ) e iqy = L . As a consequence of this we can find functions M ( x, p x ) and N ( y, p y ) where M ( x, p x ) = 14 k √L arcsin Ae − ikx + ˆ B p L ˆ A − ˆ B ! , where ˆ A = A − EA ′ , ˆ B = B − EB ′ and N ( y, p y ) = 14 p √L arcsin ae − iqy + ˆ b p L ˆ a − ˆ b ! , where ˆ a = a − Ea ′ , ˆ b = b − Eb ′ . We see that if qk is rational then we can generate an extraconstant which is polynomial in the momenta.The superintegrable systems that have involved the rational functions Z , Z and Z workbecause of the integrals Z dx √ Ax + Bx + C = 1 √ A arcsinh (cid:18) Ax + C √ AB − C (cid:19) and Z dxx √ Ax + Bx + C = − √ B arcsinh (cid:18) B + Axx √ A − BC (cid:19) . We give a brief review of the construction of the canonical form for a symmetry operator [19].Consider a Schr¨odinger equation on a 2D real or complex Riemannian manifold with Laplace–Beltrami operator ∆ and potential V : H Ψ ≡ (∆ + V )Ψ = E Ψthat also admits an orthogonal separation of variables. If { u , u } is the orthogonal separablecoordinate system the corresponding Schr¨odinger operator has the form H = L = ∆ + V ( u , u ) = 1 f ( u ) + f ( u ) (cid:0) ∂ u + ∂ u + v ( u ) + v ( u ) (cid:1) . (7)2 E.G. Kalnins, J.M. Kress and W. Miller Jr.and, due to the separability, there is the second-order symmetry operator L = f ( u ) f ( u ) + f ( u ) (cid:0) ∂ u + v ( u ) (cid:1) − f ( u ) f ( u ) + f ( u ) (cid:0) ∂ u + v ( u ) (cid:1) , i.e., [ L , H ] = 0 , and the operator identities f ( u ) H + L = ∂ u + v ( u ) , f ( u ) H − L = ∂ u + v ( u ) . We look for a partial differential operator ˜ L ( H, L , u , u ) that satisfies[ H, ˜ L ] = 0 . We require that the symmetry operator take the standard form˜ L = X j,k (cid:0) A j,k ( u , u ) ∂ u u + B j,k ( u , u ) ∂ u + C j,k ( u , u ) ∂ u + D j,k ( u , u ) (cid:1) H j L k . (8)We have shown that we can write˜ L ( H, L , u , u ) = A ( u , u , H, L ) ∂ + B ( u , u , H, L ) ∂ + C ( u , u , H, L ) ∂ + D ( u , u , H, L ) , (9)and consider ˜ L as an at most second-order order differential operator in u , u that is analyticin the parameters H , L . Then the conditions for a symmetry can be written in the compactform A u u + A u u + 2 B u + 2 C u = 0 , (10) B u u + B u u − A u v + 2 D u − Av ′ + (2 A u f + Af ′ ) H − A u L = 0 , (11) C u u + C u u − A u v + 2 D u − Av ′ + (2 A u f + Af ′ ) H + 2 A u L = 0 , (12) D u u + D u u − B u v − C u v − Bv ′ − Cv ′ + (2 B u f + 2 C u f + Bf ′ + Cf ′ ) H + (2 B u − C u ) L = 0 . (13)We can further simplify this system by noting that there are two functions F ( u , u , H, L ), G ( u , u , H, L ) such that (10) is satisfied by A = F, B = − ∂ F − ∂ G, C = − ∂ F + ∂ G. (14)Then the integrability condition for (11), (12) is (with the shorthand ∂ j F = F j , ∂ jℓ F = F jℓ ,etc., for F and G ),2 G + 12 F + 2 F ( v − f H + L ) + 3 F ( v ′ − f ′ H ) + F ( v ′′ − f ′′ H ) − G + 12 F + 2 F ( v − f H − L ) + 3 F ( v ′ − f ′ H ) + F ( v ′′ − f ′′ H ) , (15)and equation (13) becomes12 F + 2 F ( v − f H ) + F ( v ′ − f ′ H ) + 12 G + 2 G ( v − f H − L )+ G ( v ′ − f ′ H ) = − F − F ( v − f H ) − F ( v ′ − f ′ H ) + 12 G + 2 G ( v − f H + L ) + G ( v ′ − f ′ H ) . (16)ools for Verifying Classical and Quantum Superintegrability 13Here, any solution of (15), (16) with A , B , C not identically 0 corresponds to a symmetryoperator that does not commute with L , hence is algebraically independent of the symmet-ries H , L . (Informally, this follows from the construction and uniqueness of the canonical formof a symmetry operator. The operators ˜ L = g ( H, L ) algebraically dependent on H and L areexactly those such that A = B = C = 0, D = g ( H, L ). A formal proof is technical.) Note alsothat solutions of the canonical equations, with H , L treated as parameters, must be interpretedin the form (8) with H and L on the right, to get the explicit symmetry operators. In [19] we used the canonical form for symmetry operators to demonstrate the quantum superin-tegrability of the TTW system [12, 13] for all rational k , as well as a system on the 2-hyperboloidof two sheets [19]. Here we give another illustration of this construction by applying it to the 2Dcaged oscillator [11]. For p = q This is the second order superintegrable system [E1] on complexEuclidean space, as listed in [4]. Here, H Ψ = ( ∂ + ∂ + V ( u , u ))Ψ , (17)where V ( u , u ) = ω (cid:0) p u + q u (cid:1) + α u + α u , in Cartesian coordinates. We take p , q to be relatively prime positive integers. Thus f = 1 , f = 0 , v = ω p u + α u , v = ω q u + α u . The 2nd order symmetry operator is L = − (cid:0) ∂ + v ( u ) (cid:1) , and we have the operator identities ∂ = − ( v ( u ) + H + L ) , ∂ = − v ( u ) + L . Based on the results of [18] for the classical case, we postulate expansions of F , G in finiteseries F = X a,b A a,b E a,b ( u , u ) , G = X a,b B a,b E a,b ( u , u ) , E a,b ( u , u ) = u a u b . (18)The sum is taken over terms of the form a = a + m , b = b + n , and c = 0 ,
1, where m , n are integers. The point ( a , b ) could in principle be any point in R ,Taking coefficients with respect to the basis (18) in each of equation (15) and (16) givesrecurrence relations for these coefficients. The shifts in the indices of A and B are integers andso we can view this as an equation on a two-dimensional lattice with integer spacings. Whilethe shifts in the indices are of integer size, we haven’t required that the indices themselves beintegers, although they will turn out to be so in our solution. The 2 recurrence relations areof a similar complexity, but rather than write them out separately, we will combine them intoa matrix recurrence relation by defining C a,b = (cid:18) A a,b B a,b (cid:19) . · · · · · · · ·· · • · · · · ·· · · • · · · ·· · • · · · · ·· • · • · • · ·· · ◦ · • · • ·· · · • · · · ·· · · · · · · · Figure 1.
The template. Points contributing to the recurrence relation are marked with large dots ( • , ◦ ).The large dot on the bottom center corresponds to the position ( a − , b + 1), the large dot at the topcorresponds to the position ( a + 4 , b ) and ◦ corresponds to position (0 , We write the 2 recurrence relations in matrix form as = M a,b C a,b + M a − ,b +1 C a − ,b +1 + M a,b +2 C a,b +2 + M a,b +4 C a,b +4 + M a +1 ,b − C a +1 ,b − + M a +1 ,b +1 C a +1 ,b +1 + M a +1 ,b +3 C a +1 ,b +3 + M a +2 ,b C a +2 ,b + M a +3 ,b +1 C a +3 ,b +1 + M a +4 ,b C a +4 ,b , where each M i,j is a 2 × M a,b = (cid:18) ω ([ b + 1] q − [ a + 1] p ) ([ b + 1] q + [ a + 1] p ) 00 − ω ( bq − ap ) ( bq + ap ) (cid:19) ,M a − ,b +1 = (cid:18) ω p a ( b + 1) 0 (cid:19) ,M a,b +2 = (cid:18) L ( b + 2)( b + 1) 00 − L ( b + 2)( b + 1) (cid:19) ,M a,b +4 = (cid:18) ( b + 3)( b + 1)( b + 6 b + 4 α + 8) 00 − ( b + 4)( b + 2)( b + 4 b + 4 α + 3) (cid:19) ,M a +1 ,b − = (cid:18) ω q b ( a + 1) 0 (cid:19) ,M a +1 ,b +1 = (cid:18) − H ( a + 1)( b + 1) 0 (cid:19) ,M a +1 ,b +3 = (cid:18) a + 1)( b + 3)( b + 2)( b + 1) ( a + 1)( b + 2)( b + 4 b + 4 α + 3) 0 (cid:19) ,M a +2 ,b = (cid:18) H + L )( a + 2)( a + 1) 00 − H + L )( a + 2)( a + 1) (cid:19) ,M a +3 ,b +1 = (cid:18) a + 3)( a + 2)( a + 1)( b + 1) ( a + 2)( b + 1)( a + 4 a + 4 α + 3) 0 (cid:19) ,M a +4 ,b = (cid:18) − ( a + 3)( a + 1)( a + 6 a + 4 α + 8) 00 ( a + 4)( a + 2)( a + 4 a + 4 α + 3) (cid:19) . It is useful to visualize the the set of points in the lattice which enter into this recurrencefor a given choice of ( a, b ). These are represented in Fig. 1. Although the recurrence relates10 distinct points, some major simplifications are immediately apparent. We say that thelattice point ( a, b ) has even parity if a + b is an even integer and odd parity if a + b is odd. Eachrecurrence relates only lattice points of the same parity. Because of this we can assume that thenonzero terms C a,b will occur for points of one parity, while only the zero vector will occur forools for Verifying Classical and Quantum Superintegrability 15points with the opposite parity. A second simplification results from the fact that the recurrencematrices M a + m,b + n are of two distinct types. Either m , n are both even, in which case M a + m,b + n is diagonal, or m , n are both odd, in which case the diagonal elements of M a + m,b + n are zero.Another simplification follows from the observation that it is only the ratio p/q = r that mattersin our construction. We want to demonstrate that the caged anisotropic oscillator is operatorsuperintegrable for any rational r . The construction of any symmetry operator independentof H and L will suffice. By writing ω = ω ′ / p ′ = 2 p , q ′ = 2 q in H , we see that, without lossof generality, we can always assume that p , q are both even positive integers with a single 2 astheir only common factor.If for a particular choice of p , q we can find a solution of the recurrence relations with onlya finite number of nonzero vectors C a,b then there will be a minimal lattice rectangle withvertical sides and horizontal top and bottom that encloses the corresponding lattice points.The top row a of the rectangle will be the highest row in which nonzero vectors C a ,b occur.The bottom row a will be the lowest row in which nonzero vectors C a ,b occur. Similarly,the minimal rectangle will have right column b and left column b . Now slide the templatehorizontally across the top row such that only the lowest point on the template lies in the toprow. The recurrence gives ( a + 1) bA a ,b = 0 for all columns b . Based on examples for theclassical system, we expect to find solutions for the quantum system such that a ≥ b ≥
0. Thus we require A a ,b = 0 along the top of the minimal lattice rectangle, so that allvectors in the top row take the form C a ,b = (cid:18) B a ,b (cid:19) . Now we move the template such that the recurrence M a,b , second row from the bottom and onthe left, lies on top of the lattice point ( a , b ). This leads to the requirement ( b q − a p )( b q + a p ) B a ,b = 0. Again, based on hints from specific examples, we postulate B a ,b = 0, a = q , b = p . Since we can assume that p , q are even, this means that all odd lattice points correspondto zero vectors. Next we slide the template vertically down column b such that only the lefthand point on the template lies in column b . The recurrence gives ( b − aA a,b = 0 for allrows a . Since b is even, we postulate that C a,b = (cid:18) B a,b (cid:19) for all lattice points in column b .Note that the recurrence relations preserve the following structure which we will require:1. There is only the zero vector at any lattice point ( a, b ) with a + b odd.2. If the row and column are both even then C a,b = (cid:18) B a,b (cid:19) .3. If the row and column are both odd then C a,b = (cid:18) A a,b (cid:19) .This does not mean that all solutions have this form, only that we are searching for at least onesuch solution.To finish determining the size of the minimum lattice rectangle we slide the template verticallydownward such that only the right hand point on the template lies in column b . The recurrencegives ( b − b − A ( a, b ) = 0, b ( b − B a,b = 0 for all rows a . There are several possiblesolutions that are in accordance with our assumptions, the most conservative of which is b = 0.In the following we assume only that B a ,b = 0, the structure laid out above, and the necessaryimplications of these that follow from a step by step application of the recurrences. If we find6 E.G. Kalnins, J.M. Kress and W. Miller Jr.a vector at a lattice point that is not determined by the recurrences we shall assume it to bezero. Our aim is to find one nonzero solution with support in the minimum rectangle, not toclassify the multiplicity of all such solutions.Now we carry out an iterative procedure that calculates the values of C a,b at points in thelattice using only other points where the values of C i,j are already known. We position thetemplate such that the recurrence M a,b , second row from the bottom and on the left (of thetemplate), lies above the lattice point ( a , b ), a = q , and slide it from right to left along the toprow. The case b = b = p has already been considered. For the remaining cases we have2 ω p a ( b + 1) A a − ,b +1 = 2 ω ( bq − a p )( bq + a p ) B a ,b + 2 L ( b + 2)( b + 1) B a ,b +2 + 12 ( b + 4)( b + 2)( b + 4 b + 4 α + 3) B a ,b +4 . (19)Only even values of b need be considered. Now we lower the template one row and again slide itfrom right to left along the row. The contribution of the lowest point on the template is 0 andwe find2 ω ( q [ b + 2] − p [ a ])( q [ b + 2] + p [ a ]) A a − ,b +1 = − a ( b + 4)( b + 3)( b + 2) B a ,b +4 − L ( b + 3)( b + 2) A a − ,b +3 −
12 ( b + 4)( b + 2)( b + 8 b + 11 α + 8) A a − ,b +5 , (20)where we have replaced b by b + 1 so that again only even values of b need be considered.For the first step we take b = p −
2. Then equation (19) becomes2 ω p qA q − ,p − = − ω q ( p − B q,p − + 2 L p ( p − B q,p , and (20) is vacuous in this case, leaving A q − ,p − undetermined. Note that several of the termslie outside the minimal rectangle. From these two equations we can solve for B q,p − in terms ofour given B q,p , A q − ,p . Now we march across both rows from right to left. At each stage our twoequations now allow us to solve uniquely for A q − ,b +1 and B q,b in terms of A ’s and B ’s to theright (which have already been computed). We continue this until we reach b = 0 and then stop.At this point the top two rows of the minimal rectangle have been determined by our choiceof B q,p and A q − ,p − . We repeat this construction for the third and fourth rows down, then thenext two rows down, etc. The recurrence relations grow more complicated as the higher rows ofthe template give nonzero contributions. However, at each step we have two linear relations (cid:18) ω p a ( b + 1) − ω ( bq − ap )( bq + ap )2 ω ( q [ b + 2] − pa )( q [ b + 2] + pa ) 0 (cid:19) (cid:18) A a − ,b +1 B a,b (cid:19) = · · · , where the right hand side is expressed in terms of A ’s and B ’s either above or on the same linebut to the right of A a − ,b +1 , B a,b , hence already determined. Since the determinant of the 2 × A a − ,b +1 , B a,b uniquely in terms ofquantities already determined. This process stops with rows a = 1 , a = 0, the bottom row, needs special attention. We position the template such thatthe recurrence M a,b lies above the lattice point (0 , b ), and slide it from right to left along thebottom row. The bottom point in the template now contributes 0 so at each step we obtain anexpression for B ,b in terms of quantities A , B from rows either above row 0 or to the right ofcolumn b in row 0, hence already determined. Thus we can determine the entire bottom row,with the exception of the value at (0 , M a,b onthe template is above (0 ,
0) the coefficient of B , vanishes, so the value of B , is irrelevantand we have a true condition on the remaining points under the template. However, this isa linear homogeneous equation in the parameters B q,p , A q − ,p − : χB q,p + ηA q − ,p − = 0 forools for Verifying Classical and Quantum Superintegrability 17constants χ , η . Hence if, for example, χ = 0 we can require B q,p = − ( η/χ ) A q − ,p − and satisfythis condition while still keeping a nonzero solution. Thus after satisfying this linear conditionwe still have at least a one parameter family of solutions, along with the arbitrary B , (whichis clearly irrelevant since it just adds a constant to the function G ). However, we need to checkthose recurrences where the point M a,b on the template slides along rows a = − , − , − , − a = −
4, and for cases a = − , − , − M a,b on thetemplate slides along columns b = − , − , − , −
4. However again the recurrences are vacuous.We conclude that there is a one-parameter (at least) family of solutions to the caged aniso-tropic oscillator recurrence with support in the minimal rectangle. By choosing the arbitraryparameters to be polynomials in H , L we get a finite order constant of the motion. Thus thequantum caged anisotropic oscillator is superintegrable. We note that the canonical operatorconstruction permits easy generation of explicit expressions for the defining operators in a largenumber of examples. Once the basic rectangle of nonzero solutions is determined it is easy tocompute dozens of explicit examples via Maple and simple Gaussian elimination. The generationof explicit examples is easy; the proof that the method works for all orders is more challenging. Example.
Taking p = 6 and q = 4, we find (via Gaussian elimination) a solution of therecurrence with nonzero coefficients A , , A , , A , , A , , A , , A , ,B , , B , , B , , B , , B , , B , , B , , B , , B , , B , , B , . We can take A , = a and A , = a and then all other coefficients will depend linearly on a and a . Calculating the A , B , C and D from equations (11), (12), (14) we find the coefficientof H − has a factor of 2 a + 9 a and so we set a = ω and a = − / a to obtain the symmetryoperator ˜ L , (9), where u , u are Cartesian coordinates and A = 94096 ω u u L + 32048 ω u u HL − ω u u (cid:0) u − u (cid:1) L + 116 ω u u H + 164 ω α u u − ω u u (cid:0) u − u (cid:1) − ω u u ,B = − u L − u HL + 332768 ω u (cid:0) u − u (cid:1) L − ω u u HL + (cid:18) ω u − ω α u + 11024 ω u u (cid:0) u − u (cid:1)(cid:19) L + (cid:18) ω u − ω α u − ω u u (cid:19) HL + (cid:18) ω u u − ω α u u − ω u u (cid:19) H + (cid:18) ω α u (cid:0) u − u (cid:1) + 124 ω u u (cid:0) u − u (cid:1) − ω u (cid:0) u − u (cid:1)(cid:19) L + 916 ω α u u + 17128 ω α u − u u (cid:0) u − u (cid:1) ω + 12 ω u u − ω u ,C = − u L − u H L − u HL + 173728 ω u (cid:0) u − u (cid:1) L + 173728 ω u (cid:0) u − u (cid:1) HL − ω u H L + (cid:18) ω α u − ω α u + 601196608 ω u − u (cid:0) u − u u + 32 u (cid:1) ω (cid:19) L (cid:18) ω u − ω α u + 1288 ω u (cid:0) u − u (cid:1)(cid:19) HL + (cid:18) ω u − ω α u − ω u (cid:19) H + (cid:18) u u ω α + 12 ω u u − u (cid:0) − u + 31 u (cid:1) ω (cid:19) H + (cid:18) ω α u u − ω α u − u u (cid:0) u − u (cid:1) ω − u (cid:0) u − u (cid:1) ω (cid:19) L − ω α α u − ω α u + 931024 ω α u − ω α u u + 51024 ω α u + 1128 ω u (cid:0) u + 837 u − u u (cid:1) − ω u u − ω u ,D = ω (cid:18) u − u (cid:19) L + ω (cid:18) u − u (cid:19) HL − ω u H L + ω (cid:18) u u − u − u (cid:19) L + 11152 ω u (cid:0) u − u (cid:1) HL − ω u H + (cid:18) ω α (cid:18) u − u (cid:19) − ω α u − ω u (cid:0) u − u u + 189 u (cid:1) + (cid:18) u − u (cid:19) ω (cid:19) L + (cid:18) ω α (cid:18) u − u (cid:19) + 14 ω u (cid:0) u − u (cid:1) + ω (cid:18) u − u (cid:19)(cid:19) H − ω α u − ω α u (cid:0) u − u (cid:1) + ω (cid:18) u − u u + 6532 u (cid:19) − u ω u (cid:0) u − u (cid:1) . Using Maple, we have checked explicitly that the operator ˜ L commutes with H . Note thatit is of 6th order. Taking the formal adjoint ˜ L ∗ , [19], we see that S = ( ˜ L + ˜ L ∗ ) is a 6th order,formally self-adjoint symmetry operator. This second proof is very special for the oscillator and exploits the fact that separation ofvariables in Cartesian coordinates is allowed Here we write the Hamiltonian in the form H = ∂ x + ∂ y − µ x − µ y + − a x + − a y . This is the same as (17) with u = x , u = y , µ = − p ω , µ = − q ω , α = − a , α = − a . We look for eigenfunctions for the equation H Ψ = λ Ψ of the form Ψ = XY . Wefind the normalized solutions X n = e − µ x x a + L a n ( µ x ) , Y m = e − µ y y a + L a m ( µ y ) , where the L αn ( x ) are associated Laguerre polynomials [24]. For the corresponding separationconstants we obtain λ x = − µ (2 n + a + 1) , λ y = − µ (2 m + a + 1) . ools for Verifying Classical and Quantum Superintegrability 19The total energy is − λ x − λ y = E . Taking µ = pµ and µ = qµ where p and q are integers wefind that the total energy is E = − µ ( pn + qm + pa + p + qa + q ) . Therefore, in order that E remain fixed we can admit values of integers m and n such that pn + qm is a constant. One possibility that suggests itself is that n → n + q , m → m − p . Tosee that this is achievable via differential operators we need only considerΨ = L a n ( z ) L a m ( z ) , z = µ x , z = µ y . We now note the recurrence formulas for Laguerre polynomials viz. x ddx L αp ( x ) = pL αp ( x ) − ( p + α ) L αp − ( x ) = ( p + 1) L αp +1 ( x ) − ( p + 1 + α − x ) L αp ( x ) . Because of the separation equations we can associate p with a differential operator for both m and n in the expression for Ψ. We do not change the coefficients of L αp ± ( x ). Therefore wecan raise or lower the indices m and n in Ψ using differential operators. In particular we canperform the transformation n → n + q , m → m − p and preserve the energy eigenvalue. To seehow this works we observe the formulas D + ( µ , x ) X n = ∂ x − xµ ∂ x − µ + µ x + − α x ! X n = − µ ( n + 1) X n +1 ,D − ( µ , y ) Y m = ∂ y + 2 yµ ∂ y + µ + µ y + − α y ! Y m = − µ ( m + α ) Y m − . In particular, if we make the choice µ = 2 µ , µ = µ then the operator D + (2 µ, x ) D − ( µ, y ) transforms X n Y m to − µ ( n +1)( m + α )( m − α ) X n +1 Y m − . We see that this preserves theenergy eigenspace. Thus can easily be extended to the case when µ = pµ and µ = qµ for p and q integers. A suitable operator is D + ( pµ, x ) q D − ( qµ, y ) p . Hence we have constructed a differentialoperator that commutes with H ! The caged oscillator is quantum superintegrable. This worksto prove superintegrability in all dimensions as we need only take coordinates pairwise. Remark.
This second proof of superintegrability, using differential recurrence relations forLaguerre polynomials is much more transparent than the canonical operator proof, and it gen-eralizes immediately to all dimensions. Unfortunately, this oscillator system is very simple andthe recurrence relation approach is more difficult to implement for more complicated potentials.Special function recurrence relations have to be worked out. Also, we only verified explicitlythe commutivity on an eigenbasis for a bound state.
In [16] there is introduced a new family of Hamiltonians with a deformed Kepler–Coulombpotential dependent on an indexing parameter k which is shown to be related to the TTWoscillator system system via coupling constant metamorphosis. The authors showed that thissystem is classically superintegrable for all rational k . Here we demonstrate that this systemis also quantum superintegrable. The proof follows easily from the canonical equations for thesystem.The quantum TTW system is H Ψ = E Ψ with H given by (7) where u = R, u = θ, f = e R , f = 0 , v = αe R , (21)0 E.G. Kalnins, J.M. Kress and W. Miller Jr. v = β cos ( kθ ) + γ sin ( kθ ) = 2( γ + β )sin (2 kθ ) + 2( γ − β ) cos(2 kθ )sin (2 kθ ) . (Setting r = e R we get the usual expression for this system in polar coordinates r , θ .) In ourpaper [19] we used the canonical form for symmetry operators to establish the superintegrabilityof this system. Our procedure was, based on the results of [20] for the classical case, to postulateexpansions of F , G in finite series F = X a,b,c A a,b,c E a,b,c ( R, θ ) , G = X a,b,c B a,b,c E a,b,c ( R, θ ) ,E a,b, = e aR sin b (2 kθ ) , E a,b, = e aR sin b (2 kθ ) cos(2 kθ ) . The sum is taken over terms of the form a = a + m , b = b + n , and c = 0 ,
1, where m , n areintegers, a is a positive integer and b is a negative integer. Here F , G are the solutions of thecanonical form equations (14), (15), (16) for the TTW system. We succeeded in finding finiteseries solutions for all rational k , and this proved superintegrability.In [16] the authors point out that under the St¨ackel transformation determined by the po-tential U = e R the TTW system transforms to the equivalent system ˆ H Ψ = − α Ψ, whereˆ H = 1 e R (cid:18) ∂ R + ∂ θ − E exp(2 R ) + β cos ( kθ ) + γ sin ( kθ ) (cid:19) . (22)Then, setting r = e R , φ = 2 θ , we find the deformed Kepler–Coulomb system. (cid:18) ∂ r + 1 r + 1 r ∂ φ − E r + β r cos ( kφ/
2) + γ r sin ( kφ/ (cid:19) Ψ = − α . From the canonical equations, it is virtually immediate that system (22) is superintegrable.Indeed the only difference between the functions (21) defining the TTW system and the functionsdefining the St¨ackel transformed system is that f and v are replaced by f = exp(4 R ) and v = − E exp(2 R ) (we can set E = H ) and the former H is replaced by − α . From this we findthat the only difference between the canonical equations for the TTW system and the canonicalequations for the transformed system is that α becomes − ˆ H and H becomes − α . Since ourproof of the superintegrability of the TTW system did not depend on the values of H and α ,the same argument shows the superintegrability of the transformed system. Also, any solution(i.e., a second commuting operator) for the TTW system gives rise to a corresponding operatorfor the transformed system by replacing α and H with − ˆ H and − α , respectively. Example.
For the deformed Kepler system with k = 2, we take p = 2 and q = 1 and useGaussian elimination to find a solution with nonzero coefficients A − , , , A − , , , B − , , , B − , , , B − , , , B − , , , B , , , in which A − , , and A − , , are two independent parameters. To achieve the lowest ordersymmetry and ensure that the A , B , C and D are polynomial in ˆ L and ˆ H , we choose A − , , =32( ˆ L −
4) and A − , , = 0 and find, A = 32 e − R sin(4 θ ) ˆ L − e − R sin(4 θ ) ,B = 8 e − R cos(4 θ ) ˆ L + (8( β − γ ) e − R + 4 Ee − R cos(4 θ ) − e − R cos(4 θ )) ˆ L − e − R cos(4 θ ) − Ee − R cos(4 θ ) + 40( γ − β ) e − R + 4 E ( β − γ ) e − R ,C = − e − R sin(4 θ ) ˆ L − θ ) ˆ H ˆ L + 8( e − R − Ee − R ) sin(4 θ ) ˆ L + 20 sin(4 θ ) ˆ H + 96 e − R sin(4 θ ) + 32 Ee − R sin(4 θ ) − E sin(4 θ ) , ools for Verifying Classical and Quantum Superintegrability 21 D = − e − R cos(4 θ ) L − θ ) ˆ H ˆ L + (128 e − R cos(4 θ ) − Ee − R cos(4 θ ) + 16( γ − β ) e − R ) ˆ L + 24 cos(4 θ ) ˆ H + 48( β − γ ) e − R + 48 Ee − R cos(4 θ ) − E ( γ − β ) e − R + 2 E cos(4 θ ) . It has been explicitly checked, using Maple, that the fifth order operator obtained from theseexpressions commutes with ˆ H . Using this same idea we can find a new superintegrable system on the 2-sheet hyperboloid bytaking a St¨ackel transformation of the quantum caged oscillator H Ψ = E Ψ, (17) in Cartesiancoordinates u , u by multiplying with the potential U = 1 /u . The transformed system is u (cid:18) ∂ + ∂ + ω u ( p u + q u ) − Eu + α u u (cid:19) Ψ = − α Ψ . We embed this system as the upper sheet s > s − s − s = 1 in 3-dimensional Minkowski space with Minkowski metric ds = − ds + ds + ds , via the coordinatetransformation s = 1 + u + u u , s = 1 − u − u u , s = u u , u > . Then the potential for the transformed system is˜ V = α s − E ( s + s ) + ω ( p − q )( s + s ) + ω q ( s − s )( s + s ) . (23)This is an extension of the complex sphere system [S2], distinct from the system (6) that weproved classically superintegrable. It is an easy consequence of the results of [17] that theclassical version of system (23) is also superintegrable for all rationally related p , q . However,quantum superintegrability isn’t obvious. However, from the results of Section 2.2 it is virtuallyimmediate that this new system is quantum superintegrable for all relatively prime positiveintegers p , q . This follows from writing down the canonical equations (15), (16), first for thecaged oscillator where f = 1 , f = 0 , v = ω p u + α u , v = ω q u + α u , and then for the St¨ackel transformed system where f = 1 u , f = 0 , v = ω p u − E, v = ω q u + α u . The equations are identical except for the switches α → − E , H → − α . Thus our proof ofquantum superintegrability for the caged oscillator carries over to show that the system on thehyperboloid is also quantum superintegrable. A basic issue in discovering and verifying higher order superintegrabily of classical and quantumsystems is the difficulty of manipulating high order constants of the motion and, particularly,higher order partial differential operators. We have described several approaches to simplify such2 E.G. Kalnins, J.M. Kress and W. Miller Jr.calculations. Although our primary emphasis in this paper was to develop tools for verifyingclassical and quantum superintegrabity at all orders, we have presented many new results. Theclassical Eucidean systems [E8], [E17] and most of the examples of superintegrability for spaceswith non-zero scalar curvature are new. We have explored the limits of the construction ofclassical superintegrable systems via the methods of Section 1.1 for the case n = 2. Furtheruse of this method will require looking at n >
2, where new types of behavior occur, such asappearance of superintegrable systems that are not conformally flat. We also developed theuse of the canonical form for a symmetry operator to prove quantum superintegrability. Weapplied the method to the n = 2 caged anisotropic oscillator to give the first proof of quantumsuperintegrability for all rational k . We used the St¨ackel transform together with the canonicalequations to give the first proofs of quantum superintegrability for all rational k of a 2D deformedKepler–Coulomb system, and of the caged anisotropic oscillator on the 2-hyperboloid. Thenwe introduced a new approach to proving quantum superintegrability via recurrence relationsobeyed by the energy eigenfunctions of a quantum system, and gave an alternate proof of thesuperintegrabilty for all rational k of the caged anisotropic oscillator. This proof clearly extendsto all n . The recurrences for the caged oscillator are particularly simple, but the method wepresented shows great promise for broader application. Clearly, we are just at the beginningof the process of discovery and classification of higher order superintegrable systems in alldimensions n > References [1] Tempesta P., Winternitz P., Harnad J., Miller W. Jr., Pogosyan G. (Editors), Superintegrability in classicaland quantum systems (Montr´eal, 2002),
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