Tools for Verifying Classical and Quantum Superintegrability

Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n−1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are al...

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Дата:	2010
Автори:	Kalnins, E.G., Kress, J.M., Willard Miller, Jr.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Tools for Verifying Classical and Quantum Superintegrability / E.G. Kalnins, J.M. Kress, Jr. Willard Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	nasplib_isofts_kiev_ua-123456789-1465032025-02-23T20:05:33Z Tools for Verifying Classical and Quantum Superintegrability Kalnins, E.G. Kress, J.M. Willard Miller, Jr. Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n−1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the constructions to date are for n=2 but cases where n>2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stäckel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential. 2010 Article Tools for Verifying Classical and Quantum Superintegrability / E.G. Kalnins, J.M. Kress, Jr. Willard Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C99; 20C35; 22E70 DOI:10.3842/SIGMA.2010.066 https://nasplib.isofts.kiev.ua/handle/123456789/146503 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n−1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the constructions to date are for n=2 but cases where n>2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stäckel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential.
format	Article
author	Kalnins, E.G. Kress, J.M. Willard Miller, Jr.
spellingShingle	Kalnins, E.G. Kress, J.M. Willard Miller, Jr. Tools for Verifying Classical and Quantum Superintegrability Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kalnins, E.G. Kress, J.M. Willard Miller, Jr.
author_sort	Kalnins, E.G.
title	Tools for Verifying Classical and Quantum Superintegrability
title_short	Tools for Verifying Classical and Quantum Superintegrability
title_full	Tools for Verifying Classical and Quantum Superintegrability
title_fullStr	Tools for Verifying Classical and Quantum Superintegrability
title_full_unstemmed	Tools for Verifying Classical and Quantum Superintegrability
title_sort	tools for verifying classical and quantum superintegrability
publisher	Інститут математики НАН України
publishDate	2010
url	https://nasplib.isofts.kiev.ua/handle/123456789/146503
citation_txt	Tools for Verifying Classical and Quantum Superintegrability / E.G. Kalnins, J.M. Kress, Jr. Willard Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 24 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kalninseg toolsforverifyingclassicalandquantumsuperintegrability AT kressjm toolsforverifyingclassicalandquantumsuperintegrability AT willardmillerjr toolsforverifyingclassicalandquantumsuperintegrability
first_indexed	2025-11-24T21:43:24Z
last_indexed	2025-11-24T21:43:24Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 066, 23 pages Tools for Verifying Classical and Quantum Superintegrability Ernest G. KALNINS †, Jonathan M. KRESS ‡ and Willard MILLER Jr.§ † Department of Mathematics, University of Waikato, Hamilton, New Zealand E-mail: math0236@math.waikato.ac.nz URL: http://www.math.waikato.ac.nz ‡ School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia E-mail: j.kress@unsw.edu.au URL: http://web.maths.unsw.edu.au/∼jonathan/ § School of Mathematics, University of Minnesota, Minneapolis, Minnesota,55455, USA E-mail: miller@ima.umn.edu URL: http://www.ima.umn.edu/∼miller/ Received June 04, 2010, in final form August 06, 2010; Published online August 18, 2010 doi:10.3842/SIGMA.2010.066 Abstract. Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n − 1 symmetries polynomial in the canonical momenta, so that they are in fact su- perintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the con- structions to date are for n = 2 but cases where n > 2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mecha- nisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stäckel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler–Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential. Key words: superintegrability; hidden algebras; quadratic algebras 2010 Mathematics Subject Classification: 20C99; 20C35; 22E70 1 Introduction We define an n-dimensional classical superintegrable system to be an integrable Hamiltonian system that not only possesses n mutually Poisson – commuting constants of the motion, but in addition, the Hamiltonian Poisson-commutes with 2n− 1 functions on the phase space that are globally defined and polynomial in the momenta. Similarly, we define a quantum superintegrable system to be a quantum Hamiltonian which is one of a set of n independent mutually commuting differential operators, and that commutes with a set of 2n − 1 independent differential opera- tors of finite order. We restrict to classical systems of the form H = ∑n i,j=1 g ijpipj + V and corresponding quantum systems H = ∆n+ Ṽ . These systems, including the classical Kepler and anisotropic oscillator problems and the quantum anisotropic oscillator and hydrogen atom have great historical importance, due to their remarkable properties, [1, 2]. The order of a classical superintegrable system is the maximum order of the generating constants of the motion (with the Hamiltonian excluded) as a polynomial in the momenta, and the maximum order of the mailto:math0236@math.waikato.ac.nz http://www.math.waikato.ac.nz mailto:j.kress@unsw.edu.au http://web.maths.unsw.edu.au/~jonathan/ mailto:miller@ima.umn.edu http://www.ima.umn.edu/~miller/ http://dx.doi.org/10.3842/SIGMA.2010.066 2 E.G. Kalnins, J.M. Kress and W. Miller Jr. quantum symmetries as differential operators. Systems of 2nd order have been well studied and there is now a structure and classification theory [3, 4, 5, 6, 7, 8]. For 3rd and higher order superintegrable systems much less is known. In particular there have been relatively few examples and there is almost no structure theory. However, within the last three years there has been a dramatic increase in discovery of new families of possible higher order superintegrable classical and quantum systems [9, 10, 11, 12, 13, 14, 15, 16]. The authors and collaborators have developed 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 recent interest. Here, however, the emphasis is on the methods themselves. In Section 2 we review a method for constructing classical constants of the motion of all orders for n-dimensional Hamiltonians that admit a separation of variables in some orthogonal coordinate system. Then we apply it to the case n = 2 to show how to derive superintegrable systems 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 Schrödinger operators such that the Schrödinger eigenvalue equation admits a separation of variables in some orthogonal coordinate system. Then we apply this method to establish the quantum superintegrability of the caged anisotropic oscillator for all frequencies that are rationally related, and for a Stäckel transformed version of this system on the 2-sheet hyperboloid. We give a second proof for the caged oscillator in all dimensions n that relies on recurrence relations for associated Laguerre polynomials. We also apply the canonical equations to establish the quantum superintegrability of a deformed Kepler–Coulomb system for all rational values of a parameter k in the potential. 1.1 The construction tool for classical systems There are far more verified superintegrable Hamiltonian systems in classical mechanics than was the case 3 years ago. The principal method for constructing and verifying these new systems requires that the system is already integrable, in particular, that it admits a separation of variables in some coordinate system. For a Hamiltonian system in 2n-dimensional phase space the separation gives us n second order constants of the motion in involution. In this paper we first review a general procedure, essentially the construction of action angle variables, which yields an additional n − 1 constants, such that the set of 2n − 1 is functionally independent. This is of little interest unless it is possible to extract n new constants of the motion that are polynomial in the momenta, so that the system is superintegrable. We will show how this can be done in many cases. We start first with the construction of action angle variables for Hamiltonian systems 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 admits separation of variables in orthogonal separable coordinates x1, . . . , xn. Then there is an n × n Stäckel matrix S = (Sij(xi)) such that Φ = detS 6= 0 and the Hamiltonian is H = L1 = n∑ i=1 T1i ( p2 i + vi(xi) ) = n∑ i=1 T1i p 2 i + V (x1, . . . , xn), where V = ∑n i=1 T1i vi(xi), T is the matrix inverse to S: n∑ j=1 TijSjk = n∑ j=1 SijTjk = δik, 1 ≤ i, k ≤ n, (1) Tools for Verifying Classical and Quantum Superintegrability 3 and δik is the Kronecker delta. Here, we must require Πn i=1T1i 6= 0. We define the quadratic constants of the motion Lk, k = 1, . . . , n by Lk = n∑ i=1 Tki ( p2 i + vi ) , k = 1, . . . , n, or p2 i + vi = n∑ j=1 SijLj , 1 ≤ i ≤ n. (2) As is well known, {Lj ,Lk} = 0, 1 ≤ j, k ≤ n. Here, {A(x,p),B(x,p)} = n∑ i=1 (∂iA ∂piB − ∂piA ∂iB). Furthermore, by differentiating identity (1) with respect to xh we obtain ∂hTi` = − n∑ j=1 TihS ′ hjTj`, 1 ≤ h, i, ` ≤ n, where S′hj = ∂hShj . Now we define nonzero functions Mkj(xj , pj ,L1, . . . ,Ln) on the manifold by the requirement {Mkj ,L`} = T`jSjk, 1 ≤ k, j, ` ≤ n. It is straightforward to check that these conditions are equivalent to the differential equations 2pj∂jMkj + ( − v′j + n∑ q=1 S′jqLq ) ∂pjMkj = Sjk, 1 ≤ j, k ≤ n. We can use the equalities (2) to consider Mkj either as a function of xj alone, so that d dxj Mkj = Sjk/2pj or as a function of pj alone. In this paper we will take the former point of view. Now define functions L̃q = n∑ j=1 Mqj , 1 ≤ q ≤ n. Then we have {L̃q,L`} = n∑ j=1 T`jSjq = δ`q. (3) This shows that the 2n− 1 functions H = L1,L2, . . . ,Ln, L̃2, . . . , L̃n, are constants of the motion and, due to relations (3), they are functionally independent. 4 E.G. Kalnins, J.M. Kress and W. Miller Jr. Now let’s consider how this construction works in n = 2 dimensions. By replacing each separable coordinate by a suitable function of itself and the constants of the motion by suitable linear combinations of themselves, if necessary, e.g. [21], we can always assume that the Stäckel matrix and its inverse are of the form S = ( f1 1 f2 −1 ) , T = 1 f1 + f2 ( 1 1 f2 −f1 ) , where fj is a function of the variable xj alone. The constants of the motion L1 = H and L2 are given to us via variable separation. We want to compute a new constant of the motion L̃2 functionally independent of L1, L2. Setting M21 = M, M22 = −N , we see that 2p1 d dx1 M = 1, 2p2 d dx2 N = 1, (4) from which we can determine M , N . Then L̃2 = M −N is the constant of the motion that we seek. The treatment of subgroup separable superintegrable systems in n dimensions, [18], is also a special case of the above construction. Suppose the Stäckel matrix takes the form S =  1 −f1, 0 · · · · · · 0 0 1 −f2 0 · · · 0 0 0 . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . 0 0 0 · · · 1 −fn−1 0 0 0 · · · 0 1  where fi = fi(xi), i = 1, . . . , n. The inverse matrix in this case is T =  1 f1 f1f2 · · · · · · f1f2 · · · fn−1 0 1 f2 f2f3 · · · f2 · · · fn−1 0 0 . . . . . . . . . ... . . . . . . . . . . . . . . . . . . 0 0 0 · · · 1 fn−1 0 0 0 · · · 0 1  , which leads exactly to the construction found in [18]. 1.2 Application of the construction for n = 2 As we have seen, for n = 2 and separable coordinates u1 = x, u2 = y we have H = L1 = 1 f1(x) + f2(y) ( p2 x + p2 y + v1(x) + v2(y) ) , L2 = f2(y) f1(x) + f2(y) ( p2 x + v1(x) ) − f1(x) f1(x) + f2(y) ( p2 y + v2(y) ) . We will present strategies for determining functions f1, f2, v1, v1 such that there exists a 3rd constant of the motion, polynomial in the momenta. The constant of the motion L̃2 = M −N constructed by solving equations (4) is usually not a polynomial in the momenta, hence not directly useful in verifying superintegrability. We describe a procedure for obtaining a polynomial constant from M −N , based on the observation that the integrals M = 1 2 ∫ dx1√ f1H+ L2 − v1 , N = 1 2 ∫ dx2√ f2H−L2 − v2 , Tools for Verifying Classical and Quantum Superintegrability 5 can often be expressed in terms of multiples of the inverse hyperbolic sine or cosine (or the ordinary inverse sine or cosine), and the hyperbolic sine and cosine satisfy addition formulas. Thus we will search for functions fj , vj such that M and N possess this property. There is a larger class of prototypes for this construction, namely the second order superintegrable systems. 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 space in two dimensions is Stäckel equivalent to a constant curvature superintegrable system [22, 23], this list includes all cases. We will typically start our construction with one of these second order 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 choose the potential functions f(z), v(z) from the list Z1(z) = Az2 +Bz + C, Z2(z) = A+B sin pz cos2 py , Z3(z) = A cos2 pz + B sin2 pz , Z4(z) = Ae2ipz +Beipz, Z5(z) = Az2 + B z2 , Z6(z) = A z + B z2 , sometimes restricting the parameters to special cases. Many of these cases actually occur in the lists [4] so we are guaranteed that it will be possible to construct at least one second order polynomial constant of the motion from such a selection. However, some of the cases lead only to higher order constants. Cartesian type systems For simplicity, we begin with Cartesian coordinates in flat space. The systems of this type are related to oscillators and are associated with functions Zj for j = 1, 5, 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 superintegrable system [E1], [4], by looking at the potential V = ω2 1x 2 + ω2 2y 2 + β x2 + γ y2 , with L2 = p2 x + ω2 1x 2 + β x2 . (For ω1 = ω2 this is just [E1].) It corresponds to the choice of a function of the form Z5 for each of v1, v2. Evaluating the integrals, we obtain the solutions M(x, px) = i 4ω1 A, sinhA = i(2ω2 1x 2 − L2)√ L2 2 − 4ω2 1β , N(y, py) = −i 4ω2 B, sinhB = i(2ω2 2y 2 −H+ L2)√ (H−L2)2 − 4ω2 2γ , to within arbitrary additive constants. We choose these constants so that M , N are proportional to inverse hyperbolic sines. Then, due to the formula cosh2 u − sinh2 u = 1, we can use the identities (2) to compute coshA and coshB: coshA = 2ω1xpx√ L2 2 − 4ω2 1β , coshB = 2ω2ypy√ (H−L2)2 − 4ω2 2γ . Now suppose that ω1/ω2 = k is rational, i.e. k = p q where p, q are relatively prime integers. Then ω1 = pω, ω2 = qω and sinh(−4ipqω[M −N ]) = sinh(qA+ pB), cosh(−4ipqω[M −N ]) = cosh(qA+ pB), 6 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 (coshx± sinhx)n = coshnx± sinhnx, cosh(x+ y) = coshx cosh y + sinhx sinh y, sinh(x+ y) = coshx sinh y + sinhx cosh y, coshnx = [n/2]∑ j=0 ( n 2j ) sinh2j x coshn−2j x, sinhnx = sinhx [(n=1)/2]∑ j=1 ( n 2j − 1 ) sinh2j−2 x coshn−2j−1 x. recursively to express each constant as a polynomial in coshA, sinhA, coshB, sinhB. Then, writing each constant as a single fraction with denominator of the form ((H−L2)2 − 4ω2 2γ) n1/2(L2 2 − 4ω2 1β)n2/2 we see that the numerator is a polynomial constant of the motion. Note that, by construction, both sinh(−4ipqω[M−N ]) and cosh(−4ipqω[M−N ]) will have nonzero Poisson brackets with L2, hence they are each functionally independent of H, L2. Since each of our polynomial constants of the motion differs from these by a factor that is a function of H, L2 alone, each polynomial constant of the motion is also functionally independent of H, L2. Similar remarks apply to all of our examples. [E2]. There are several proofs of superintegrability for this system, but we add another. Here, we make the choice v1 = Z1, v2 = Z5, corresponding to the second order superintegrable system [E2] in [4]. The potential is V = ω2 1x 2 + ω2 2y 2 + αx+ β y2 , where L2 = p2 x+ω2 1x 2 +αx and the system is second order superintegrable for the case ω1 = ω1. Applying our method we obtain M(x, px) = i 2ω1 A, sinhA = i(ω2 1x+ α)√ 4ω2 1L2 + α2 , coshA = 2ω1px√ 4ω2 1L2 + α2 , N(y, py) = i 4ω2 B, sinhB = i(2ω2 2y 2 −H+ L2)√ (H−L2)2 − 4ω2 2β , coshB = 2ω2ypy√ (H−L2)2 − 4ω2 2β . Thus if ω1/2ω2 is rational we obtain a constant of the motion which is polynomial in the momenta. Polar type systems Next we look at flat space systems that separate in polar coordinates. The Hamiltonian is of the form H = p2 r + 1 r2 ( p2 θ + f(r) + g(θ) ) = e−2R ( p2 R + p2 θ + v1(R) + v2(θ) ) , x = R = ln r, y = θ, f1 = e2R, f2 = 0. Tools for Verifying Classical and Quantum Superintegrability 7 Cases for which the whole process works can now be evaluated. Possible choices of f and g are (1) f(r) = αr2, g(θ) = Z3(kθ), (2) f(r) = α r , g(θ) = Z3(kθ), (3) f(r) = αr2, g(θ) = Z4(kθ), (4) f(r) = α r , g(θ) = Z4(kθ), (5) f(r) = αr2, g(θ) = Z2(kθ), (6) f(r) = α r , g(θ) = Z2(kθ). In each case if p is rational there is an extra constant of the motion that is polynomial in the canonical momenta. [E1]. Case (1) is system [E1] for k = 1. For general k this is the TTW system [12], which we have shown to be supperintegrable for k rational. Case (2). This case is not quadratic superintegrable, but as shown in [19] it is superintegrable for all rational k. [E8]. For our next example we take Case (3) where z = x+ iy: V = αzz̄ + β zk−1 z̄k+1 + γ zk/2−1 z̄k/2+1 . for arbitrary k. (If k = 2 this is the nondegenerate superintegrable system [E8] listed in [4].) In polar coordinates, with variables r = eR and z = eR+iθ. Then we have H = e−2R ( p2 R + p2 θ + 4αe4R + βe2ikθ + γeikθ ) , −L2 = p2 θ + βe2ikθ + γeikθ, H = e−2R ( p2 R − L2 + 4αe4R ) . Our method yields N(θ, pθ) = i k √ −L2 B, sinhB = i(2L2e −ikθ + γ)√ −4βL2 + γ2 , coshB = 2ipθ √ −L2√ −4βL2 + γ2 e−ikθ, M(R, pR) = i√ −L2 A, sinhA = (−2L2e −2R −H)√ −4αL2 −H2 , coshA = 2ipR √ −L2√ −4αL2 −H2 e−2R. This system is superintegrable for all rational k. [E17]. Taking Case (4) we have, for z = x+ iy: V = α√ zz̄ + β z̄k−1 zk+1 + γ z̄k/2−1 zk/2+1 . (For k = 1 this is the superintegrable system [E17] in [4].) Then we have H = e−2R ( p2 R + p2 θ + αeR + βe−2ikθ + γe−ikθ ) , (5) −L2 = p2 θ + βe−2ikθ + γe−ikθ, H = e−2R ( p2 R − L2 + αeR ) . Applying our procedure we find the functions N(θ, pθ) = 1 k √ L2 B, sinhB = (2L2e ikθ + γ)√ 4βL2 − γ2 , coshB = − 2ipθ √ L2√ 4βL2 − γ2 eikθ, M(R, pR) = 1√ L2 A, sinhA = (−2L2e −R + α)√ 4HL2 − α2 , coshA = 2 √ L2pR√ 4HL2 − α2 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 superintegrable for all rational k. Case (6). This case is not quadratic superintegrable, but it is superintegrable for all rational k. 8 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. The Hamiltonian is of the form H = cosh2 ψ ( p2 ψ + p2 ϕ + v1(ϕ) + v2(ψ) ) , x = ϕ, y = ψ, f1 = 0, f2 = 1 cosh2 ψ . Embedded in complex Euclidean 3-space with Cartesian coordinates, such 2-sphere systems can be written in the form H = J 2 1 + J 2 2 + J 2 3 + V (s), where J3 = s1ps2 − s2ps1 , J1 = s2ps3 − s3ps2 , J2 = s3ps1 − s1ps3 , s21 + s22 + s33 = 1. [S9]. Here, we have the case v1(ϕ) = Z3 and v2(ψ) is a special case of Z3. V = α s21 + β s22 + γ s23 . It is convenient to choose spherical coordinates s1 = cosϕ sinhψ , s2 = sinϕ coshψ, s3 = tanhψ. In terms of these coordinates the Hamiltonian has the form H = cosh2 ψ [ p2 ψ + p2 ϕ + α cos2 ϕ + β sin2 ϕ + γ sinh2 ψ ] , L2 = p2 ϕ + α cos2 ϕ + β sin2 ϕ , H = cosh2 ψ [ p2 ψ + L2 + γ sinh2 ψ ] , system [S9] in [4]. We extend this Hamiltonian via the replacement ϕ → kϕ and proceed with our method. Thus H = cosh2 ψ [ p2 ψ + p2 ϕ + α cos2 kϕ + β sin2 kϕ + γ sinh2 ψ ] , L2 = p2 ϕ + α cos2 kϕ + β sin2 kϕ with H expressed as above. The functions that determine the extra constant are M(ϕ, pϕ) = i 4k √ L2 A, sinhA = i(L2 cos(2kϕ)− α+ β)√ (L2 − α− β)2 − 4αβ , coshA = sin(2kϕ)pϕ√ (L2 − α− β)2 − 4αβ , N(ψ, pψ) = i 4 √ L2 B, sinhB = i(L2 cosh(2ψ) + γ −H)√ (H+ L2 − γ)2 + 4L2γ , coshB = i sinh(2ψ)pψ√ (H+ L2 − γ)2 + 4L2γ . Thus this system is superintegrable for all rational k. Tools for Verifying Classical and Quantum Superintegrability 9 [S7]. This system corresponds to v1(φ) = Z2 and v2(ψ) a variant of Z3. The system is second order superintegrable: V = αs3√ s21 + s22 + βs1 s22 √ s21 + s22 + γ s22 . Choosing the coordinates ψ and ϕ we find H = cosh2 ψ ( p2 ψ + p2 ϕ + α sinhψ cosh2 ψ + β cosϕ sin2 ϕ + γ sin2 ϕ ) , L2 = p2 ϕ + β cosϕ sin2 ϕ + γ sin2 ϕ , H = cosh2 ψ ( p2 ψ + L2 + α sinhψ cosh2 ψ ) . We make the transformation ϕ→ kϕ and obtain H = cosh2 ψ ( p2 ψ + p2 ϕ + α sinhψ cosh2 ψ + β cos kϕ sin2 kϕ + γ sin2 kϕ ) , L2 = p2 ϕ + β cos kϕ sin2 kϕ + γ sin2 kϕ , with H as before. The functions that determine the extra constants are M(ϕ, pϕ) = 1√ L2k A, sinhA = i(L2 cos(kϕ) + β)√ β2 + 4L2 2 − 4L2γ , coshA = 2 √ L2 sin(kϕ)pϕ√ β2 + 4L2 2 − 4L2γ , N(ψ, pψ) = i√ L2 B, sinhB = 2L2 sinhψ + α√ −α2 + 4L2 2 − 4L2H , coshB = 2i coshψpψ√ −α2 + 4L2 2 − 4L2H . Thus this system is superintegrable for all rational k. [S4]. Here v1(ϕ) = Z4 and v2(ψ) is a variant of Z2. This is another system on the sphere that is second order superintegrable and separates in polar coordinates: V = α (s1 − is2)2 + βs3√ s21 + s22 + γ (s1 − is2) √ s21 + s22 . In terms of angular coordinates ψ, ϕ the Hamiltonian is H = cosh2 ψ ( p2 ψ + p2 ϕ + αe2ikϕ + γikϕ + β sinhψ cosh2 ψ ) . After the substitution ϕ→ kϕ we have L2 = p2 ϕ + αe2ikϕ + γikϕ, H = cosh2 ψ ( p2 ψ + L2 + β sinhψ cosh2 ψ ) . The functions that determine the extra constants are M(ϕ, pϕ) = i 2 √ L2k A, sinhA = i(2L2e −ikϕ − γ)√ 4L2α+ γ2 , coshA = 2i √ L2pϕ√ 4L2α+ γ2 e−ikϕ, N(ψ, pψ) = i 2 √ L2 B, sinhB = L2 sinhψ − β√ 4L2 2 + 4L2 − β2 , coshB = 2i coshψpψ√ 4L2 2 + 4L2 − β2 , so this systems is also superintegrable for all rational k. 10 E.G. Kalnins, J.M. Kress and W. Miller Jr. [S2]. Here v1(ϕ) = Z4 and v2(ψ) is a variant of Z3: V = α s23 + β (s1 − is2)2 + γ(s1 + is2) (s1 − is2)3 . which is is second order superintegrable. After the substitution ϕ→ kϕ we obtain the system H = cosh2 ψ ( p2 ψ + p2 ϕ + α sinh2 ψ + βe2ikϕ + γe4ikϕ ) , (6) L2 = p2 ϕ + βe2ikϕ + γe4ikϕ, H = cosh2 ψ ( p2 ψ + L2 + α sinh2 ψ ) . Applying our procedure we find M(ϕ, pϕ) = i 4 √ L2k A, sinhA = (2L2e −ikϕ − β)√ 4L2γ − β2 , coshA = 2 √ L2pϕ√ 4L2γ − β2 e−ikϕ, N(ψ, pψ) = i 4 √ L2 B, sinhB = i(L2 cosh(2ψ)− α+H)√ (L2 − α−H)2 − 4αH , coshB = √ L2 sinh(2ψ)pψ√ (L2 − α−H)2 − 4αH . Thus the system is superintegrable for all rational k. 1.3 Horospherical systems In terms of horospherical coordinates on the complex sphere we can construct the Hamiltonian H = y2 ( p2 x + p2 y + ω2 1x 2 + ω2 2y 2 + α+ βx ) . If ω1/ω2 is rational then this system is superintegrable. However, there is no need to go into much detail for the construction because a Stäckel transform, essentially multiplication by 1/y2, 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 = y2 ( p2 x + p2 y + α x2 + β + ω2 1x 2 + ω2 2y 2 ) . It is superintegrable for ω1/ω2 rational, as follows from the Stäckel transform 1/y2 from the sphere to flat space. If ω1 = ω2 then we obtain the system [S2]. We note that each of the potentials associated with horospherical coordinates can quite generally be written in terms of s1, s2, s3 coordinates using the relations y = −i s1 − is2 , x = −s3 s1 − is2 , s21 + s22 + s23 = 1. 1.4 Generic systems These are systems of the form H = 1 Zj(x)− Zk(y) ( p2 x − p2 y + Ẑj(x)− Ẑk(y) ) , where j, k can independently take the values 2, 3, 4 and Zj , Ẑj depend on distinct parameters. Superintegrability is possible because of the integrals∫ dx√ C +Be2ikx +Ae4ikx = 1 2k √ C arcsin ( 2Ce−2ikx +B√ 4AC −B2 ) , Tools for Verifying Classical and Quantum Superintegrability 11∫ dx√ A+B sin kx cos2 kx + C = − 1 k √ C arcsin ( −2C sin kx+B√ 4C2 + 4CA+B2 ) , ∫ dx√ A cos2 kx + B sin2 kx + C = − 1 2k √ C arcsin ( C cos 2kx+A−B√ (A+B + C)2 − 4AB ) . Consider an example of this last type of system: H = 1 A′e2ikx +B′e4ikx − a′e2iqy − b′e4iqy ( p2 x − p2 y +Ae2ikx +Be4ikx − ae2iqy − be4iqy ) . The equation H = E admits a separation constant p2 x + (A− EA′)e2ikx + (B − EB′)e4ikx = p2 y + (a− Ea′)e2iqy + (b− Eb′)e4iqy = L. As a consequence of this we can find functions M(x, px) and N(y, py) where M(x, px) = 1 4k √ L arcsin ( 2Âe−2ikx + B̂√ 4LÂ− B̂2 ) , where Â = A− EA′, B̂ = B − EB′ and N(y, py) = 1 4p √ L arcsin ( 2âe−2iqy + b̂√ 4Lâ− b̂2 ) , where â = a − Ea′, b̂ = b − Eb′. We see that if q k is rational then we can generate an extra constant which is polynomial in the momenta. The superintegrable systems that have involved the rational functions Z1, Z5 and Z6 work because of the integrals∫ dx√ Ax2 +Bx+ C = 1√ A arcsinh ( 2Ax+ C√ 4AB − C2 ) and ∫ dx x √ Ax2 +Bx+ C = − 1√ B arcsinh ( 2B +Ax x √ A2 − 4BC ) . 2 Quantum superintegrability 2.1 The canonical form for a symmetry operator We give a brief review of the construction of the canonical form for a symmetry operator [19]. Consider a Schrödinger equation on a 2D real or complex Riemannian manifold with Laplace– Beltrami operator ∆2 and potential V : HΨ ≡ (∆2 + V )Ψ = EΨ that also admits an orthogonal separation of variables. If {u1, u2} is the orthogonal separable coordinate system the corresponding Schrödinger operator has the form H = L1 = ∆2 + V (u1, u2) = 1 f1(u1) + f2(u2) ( ∂2 u1 + ∂2 u2 + v1(u1) + v2(u2) ) . (7) 12 E.G. Kalnins, J.M. Kress and W. Miller Jr. and, due to the separability, there is the second-order symmetry operator L2 = f2(u2) f1(u1) + f2(u2) ( ∂2 u1 + v1(u1) ) − f1(u1) f1(u1) + f2(u2) ( ∂2 u2 + v2(u2) ) , i.e., [L2,H] = 0, and the operator identities f1(u1)H + L2 = ∂2 u1 + v1(u1), f2(u2)H − L2 = ∂2 u2 + v2(u2). We look for a partial differential operator L̃(H,L2, u1, u2) that satisfies [H, L̃] = 0. We require that the symmetry operator take the standard form L̃ = ∑ j,k ( Aj,k(u1, u2)∂u1u2 +Bj,k(u1, u2)∂u1 + Cj,k(u1, u2)∂u2 +Dj,k(u1, u2) ) HjLk2. (8) We have shown that we can write L̃(H,L2, u1, u2) = A(u1, u2,H, L2)∂12 +B(u1, u2,H, L2)∂1 + C(u1, u2,H, L2)∂2 +D(u1, u2,H, L2), (9) and consider L̃ as an at most second-order order differential operator in u1, u2 that is analytic in the parameters H, L2. Then the conditions for a symmetry can be written in the compact form Au1u1 +Au2u2 + 2Bu2 + 2Cu1 = 0, (10) Bu1u1 +Bu2u2 − 2Au2v2 + 2Du1 −Av′2 + (2Au2f2 +Af ′2)H − 2Au2L2 = 0, (11) Cu1u1 + Cu2u2 − 2Au1v1 + 2Du2 −Av′1 + (2Au1f1 +Af ′1)H + 2Au1L2 = 0, (12) Du1u1 +Du2u2 − 2Bu1v1 − 2Cu2v2 −Bv′1 − Cv′2 + (2Bu1f1 + 2Cu2f2 +Bf ′1 + Cf ′2)H + (2Bu1 − 2Cu2)L2 = 0. (13) We can further simplify this system by noting that there are two functions F (u1, u2,H, L2), G(u1, u2,H, L2) such that (10) is satisfied by A = F, B = −1 2 ∂2F − ∂1G, C = −1 2 ∂1F + ∂2G. (14) Then the integrability condition for (11), (12) is (with the shorthand ∂jF = Fj , ∂j`F = Fj`, etc., for F and G), 2G1222 + 1 2 F2222 + 2F22(v2 − f2H + L2) + 3F2(v′2 − f ′2H) + F (v′′2 − f ′′2H) − 2G1112 + 1 2 F1111 + 2F11(v1 − f1H − L2) + 3F1(v′1 − f ′1H) + F (v′′1 − f ′′1H), (15) and equation (13) becomes 1 2 F1112 + 2F12(v1 − f1H) + F1(v′2 − f ′2H) + 1 2 G1111 + 2G11(v1 − f1H − L2) +G1(v′1 − f ′1H) = −1 2 F1222 − 2F12(v2 − f2H)− F2(v′1 − f ′1H) + 1 2 G2222 + 2G22(v2 − f2H + L2) +G2(v′2 − f ′2H). (16) Tools for Verifying Classical and Quantum Superintegrability 13 Here, any solution of (15), (16) with A, B, C not identically 0 corresponds to a symmetry operator that does not commute with L2, hence is algebraically independent of the symmet- ries H, L2. (Informally, this follows from the construction and uniqueness of the canonical form of a symmetry operator. The operators L̃ = g(H,L2) algebraically dependent on H and L2 are exactly those such that A = B = C = 0, D = g(H,L2). A formal proof is technical.) Note also that solutions of the canonical equations, with H, L2 treated as parameters, must be interpreted in the form (8) with H and L2 on the right, to get the explicit symmetry operators. 2.2 The caged anisotropic oscillator 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-hyperboloid of two sheets [19]. Here we give another illustration of this construction by applying it to the 2D caged oscillator [11]. For p = q This is the second order superintegrable system [E1] on complex Euclidean space, as listed in [4]. Here, HΨ = (∂11 + ∂22 + V (u1, u2))Ψ, (17) where V (u1, u2) = ω2 ( p2u2 1 + q2u2 2 ) + α1 u2 1 + α2 u2 2 , in Cartesian coordinates. We take p, q to be relatively prime positive integers. Thus f1 = 1, f2 = 0, v1 = ω2p2u2 1 + α1 u2 1 , v2 = ω2q2u2 2 + α2 u2 2 . The 2nd order symmetry operator is L2 = − ( ∂2 2 + v2(u2) ) , and we have the operator identities ∂2 1 = −(v1(u1) +H + L2), ∂2 2 = −v2(u2) + L2. Based on the results of [18] for the classical case, we postulate expansions of F , G in finite series F = ∑ a,b Aa,bEa,b(u1, u2), G = ∑ a,b Ba,bEa,b(u1, u2), Ea,b(u1, u2) = ua1u b 2. (18) The sum is taken over terms of the form a = a0 +m, b = b0 + n, and c = 0, 1, where m, n are integers. The point (a0, b0) could in principle be any point in R2, Taking coefficients with respect to the basis (18) in each of equation (15) and (16) gives recurrence relations for these coefficients. The shifts in the indices of A and B are integers and so we can view this as an equation on a two-dimensional lattice with integer spacings. While the shifts in the indices are of integer size, we haven’t required that the indices themselves be integers, although they will turn out to be so in our solution. The 2 recurrence relations are of a similar complexity, but rather than write them out separately, we will combine them into a matrix recurrence relation by defining Ca,b = ( Aa,b Ba,b ) . 14 E.G. Kalnins, J.M. Kress and W. Miller Jr. · · · · · · · · · · • · · · · · · · · • · · · · · · • · · · · · · • · • · • · · · · ◦ · • · • · · · · • · · · · · · · · · · · · 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 − 1, b + 1), the large dot at the top corresponds to the position (a+ 4, b) and ◦ corresponds to position (0, 0). We write the 2 recurrence relations in matrix form as 0 = Ma,bCa,b +Ma−1,b+1Ca−1,b+1 +Ma,b+2Ca,b+2 +Ma,b+4Ca,b+4 +Ma+1,b−1Ca+1,b−1 +Ma+1,b+1Ca+1,b+1 +Ma+1,b+3Ca+1,b+3 +Ma+2,bCa+2,b +Ma+3,b+1Ca+3,b+1 +Ma+4,bCa+4,b, where each Mi,j is a 2× 2 matrix given below. Ma,b = ( 2ω2 ([b+ 1]q − [a+ 1]p) ([b+ 1]q + [a+ 1]p) 0 0 −2ω2 (bq − ap) (bq + ap) ) , Ma−1,b+1 = ( 0 0 2ω2p2a(b+ 1) 0 ) , Ma,b+2 = ( 2L2(b+ 2)(b+ 1) 0 0 −2L2(b+ 2)(b+ 1) ) , Ma,b+4 = ( 1 2(b+ 3)(b+ 1)(b2 + 6b+ 4α2 + 8) 0 0 −1 2(b+ 4)(b+ 2)(b2 + 4b+ 4α2 + 3) ) , Ma+1,b−1 = ( 0 0 2ω2q2b(a+ 1) 0 ) , Ma+1,b+1 = ( 0 0 −2H(a+ 1)(b+ 1) 0 ) , Ma+1,b+3 = ( 0 2(a+ 1)(b+ 3)(b+ 2)(b+ 1) 1 2(a+ 1)(b+ 2)(b2 + 4b+ 4α2 + 3) 0 ) , Ma+2,b = ( 2(H + L2)(a+ 2)(a+ 1) 0 0 −2(H + L2)(a+ 2)(a+ 1) ) , Ma+3,b+1 = ( 0 2(a+ 3)(a+ 2)(a+ 1)(b+ 1) 1 2(a+ 2)(b+ 1)(a2 + 4a+ 4α1 + 3) 0 ) , Ma+4,b = ( −1 2(a+ 3)(a+ 1)(a2 + 6a+ 4α1 + 8) 0 0 1 2(a+ 4)(a+ 2)(a2 + 4a+ 4α1 + 3) ) . It is useful to visualize the the set of points in the lattice which enter into this recurrence for a given choice of (a, b). These are represented in Fig. 1. Although the recurrence relates 10 distinct points, some major simplifications are immediately apparent. We say that the lattice point (a, b) has even parity if a+ b is an even integer and odd parity if a+ b is odd. Each recurrence relates only lattice points of the same parity. Because of this we can assume that the nonzero terms Ca,b will occur for points of one parity, while only the zero vector will occur for Tools for Verifying Classical and Quantum Superintegrability 15 points with the opposite parity. A second simplification results from the fact that the recurrence matrices Ma+m,b+n are of two distinct types. Either m, n are both even, in which case Ma+m,b+n is diagonal, or m, n are both odd, in which case the diagonal elements of Ma+m,b+n are zero. Another simplification follows from the observation that it is only the ratio p/q = r that matters in our construction. We want to demonstrate that the caged anisotropic oscillator is operator superintegrable for any rational r. The construction of any symmetry operator independent of H and L2 will suffice. By writing ω = ω′/2, p′ = 2p, q′ = 2q in H, we see that, without loss of generality, we can always assume that p, q are both even positive integers with a single 2 as their only common factor. If for a particular choice of p, q we can find a solution of the recurrence relations with only a finite number of nonzero vectors Ca,b then there will be a minimal lattice rectangle with vertical sides and horizontal top and bottom that encloses the corresponding lattice points. The top row a0 of the rectangle will be the highest row in which nonzero vectors Ca0,b occur. The bottom row a1 will be the lowest row in which nonzero vectors Ca1,b occur. Similarly, the minimal rectangle will have right column b0 and left column b1. Now slide the template horizontally across the top row such that only the lowest point on the template lies in the top row. The recurrence gives (a0 + 1)bAa0,b = 0 for all columns b. Based on examples for the classical system, we expect to find solutions for the quantum system such that a0 ≥ 0 and b ≥ 0. Thus we require Aa0,b = 0 along the top of the minimal lattice rectangle, so that all vectors in the top row take the form Ca0,b = ( 0 Ba0,b ) . Now we move the template such that the recurrence Ma,b, second row from the bottom and on the left, lies on top of the lattice point (a0, b0). This leads to the requirement (b0q − a0p)(b0q + a0p)Ba0,b0 = 0. Again, based on hints from specific examples, we postulate Ba0,b0 6= 0, a0 = q, b0 = p. Since we can assume that p, q are even, this means that all odd lattice points correspond to zero vectors. Next we slide the template vertically down column b0 such that only the left hand point on the template lies in column b0. The recurrence gives (b0 − 1)aAa,b0 = 0 for all rows a. Since b0 is even, we postulate that Ca,b0 = ( 0 Ba,b0 ) for all lattice points in column b0. 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 Ca,b = ( 0 Ba,b ) . 3. If the row and column are both odd then Ca,b = ( Aa,b 0 ) . This does not mean that all solutions have this form, only that we are searching for at least one such solution. To finish determining the size of the minimum lattice rectangle we slide the template vertically downward such that only the right hand point on the template lies in column b1. The recurrence gives (b1 − 1)(b1 − 3)A(a, b1) = 0, b1(b1 − 2)Ba,b1 = 0 for all rows a. There are several possible solutions that are in accordance with our assumptions, the most conservative of which is b1 = 0. In the following we assume only that Ba0,b0 6= 0, the structure laid out above, and the necessary implications of these that follow from a step by step application of the recurrences. If we find 16 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 be zero. Our aim is to find one nonzero solution with support in the minimum rectangle, not to classify the multiplicity of all such solutions. Now we carry out an iterative procedure that calculates the values of Ca,b at points in the lattice using only other points where the values of Ci,j are already known. We position the template such that the recurrence Ma,b, second row from the bottom and on the left (of the template), lies above the lattice point (a0, b), a0 = q, and slide it from right to left along the top row. The case b = b0 = p has already been considered. For the remaining cases we have 2ω2p2a0(b+ 1)Aa0−1,b+1 = 2ω2(bq − a0p)(bq + a0p)Ba0,b + 2L2(b+ 2)(b+ 1)Ba0,b+2 + 1 2 (b+ 4)(b+ 2)(b2 + 4b+ 4α2 + 3)Ba0,b+4. (19) Only even values of b need be considered. Now we lower the template one row and again slide it from right to left along the row. The contribution of the lowest point on the template is 0 and we find 2ω2(q[b+ 2]− p[a0])(q[b+ 2] + p[a0])Aa0−1,b+1 = −2a0(b+ 4)(b+ 3)(b+ 2)Ba0,b+4 − 2L2(b+ 3)(b+ 2)Aa0−1,b+3 − 1 2 (b+ 4)(b+ 2)(b2 + 8b+ 11α2 + 8)Aa0−1,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) becomes 2ω2p3qAq−1,p−1 = −8ω2q2(p− 1)Bq,p−2 + 2L2p(p− 1)Bq,p, and (20) is vacuous in this case, leaving Aq−1,p−1 undetermined. Note that several of the terms lie outside the minimal rectangle. From these two equations we can solve for Bq,p−2 in terms of our given Bq,p, Aq−1,p. Now we march across both rows from right to left. At each stage our two equations now allow us to solve uniquely for Aq−1,b+1 and Bq,b in terms of A’s and B’s to the right (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 choice of Bq,p and Aq−1,p−1. We repeat this construction for the third and fourth rows down, then the next two rows down, etc. The recurrence relations grow more complicated as the higher rows of the template give nonzero contributions. However, at each step we have two linear relations( 2ω2p2a(b+ 1) −2ω2(bq − ap)(bq + ap) 2ω2(q[b+ 2]− pa)(q[b+ 2] + pa) 0 )( Aa−1,b+1 Ba,b ) = · · · , where the right hand side is expressed in terms of A’s and B’s either above or on the same line but to the right of Aa−1,b+1, Ba,b, hence already determined. Since the determinant of the 2× 2 matrix is nonzero over the minimal rectangle, except for the upper right corner and lower left corner, at all but those those two points we can compute Aa−1,b+1, Ba,b uniquely in terms of quantities already determined. This process stops with rows a = 1, 2. Row a = 0, the bottom row, needs special attention. We position the template such that the recurrence Ma,b lies above the lattice point (0, b), and slide it from right to left along the bottom row. The bottom point in the template now contributes 0 so at each step we obtain an expression for B0,b in terms of quantities A, B from rows either above row 0 or to the right of column b in row 0, hence already determined. Thus we can determine the entire bottom row, with the exception of the value at (0, 0), the lower left hand corner. When the point Ma,b on the template is above (0, 0) the coefficient of B0,0 vanishes, so the value of B0,0 is irrelevant and we have a true condition on the remaining points under the template. However, this is a linear homogeneous equation in the parameters Bq,p, Aq−1,p−1: χBq,p + ηAq−1,p−1 = 0 for Tools for Verifying Classical and Quantum Superintegrability 17 constants χ, η. Hence if, for example, χ 6= 0 we can require Bq,p = −(η/χ)Aq−1,p−1 and satisfy this condition while still keeping a nonzero solution. Thus after satisfying this linear condition we still have at least a one parameter family of solutions, along with the arbitrary B0,0 (which is clearly irrelevant since it just adds a constant to the function G). However, we need to check those recurrences where the point Ma,b on the template slides along rows a = −1,−2,−3,−4 to verify that these relations are satisfied. We have already utilized the case a = −4, and for cases a = −3,−2,−1 it is easy to check that the relations are vacuous. The last issue is the left hand boundary. We have determined all vectors in the minimal rectangle, but we must verify that those recurrences are satisfied where the point Ma,b on the template slides along columns b = −1,−2,−3,−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 arbitrary parameters to be polynomials in H, L2 we get a finite order constant of the motion. Thus the quantum caged anisotropic oscillator is superintegrable. We note that the canonical operator construction permits easy generation of explicit expressions for the defining operators in a large number of examples. Once the basic rectangle of nonzero solutions is determined it is easy to compute dozens of explicit examples via Maple and simple Gaussian elimination. The generation of 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 the recurrence with nonzero coefficients A1,1, A1,3, A1,5, A3,1, A3,3, A3,5, B0,2, B0,4, B2,0, B2,2, B2,4, B4,0, B4,2, B4,4, B6,0, B6,2, B6,4. We can take A1,5 = a1 and A3,3 = a2 and then all other coefficients will depend linearly on a1 and a2. Calculating the A, B, C and D from equations (11), (12), (14) we find the coefficient of H−1 has a factor of 2a2 +9a1 and so we set a1 = ω6 and a2 = −9/2a1 to obtain the symmetry operator L̃, (9), where u1, u2 are Cartesian coordinates and A = 9 4096 ω2u1u2L 2 2 + 3 2048 ω2u1u2HL2 − 1 256 ω4u1u2 ( 27u2 1 − 32u2 2 ) L2 + 1 16 ω4u1u 3 2H + 1 64 ω4α2u1u2 − 1 2 ω6u1u 3 2 ( 9u2 1 − 2u2 2 ) − 93 256 ω4u1u2, B = − 1 786432 u1L 4 2 − 1 786432 u1HL 3 2 + 3 32768 ω2u1 ( u2 1 − 4u2 2 ) L3 2 − 3 8192 ω2u1u 2 2HL 2 2 + ( 149 49152 ω2u1 − 1 4096 ω2α2u1 + 1 1024 ω4u1u 2 2 ( 27u2 1 − 16u2 2 )) L2 2 + ( 59 49152 ω2u1 − 1 4096 ω2α2u1 − 1 64 ω4u1u 4 2 ) HL2 + ( 45 512 ω4u1u 2 2 − 1 128 ω4α2u1u 2 2 − 1 6 ω6u1u 6 2 ) H + ( 1 512 ω4α2u1 ( 9u2 1 − 4u2 2 ) + 1 24 ω6u1u 4 2 ( 27u2 1 − 4u2 2 ) − 3 2048 ω4u1 ( 59u2 1 − 284u2 2 )) L2 + 9 16 ω6α2u 2 2u 3 1 + 17 128 ω4α2u1 − 1 64 u1u 2 2 ( 405u2 1 − 416u2 2 ) ω6 + 12ω8u3 1u 6 2 − 45 512 ω4u1, C = − 1 786432 u2L 4 2 − 1 786432 u2H 2L2 2 − 1 393216 u2HL 3 2 + 1 73728 ω2u2 ( 27u2 1 − 8u2 2 ) L3 2 + 1 73728 ω2u2 ( 27u2 1 − 16u2 2 ) HL2 2 − 1 9216 ω2u3 2H 2L2 + ( 3 16384 ω2α1u2 − 1 36864 ω2α2u2 + 601 196608 ω2u2 − 1 18432 u2 ( 243u4 1 − 576u2 1u 2 2 + 32u4 2 ) ω4 ) L2 2 18 E.G. Kalnins, J.M. Kress and W. Miller Jr. + ( 85 24576 ω2u2 − 1 18432 ω2α2u2 + 1 288 ω4u3 2 ( 9u2 1 − u2 2 )) HL2 + ( 31 49152 ω2u2 − 1 36864 ω2α2u2 − 1 576 ω4u5 2 ) H2 + ( 1 128 u2 1u2ω 4α2 + 1 2 ω6u5 2u 2 1 − 3 512 u2 ( −16u2 2 + 31u2 1 ) ω4 ) H + ( 1 128 ω4α2u 2 1u2 − 1 64 ω4α1u 3 2 − 1 8 u2 1u 3 2 ( 9u2 1 − 2u2 2 ) ω6 − 1 256 u2 ( 114u2 1 − 29u2 2 ) ω4 ) L2 − 1 256 ω4α1α2u2 − 1 4 ω6α1u 5 2 + 93 1024 ω4α1u2 − 9 32 ω6α2u2u 4 1 + 5 1024 ω4α2u2 + 1 128 ω6u2 ( 40u4 2 + 837u4 1 − 1440u2 2u 2 1 ) − 18ω8u5 2u 4 1 − 465 4096 ω4u2, D = ω2 ( 75 65536 u2 1 − 55 49152 u2 2 ) L3 2 + ω2 ( 15 16384 u2 1 − 83 49152 u2 2 ) HL2 2 − 7 12288 ω2u2 2H 2L2 + ω4 ( 471 2048 u2 2u 2 1 − 5 144 u4 2 − 135 4096 u4 1 ) L2 2 + 1 1152 ω4u2 2 ( 189u2 1 − 53u2 2 ) HL2 − 13 1152 ω4u4 2H 2 + ( ω4α2 ( 113 1024 u2 1 − 3 256 u2 2 ) − 21 256 ω4α1u 2 2 − 1 32 ω6u2 2 ( 8u4 2 − 194u2 2u 2 1 + 189u4 1 ) + ( 15 64 u2 2 − 1065 4096 u2 1 ) ω4 ) L2 + ( ω4α2 ( 17 256 u2 1 − 3 256 u2 2 ) + 1 4 ω6u4 2 ( 13u2 1 − u2 2 ) + ω4 ( 135 1024 u2 2 − 45 1024 u2 1 )) H − ω6α1 13 8 u4 2 − 9 64 ω6α2u 2 1 ( 17u2 1 − 10u2 2 ) + ω6 ( 405 256 u4 1 − 2025 128 u2 1u 2 2 + 65 32 u4 2 ) − 3u2 1ω 8u4 2 ( 39u2 1 − 10u2 2 ) . Using Maple, we have checked explicitly that the operator L̃ commutes with H. Note that it is of 6th order. Taking the formal adjoint L̃∗, [19], we see that S = 1 2(L̃+ L̃∗) is a 6th order, formally self-adjoint symmetry operator. 3 An alternate proof of superintegrability for the caged quantum oscillator This second proof is very special for the oscillator and exploits the fact that separation of variables in Cartesian coordinates is allowed Here we write the Hamiltonian in the form H = ∂2 x + ∂2 y − µ2 1x 2 − µ2 2y 2 + 1 4 − a2 1 x2 + 1 4 − a2 2 y2 . This is the same as (17) with u1 = x, u2 = y, µ2 1 = −p2ω2, µ2 2 = −q2ω2, α1 = 1 4 − a2 1, α2 = 1 4 − a2 2. We look for eigenfunctions for the equation HΨ = λΨ of the form Ψ = XY . We find the normalized solutions Xn = e− 1 2 µ1x2 xa1+ 1 2La1 n (µ1x 2), Ym = e− 1 2 µ2y2ya2+ 1 2La2 m (µ2y 2), where the Lαn(x) are associated Laguerre polynomials [24]. For the corresponding separation constants we obtain λx = −2µ1(2n+ a1 + 1), λy = −2µ2(2m+ a2 + 1). Tools for Verifying Classical and Quantum Superintegrability 19 The total energy is −λx − λy = E. Taking µ1 = pµ and µ2 = qµ where p and q are integers we find that the total energy is E = −2µ(pn+ qm+ pa1 + p+ qa2 + 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. To see that this is achievable via differential operators we need only consider Ψ = La1 n (z1)La2 m (z2), z1 = µ1x 2, z2 = µ2y 2. We now note the recurrence formulas for Laguerre polynomials viz. x d dx Lαp (x) = pLαp (x)− (p+ α)Lαp−1(x) = (p+ 1)Lαp+1(x)− (p+ 1 + α− x)Lαp (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 Lαp±1(x). Therefore we can raise or lower the indices m and n in Ψ using differential operators. In particular we can perform the transformation n→ n+ q, m→ m− p and preserve the energy eigenvalue. To see how this works we observe the formulas D+(µ1, x)Xn = ( ∂2 x − 2xµ1∂x − µ1 + µ2 1x 2 + 1 4 − α2 1 x2 ) Xn = −4µ1(n+ 1)Xn+1, D−(µ2, y)Ym = ( ∂2 y + 2yµ2∂y + µ2 + µ2 2y 2 + 1 4 − α2 2 y2 ) Ym = −4µ2(m+ α2)Ym−1. In particular, if we make the choice µ1 = 2µ, µ2 = µ then the operator D+(2µ, x)D−(µ, y)2 transformsXnYm to −128µ3(n+1)(m+α2)(m−1+α2)Xn+1Ym−2. We see that this preserves the energy eigenspace. Thus can easily be extended to the case when µ1 = pµ and µ2 = qµ for p and q integers. A suitable operator is D+(pµ, x)qD−(qµ, y)p. Hence we have constructed a differential operator that commutes with H! The caged oscillator is quantum superintegrable. This works to prove superintegrability in all dimensions as we need only take coordinates pairwise. Remark. This second proof of superintegrability, using differential recurrence relations for Laguerre 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 and the 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 explicitly the commutivity on an eigenbasis for a bound state. 3.1 A proof of superintegrability for a deformed Kepler–Coulomb system In [16] there is introduced a new family of Hamiltonians with a deformed Kepler–Coulomb potential dependent on an indexing parameter k which is shown to be related to the TTW oscillator system system via coupling constant metamorphosis. The authors showed that this system is classically superintegrable for all rational k. Here we demonstrate that this system is also quantum superintegrable. The proof follows easily from the canonical equations for the system. The quantum TTW system is HΨ = EΨ with H given by (7) where u1 = R, u2 = θ, f1 = e2R, f2 = 0, v1 = αe4R, (21) 20 E.G. Kalnins, J.M. Kress and W. Miller Jr. v2 = β cos2(kθ) + γ sin2(kθ) = 2(γ + β) sin2(2kθ) + 2(γ − β) cos(2kθ) sin2(2kθ) . (Setting r = eR we get the usual expression for this system in polar coordinates r, θ.) In our paper [19] we used the canonical form for symmetry operators to establish the superintegrability of this system. Our procedure was, based on the results of [20] for the classical case, to postulate expansions of F , G in finite series F = ∑ a,b,c Aa,b,cEa,b,c(R, θ), G = ∑ a,b,c Ba,b,cEa,b,c(R, θ), Ea,b,0 = e2aR sinb(2kθ), Ea,b,1 = e2aR sinb(2kθ) cos(2kθ). The sum is taken over terms of the form a = a0 +m, b = b0 + n, and c = 0, 1, where m, n are integers, a0 is a positive integer and b0 is a negative integer. Here F , G are the solutions of the canonical form equations (14), (15), (16) for the TTW system. We succeeded in finding finite series solutions for all rational k, and this proved superintegrability. In [16] the authors point out that under the Stäckel transformation determined by the po- tential U = e2R the TTW system transforms to the equivalent system ĤΨ = −αΨ, where Ĥ = 1 e4R ( ∂2 R + ∂2 θ − E exp(2R) + β cos2(kθ) + γ sin2(kθ) ) . (22) Then, setting r = e2R, φ = 2θ, we find the deformed Kepler–Coulomb system.( ∂2 r + 1 r + 1 r2 ∂2 φ − E 4r + β 4r2 cos2(kφ/2) + γ 4r2 sin2(kφ/2) ) Ψ = −α 4 Ψ. 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 functions defining the Stäckel transformed system is that f1 and v1 are replaced by f1 = exp(4R) and v1 = −E exp(2R) (we can set E = H) and the former H is replaced by −α. From this we find that the only difference between the canonical equations for the TTW system and the canonical equations for the transformed system is that α becomes −Ĥ and H becomes −α. Since our proof 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 operator for the transformed system by replacing α and H with −Ĥ and −α, respectively. Example. For the deformed Kepler system with k = 2, we take p = 2 and q = 1 and use Gaussian elimination to find a solution with nonzero coefficients A−2,1,0, A−1,1,0, B−2,0,0, B−2,0,1, B−1,0,0, B−1,0,1, B0,0,1, in which A−2,1,0 and A−1,1,0 are two independent parameters. To achieve the lowest order symmetry and ensure that the A, B, C and D are polynomial in L̂2 and Ĥ, we choose A−2,1,0 = 32(L̂2 − 4) and A−1,1,0 = 0 and find, A = 32e−4R sin(4θ)L̂2 − 128e−4R sin(4θ), B = 8e−4R cos(4θ)L̂2 2 + (8(β − γ)e−4R + 4Ee−2R cos(4θ)− 8e−4R cos(4θ))L̂2 − 96e−4R cos(4θ)− 16Ee−2R cos(4θ) + 40(γ − β)e−4R + 4E(β − γ)e−2R, C = −8e−4R sin(4θ)L̂2 2 − 4 sin(4θ)ĤL̂2 + 8(e−4R − Ee−2R) sin(4θ)L̂2 + 20 sin(4θ)Ĥ + 96e−4R sin(4θ) + 32Ee−2R sin(4θ)− E2 sin(4θ), Tools for Verifying Classical and Quantum Superintegrability 21 D = −32e−4R cos(4θ)L2 2 − 8 cos(4θ)ĤL̂2 + (128e−4R cos(4θ)− 12Ee−2R cos(4θ) + 16(γ − β)e−4R)L̂2 + 24 cos(4θ)Ĥ + 48(β − γ)e−4R + 48Ee−2R cos(4θ)− 4E(γ − β)e−2R + 2E2 cos(4θ). It has been explicitly checked, using Maple, that the fifth order operator obtained from these expressions commutes with Ĥ. 3.2 A proof of superintegrability for the caged oscillator on the hyperboloid Using this same idea we can find a new superintegrable system on the 2-sheet hyperboloid by taking a Stäckel transformation of the quantum caged oscillator HΨ = EΨ, (17) in Cartesian coordinates u1, u2 by multiplying with the potential U = 1/u2 1. The transformed system is u2 1 ( ∂2 1 + ∂2 2 + ω2u2 1(p 2u2 1 + q2u2 2)− Eu2 1 + α2 u2 1 u2 2 ) Ψ = −α1Ψ. We embed this system as the upper sheet s0 > 0 of the 2-sheet hyperboloid s20−s21−s22 = 1 in 3- dimensional Minkowski space with Minkowski metric ds2 = −ds20 +ds21 +ds22, via the coordinate transformation s0 = 1 + u2 1 + u2 2 2u1 , s1 = 1− u2 1 − u2 2 2u1 , s2 = u2 u1 , u1 > 0. Then the potential for the transformed system is Ṽ = α2 s22 − E (s0 + s1)2 + ω2(p2 − q2) (s0 + s1)4 + ω2q2(s0 − s1) (s0 + s1)3 . (23) This is an extension of the complex sphere system [S2], distinct from the system (6) that we proved classically superintegrable. It is an easy consequence of the results of [17] that the classical 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 virtually immediate that this new system is quantum superintegrable for all relatively prime positive integers p, q. This follows from writing down the canonical equations (15), (16), first for the caged oscillator where f1 = 1, f2 = 0, v1 = ω2p2u2 1 + α1 u2 1 , v2 = ω2q2u2 2 + α2 u2 2 , and then for the Stäckel transformed system where f1 = 1 u2 1 , f2 = 0, v1 = ω2p2u2 1 − E, v2 = ω2q2u2 2 + α2 u2 2 . The equations are identical except for the switches α1 → −E, H → −α1. Thus our proof of quantum superintegrability for the caged oscillator carries over to show that the system on the hyperboloid is also quantum superintegrable. 4 Discussion A basic issue in discovering and verifying higher order superintegrabily of classical and quantum systems 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 such 22 E.G. Kalnins, J.M. Kress and W. Miller Jr. calculations. Although our primary emphasis in this paper was to develop tools for verifying classical and quantum superintegrabity at all orders, we have presented many new results. The classical Eucidean systems [E8], [E17] and most of the examples of superintegrability for spaces with non-zero scalar curvature are new. We have explored the limits of the construction of classical superintegrable systems via the methods of Section 1.1 for the case n = 2. Further use of this method will require looking at n > 2, where new types of behavior occur, such as appearance of superintegrable systems that are not conformally flat. We also developed the use of the canonical form for a symmetry operator to prove quantum superintegrability. We applied the method to the n = 2 caged anisotropic oscillator to give the first proof of quantum superintegrability for all rational k. We used the Stäckel transform together with the canonical equations to give the first proofs of quantum superintegrability for all rational k of a 2D deformed Kepler–Coulomb system, and of the caged anisotropic oscillator on the 2-hyperboloid. Then we introduced a new approach to proving quantum superintegrability via recurrence relations obeyed by the energy eigenfunctions of a quantum system, and gave an alternate proof of the superintegrabilty for all rational k of the caged anisotropic oscillator. This proof clearly extends to all n. The recurrences for the caged oscillator are particularly simple, but the method we presented shows great promise for broader application. Clearly, we are just at the beginning of the process of discovery and classification of higher order superintegrable systems in all dimensions n > 2. References [1] Tempesta P., Winternitz P., Harnad J., Miller W. Jr., Pogosyan G. (Editors), Superintegrability in classical and quantum systems (Montréal, 2002), CRM Proceedings and Lecture Notes, Vol. 37, American Mathema- tical Society, Providence, RI, 2004. [2] Eastwood M., Miller W. Jr. (Editors), Symmetries and overdetermined systems of partial differential equa- tions, Summer Program (Minneapolis, 2006), The IMA Volumes in Mathematics and its Applications, Vol. 144, Springer, New York, 2008. [3] Kalnins E.G., Kress J.M., Miller W. Jr., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages. [4] Kalnins E.G., Kress J.M., Pogosyan G.S., Miller W. Jr., Completeness of superintegrability in two- dimensional constant curvature spaces, J. Phys. A: Math Gen. 34 (2001), 4705–4720, math-ph/0102006. [5] Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with inte- grals quadratic in momenta on a two dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055. [6] Kalnins E.G., Kress J.M., Miller W. Jr., Nondegenerate three-dimensional complex Euclidean superinte- grable systems and algebraic varieties, J. Math. Phys. 48 (2007), 113518, 26 pages, arXiv:0708.3044. [7] Kalnins E.G., Miller W. Jr., Post S., Wilson polynomials and the generic superintegrable system on the 2-sphere, J. Phys. A: Math. Theor. 40 (2007), 11525–11538. [8] Kalnins E.G., Miller W. 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[19] Kalnins E.G., Kress J.M., Miller W. Jr., Superintegrability and higher order integrals for quantum systems, J. Phys. A: Math. Theor. 43 (2010), 265205, 21 pages, arXiv:1002.2665. [20] Kalnins E.G., Miller W. Jr., Pogosyan G.S., Superintegrability and higher order constants for classical and quantum systems, Phys. Atomic Nuclei, to appear, arXiv:0912.2278. [21] Eisenhart L.P., Separable systems of Stäckel, Ann. of Math. (2) 35 (1934), 284–305. [22] Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages. [23] Kalnins E.G., Kress J.M., Miller W. Jr., Post S., Structure theory for second order 2D superintegrable systems with 1-parameter potentials, SIGMA 5 (2009), 008, 24 pages, arXiv:0901.3081. [24] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, UK, 1999. http://dx.doi.org/10.1088/1751-8113/43/1/015202 http://www.arxiv.org/abs/0910.0299 http://dx.doi.org/10.1088/1751-8113/43/8/082001 http://www.arxiv.org/abs/0911.4404 http://dx.doi.org/10.1088/1751-8113/43/17/175206 http://dx.doi.org/10.1088/1751-8113/43/17/175206 http://www.arxiv.org/abs/1002.1989 http://dx.doi.org/10.1088/1751-8113/43/22/222001 http://www.arxiv.org/abs/1003.5230 http://dx.doi.org/10.1088/1751-8113/43/3/035202 http://www.arxiv.org/abs/0908.4393 http://dx.doi.org/10.1088/1751-8113/43/9/092001 http://www.arxiv.org/abs/0912.3158 http://dx.doi.org/10.1088/1751-8113/43/26/265205 http://www.arxiv.org/abs/1002.2665 http://www.arxiv.org/abs/0912.2278 http://dx.doi.org/10.2307/1968433 http://dx.doi.org/10.1063/1.1894985 http://dx.doi.org/10.3842/SIGMA.2009.008 http://www.arxiv.org/abs/0901.3081 1 Introduction 1.1 The construction tool for classical systems 1.2 Application of the construction for n=2 1.3 Horospherical systems 1.4 Generic systems 2 Quantum superintegrability 2.1 The canonical form for a symmetry operator 2.2 The caged anisotropic oscillator 3 An alternate proof of superintegrability for the caged quantum oscillator 3.1 A proof of superintegrability for a deformed Kepler-Coulomb system 3.2 A proof of superintegrability for the caged oscillator on the hyperboloid 4 Discussion References

Tools for Verifying Classical and Quantum Superintegrability

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