# Superintegrability in a non-conformally-at space

Size: px
Start display at page:

## Transcription

1 (Joint work with Ernie Kalnins and Willard Miller) School of Mathematics and Statistics University of New South Wales ANU, September 2011

2 Outline Background What is a superintegrable system Extending the classical superintegrable TTW systems Extending the quantum superintegrable TTW systems

3 What is a Superintegrable Systems? What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system By classical superintegrable system is a Liouville integrable natural Hamiltonian on a 2n-dimensional phase space of the form n H = g ij p i p j + V (x 1,..., x n) i,j=1 admitting 2n 1 functionally independent globally dened functions of the coordindates and momenta {L 0 = H, L 1,..., L 2n 2} that are constants of the motion. If {, } is the usual Poisson bracket, H is integrable if and superintegrable if, in addition, {L i, L j } = 0 for i = 0... n 1. {H, L i } = 0 for i = n 2. Here we also ask that the the constants are polynomial in the momenta. For example, second order constants would have the form n L i = a jk (i) p jp k + W (i) (x 1,..., x n). j,k=1

4 What is a Superintegrable Systems? What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system By classical superintegrable system is a Liouville integrable natural Hamiltonian on a 2n-dimensional phase space of the form n H = g ij p i p j + V (x 1,..., x n) i,j=1 admitting 2n 1 functionally independent globally dened functions of the coordindates and momenta {L 0 = H, L 1,..., L 2n 2} that are constants of the motion. If {, } is the usual Poisson bracket, H is integrable if and superintegrable if, in addition, {L i, L j } = 0 for i = 0... n 1. {H, L i } = 0 for i = n 2. Here we also ask that the the constants are polynomial in the momenta. For example, second order constants would have the form n L i = a jk (i) p jp k + W (i) (x 1,..., x n). j,k=1

5 What is a Superintegrable Systems? What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system By classical superintegrable system is a Liouville integrable natural Hamiltonian on a 2n-dimensional phase space of the form n H = g ij p i p j + V (x 1,..., x n) i,j=1 admitting 2n 1 functionally independent globally dened functions of the coordindates and momenta {L 0 = H, L 1,..., L 2n 2} that are constants of the motion. If {, } is the usual Poisson bracket, H is integrable if and superintegrable if, in addition, {L i, L j } = 0 for i = 0... n 1. {H, L i } = 0 for i = n 2. Here we also ask that the the constants are polynomial in the momenta. For example, second order constants would have the form n L i = a jk (i) p jp k + W (i) (x 1,..., x n). j,k=1

6 An Example Background What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system The Hamiltonian H = p 2 x + p 2 y + α(x 2 + y 2 ) + β x 2 + γ y 2 has two constants that are second order polynomials in the momenta L 1 = p 2 x + αx 2 + β x 2, L2 = (xpy ypx)2 + β y 2 That is, {H, L 1} = {H, L 2} = 0. x 2 + γ x2 y 2. If we dene R = {L 1, L 2}, the Poisson algebra closes quadratically, {R, L 1} = 8L 2 1 8HL αL 2, {R, L 2} = 16L 1L 2 + 8HL 2 16(β + γ)l β. (This is TTW with k = 1.)

7 Killing Tensors Background What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system The space of functions polynomial in the momenta is isomorphic to the space of symmetric contravariant tensors. If H 0 and L 0 are the parts of H and L quadratic and polynomial of degree r in the momenta, then {H 0, L 0} = 0 is equivalent to where [, ] is the Schouten bracket and g = g ij x i [g, K] = 0 and K = a i 1 i r x j x i1 So K is a second rank (symmetric) Killing tensor. x ir.

8 Tremblay-Turbiner-Winternitz (TTW) system What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system Tremblay, Turbiner and Winternitz (2009) introduced a family of potentials V TTW = αr 2 + β r 2 cos 2 (kθ) + γ r 2 sin 2 (kθ). For k = 1, 2, 3 these were known systems (Smorodinsky-Winternitz, rational BC 2 model, Wolfes model). For all k N +, Tremblay, Turbiner and Winternitz showed that the quantum systems are exactly solvable and the classical systems have closed trajectories. demonstrated superintegrability for k = 1, 2, 3, 4. conjectured that the classical and quantum systems are superintegrable. Since then KMP 2009: Rational k classical TTW systems are superintegrable. KKM 2009: Extended treatment of classical TTW systems. KKM 2010: Rational k quantum TTW systems are superintegrable. Results can be extended to higher dimensions.

9 Tremblay-Turbiner-Winternitz (TTW) system What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system Tremblay, Turbiner and Winternitz (2009) introduced a family of potentials V TTW = αr 2 + β r 2 cos 2 (kθ) + γ r 2 sin 2 (kθ). For k = 1, 2, 3 these were known systems (Smorodinsky-Winternitz, rational BC 2 model, Wolfes model). For all k N +, Tremblay, Turbiner and Winternitz showed that the quantum systems are exactly solvable and the classical systems have closed trajectories. demonstrated superintegrability for k = 1, 2, 3, 4. conjectured that the classical and quantum systems are superintegrable. Since then KMP 2009: Rational k classical TTW systems are superintegrable. KKM 2009: Extended treatment of classical TTW systems. KKM 2010: Rational k quantum TTW systems are superintegrable. Results can be extended to higher dimensions.

10 Tremblay-Turbiner-Winternitz (TTW) system What is a Superintegrable System? Second Order Constants Tremblay-Turbiner-Winternitz (TTW) system Tremblay, Turbiner and Winternitz (2009) introduced a family of potentials V TTW = αr 2 + β r 2 cos 2 (kθ) + γ r 2 sin 2 (kθ). For k = 1, 2, 3 these were known systems (Smorodinsky-Winternitz, rational BC 2 model, Wolfes model). For all k N +, Tremblay, Turbiner and Winternitz showed that the quantum systems are exactly solvable and the classical systems have closed trajectories. demonstrated superintegrability for k = 1, 2, 3, 4. conjectured that the classical and quantum systems are superintegrable. Since then KMP 2009: Rational k classical TTW systems are superintegrable. KKM 2009: Extended treatment of classical TTW systems. KKM 2010: Rational k quantum TTW systems are superintegrable. Results can be extended to higher dimensions.

11 Generalized TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The classical TTW Hamiltonian for k = 1 can be written in Cartesian coordinates as and in polar coordinates as H TTW = p 2 x + p 2 y + α(x 2 + y 2 ) + β1 x 2 + β2 y 2 H TTW = p 2 r + αr r 2 A natural generalization to 3 dimensions is or in polar coordinates, ( p 2 θ + β1 cos 2 θ + ) β2 sin 2 θ H = p 2 x + p 2 y + p 2 z + α(x 2 + y 2 + z 2 ) + β1 z 2 + β2 x 2 + β3 y 2 H = p 2 r + αr r 2 ( p 2 θ1 + β1 cos ( θ 1 sin 2 p 2 θ2 θ + 1 This pattern will continue in higher dimensions. β2 cos 2 θ 2 + β3 sin 2 θ 2 ))

12 Generalized TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The classical TTW Hamiltonian for k = 1 can be written in Cartesian coordinates as and in polar coordinates as H TTW = p 2 x + p 2 y + α(x 2 + y 2 ) + β1 x 2 + β2 y 2 H TTW = p 2 r + αr r 2 A natural generalization to 3 dimensions is or in polar coordinates, ( p 2 θ + β1 cos 2 θ + ) β2 sin 2 θ H = p 2 x + p 2 y + p 2 z + α(x 2 + y 2 + z 2 ) + β1 z 2 + β2 x 2 + β3 y 2 H = p 2 r + αr r 2 ( p 2 θ1 + β1 cos ( θ 1 sin 2 p 2 θ2 θ + 1 This pattern will continue in higher dimensions. β2 cos 2 θ 2 + β3 sin 2 θ 2 ))

13 Generalized TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The classical TTW Hamiltonian for k = 1 can be written in Cartesian coordinates as and in polar coordinates as H TTW = p 2 x + p 2 y + α(x 2 + y 2 ) + β1 x 2 + β2 y 2 H TTW = p 2 r + αr r 2 A natural generalization to 3 dimensions is or in polar coordinates, ( p 2 θ + β1 cos 2 θ + ) β2 sin 2 θ H = p 2 x + p 2 y + p 2 z + α(x 2 + y 2 + z 2 ) + β1 z 2 + β2 x 2 + β3 y 2 H = p 2 r + αr r 2 ( p 2 θ1 + β1 cos ( θ 1 sin 2 p 2 θ2 θ + 1 This pattern will continue in higher dimensions. β2 cos 2 θ 2 + β3 sin 2 θ 2 ))

14 Generalized TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The classical TTW Hamiltonian for k = 1 can be written in Cartesian coordinates as and in polar coordinates as H TTW = p 2 x + p 2 y + α(x 2 + y 2 ) + β1 x 2 + β2 y 2 H TTW = p 2 r + αr r 2 A natural generalization to 3 dimensions is or in polar coordinates, ( p 2 θ + β1 cos 2 θ + ) β2 sin 2 θ H = p 2 x + p 2 y + p 2 z + α(x 2 + y 2 + z 2 ) + β1 z 2 + β2 x 2 + β3 y 2 H = p 2 r + αr r 2 ( p 2 θ1 + β1 cos ( θ 1 sin 2 p 2 θ2 θ + 1 This pattern will continue in higher dimensions. β2 cos 2 θ 2 + β3 sin 2 θ 2 ))

15 Generalized TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The classical TTW Hamiltonian for arbitrary k is written in polar coordinates as H TTW = pr 2 + αr ( ) r 2 p 2 β 1 θ + cos 2 (kθ) + β 2 sin 2 (kθ) and so we generalize this as H = p 2 r +αr r 2 and so on to higher dimensions. ( p 2 θ1 + β 1 cos 2 (k 1θ 1) + 1 sin 2 (k 1θ 1) Inside each pair of brackets is a constant of the motion. ( )) p 2 θ2 + β 2 cos 2 (k + β 3 2θ 2) sin 2 (k 2θ 2) Next we consider some machinery for dealing with a Hamiltonian having this `nested' structure. This is a special case of a more general approach for orthogonal separable coordinates. [J. Phys. A: Math. Theor. 43 (2010) ]

16 Generalized TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The classical TTW Hamiltonian for arbitrary k is written in polar coordinates as H TTW = pr 2 + αr ( ) r 2 p 2 β 1 θ + cos 2 (kθ) + β 2 sin 2 (kθ) and so we generalize this as H = p 2 r +αr r 2 and so on to higher dimensions. ( p 2 θ1 + β 1 cos 2 (k 1θ 1) + 1 sin 2 (k 1θ 1) Inside each pair of brackets is a constant of the motion. ( )) p 2 θ2 + β 2 cos 2 (k + β 3 2θ 2) sin 2 (k 2θ 2) Next we consider some machinery for dealing with a Hamiltonian having this `nested' structure. This is a special case of a more general approach for orthogonal separable coordinates. [J. Phys. A: Math. Theor. 43 (2010) ]

17 Generalized TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The classical TTW Hamiltonian for arbitrary k is written in polar coordinates as H TTW = pr 2 + αr ( ) r 2 p 2 β 1 θ + cos 2 (kθ) + β 2 sin 2 (kθ) and so we generalize this as H = p 2 r +αr r 2 and so on to higher dimensions. ( p 2 θ1 + β 1 cos 2 (k 1θ 1) + 1 sin 2 (k 1θ 1) Inside each pair of brackets is a constant of the motion. ( )) p 2 θ2 + β 2 cos 2 (k + β 3 2θ 2) sin 2 (k 2θ 2) Next we consider some machinery for dealing with a Hamiltonian having this `nested' structure. This is a special case of a more general approach for orthogonal separable coordinates. [J. Phys. A: Math. Theor. 43 (2010) ]

18 Subgroup separable classical Hamiltonians Generalized TTW Subgroup separable classical Hamiltonians A 4D example Consider the n functionally independent functions of q i and p i (i = 1,..., n), H = L 1 = p V 1(q 1) + f 1(q 1)L 2, L 2 = p V 2(q 2) + f 2(q 2)L 3,. L n 1 = p 2 n 1 + V n 1(q n 1) + f n 1(q n 1)L n, L n = p 2 n + V n(q n). The coordinates q i separate the Hamilton-Jacobi equation for H. If p i = S q i then ( ) 2 S L i = + V i (q i ) + f i (q i )L i+1 = λ i q i is an equation in q i alone. (For convenience, dene L n+1 = 0.)

19 Subgroup separable classical Hamiltonians Generalized TTW Subgroup separable classical Hamiltonians A 4D example Consider the n functionally independent functions of q i and p i (i = 1,..., n), H = L 1 = p V 1(q 1) + f 1(q 1)L 2, L 2 = p V 2(q 2) + f 2(q 2)L 3,. L n 1 = p 2 n 1 + V n 1(q n 1) + f n 1(q n 1)L n, L n = p 2 n + V n(q n). The coordinates q i separate the Hamilton-Jacobi equation for H. If p i = S q i then ( ) 2 S L i = + V i (q i ) + f i (q i )L i+1 = λ i q i is an equation in q i alone. (For convenience, dene L n+1 = 0.)

20 Additional constants Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example It is easy to check that H is integrable, that is, {L i, L j } = 0 i, j = 1,..., n. We would like to nd more constants of the motion to show superintegrability. For each i = 1,..., n 1 try to nd M i (q i, L 1, L 2,..., L n) and N i (q i+1, L 1, L 2,..., L n) satisfying i i {H, M i } = f k (q k ) and {H, N i } = f k (q k ). k=1 k=1 Then {H, N i M i } = 0 and hence L i = N i M i is in involution with H.

21 Additional constants Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example It is easy to check that H is integrable, that is, {L i, L j } = 0 i, j = 1,..., n. We would like to nd more constants of the motion to show superintegrability. For each i = 1,..., n 1 try to nd M i (q i, L 1, L 2,..., L n) and N i (q i+1, L 1, L 2,..., L n) satisfying i i {H, M i } = f k (q k ) and {H, N i } = f k (q k ). k=1 k=1 Then {H, N i M i } = 0 and hence L i = N i M i is in involution with H.

22 Additional constants Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example It is easy to check that H is integrable, that is, {L i, L j } = 0 i, j = 1,..., n. We would like to nd more constants of the motion to show superintegrability. For each i = 1,..., n 1 try to nd M i (q i, L 1, L 2,..., L n) and N i (q i+1, L 1, L 2,..., L n) satisfying i i {H, M i } = f k (q k ) and {H, N i } = f k (q k ). k=1 k=1 Then {H, N i M i } = 0 and hence L i = N i M i is in involution with H.

23 Additional constants Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The M i and N i satisfy 2p i M i q i = f i (q i ) and 2p i+1 N i q i+1 = 1. The equation can be solved by writing p i in terms of L i and integrating f i (q i )dq i M i = 2 L i V i (q i ) f i (q i )L i+1 dq i+1 N i = 2. L i+1 V i+1 (q i+1 ) f i+1 (q i+1 )L i+2 It's straightforward to check that {L i, L i 1} = 1 for i > 1 and {L i, L j} = 0 for j i 1. and deduce set {L 1,..., L n, L 1,..., L n 1} is functionally independent. They are a set of action-angle variables.

24 Additional constants Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The M i and N i satisfy 2p i M i q i = f i (q i ) and 2p i+1 N i q i+1 = 1. The equation can be solved by writing p i in terms of L i and integrating f i (q i )dq i M i = 2 L i V i (q i ) f i (q i )L i+1 dq i+1 N i = 2. L i+1 V i+1 (q i+1 ) f i+1 (q i+1 )L i+2 It's straightforward to check that {L i, L i 1} = 1 for i > 1 and {L i, L j} = 0 for j i 1. and deduce set {L 1,..., L n, L 1,..., L n 1} is functionally independent. They are a set of action-angle variables.

25 Four-dimensional TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example We now consider a 4-dimensional generalization of the TTW system. H = L 1 = p 2 r + αr 2 + L2 r 2 L 2 = p 2 θ1 + β 1 cos 2 (k 1θ 1) + L 3 sin 2 (k 1θ 1) L 3 = p 2 θ2 + β 2 cos 2 (k 2θ 2) + L 4 sin 2 (k 2θ 2) L 4 = p 2 θ3 + β 3 cos 2 (k 3θ 3) + β 4 sin 2 (k 3θ 3). This system is superintegrable for all positive rational k 1, k 2 and k 3.

26 Four-dimensional TTW Background Generalized TTW Subgroup separable classical Hamiltonians A 4D example The corresponding metric g = e 0 e 0 + e 1 e 1 + e 2 e 2 + e 3 e 3, e 0 = dr, e 1 = rdθ 1, e 2 = r sin(k 1θ 1)dθ 2, e 3 = r sin(k 1θ 1) sin(k 2θ 2)dθ 3. is not conformally at since W abcd W abcd = 4(k2 1 k 2 2 ) 2 3r 4 sin 4 (k 1θ 1). W abcd is the Weyl conformal curvature tensor. The metric g is conformally at if and only if k 2 1 = k 2 2. Next, illustrate the superintegrability with k 1 = 2, k 2 = k 3 = 1. The general case is similar.

27 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example We need M 1(r, H, L 2, L 3, L 4) and N 1(θ 1, H, L 2, L 3, L 4) such that and so M 1 = {H, M 1} = 1 r 2 and {H, N 1} = 1 r 2 dr, N 1 = 2r H 2 αr 2 L r 2 2 L 2 dθ 1 β1 cos 2 (k 1 θ1) L 3 sin 2 (k 1 θ1) which gives where sinh B 1 = i sinh A 1 = i M 1 = B1 4 A 1, N 1 =, L 2 4k 1 L2 H 2L 2 r 2 H 2 cosh B 1 = 2 L 2p r 4αL 2 r H 2 4αL 2 L 2 cos(2k 1θ 1) + L 3 β 1 (β1 L 2 L 3) 2 4L 2L 3 cosh A 1 = L2 sin(2k 1θ 1)p θ1 (β1 L 2 L 3) 2 4L 2L 3

28 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example We need M 1(r, H, L 2, L 3, L 4) and N 1(θ 1, H, L 2, L 3, L 4) such that and so M 1 = {H, M 1} = 1 r 2 and {H, N 1} = 1 r 2 dr, N 1 = 2r H 2 αr 2 L r 2 2 L 2 dθ 1 β1 cos 2 (k 1 θ1) L 3 sin 2 (k 1 θ1) which gives where sinh B 1 = i sinh A 1 = i M 1 = B1 4 A 1, N 1 =, L 2 4k 1 L2 H 2L 2 r 2 H 2 cosh B 1 = 2 L 2p r 4αL 2 r H 2 4αL 2 L 2 cos(2k 1θ 1) + L 3 β 1 (β1 L 2 L 3) 2 4L 2L 3 cosh A 1 = L2 sin(2k 1θ 1)p θ1 (β1 L 2 L 3) 2 4L 2L 3

29 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example We need M 1(r, H, L 2, L 3, L 4) and N 1(θ 1, H, L 2, L 3, L 4) such that and so M 1 = {H, M 1} = 1 r 2 and {H, N 1} = 1 r 2 dr, N 1 = 2r H 2 αr 2 L r 2 2 L 2 dθ 1 β1 cos 2 (k 1 θ1) L 3 sin 2 (k 1 θ1) which gives where sinh B 1 = i sinh A 1 = i M 1 = B1 4 A 1, N 1 =, L 2 4k 1 L2 H 2L 2 r 2 H 2 cosh B 1 = 2 L 2p r 4αL 2 r H 2 4αL 2 L 2 cos(2k 1θ 1) + L 3 β 1 (β1 L 2 L 3) 2 4L 2L 3 cosh A 1 = L2 sin(2k 1θ 1)p θ1 (β1 L 2 L 3) 2 4L 2L 3

30 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example sinh(8 L 2(N 1 M 1)) = sinh(a 1 2B 1) = 2 cosh A 1 sinh B 1 cosh B sinh A 1 cosh 2 B 1 sinh A 1 = (( ) ) 4iL 2 H 2L2 sin(4θ1) 2(L2 cos(4θ1) + L3 β1) r 2 p θ1 p r + r r 2 pr 2 (H 2 4αL 2) (β 1 L 2 L 3) 2 4L 2L 3 i (H 2 4αL 2)(L 2 cos(4θ 1) + L 3 β 1) (H 2 4αL 2) (β 1 L 2 L 3) 2 4L 2L 3 = 4iL 2L 1 ih 2 (L 3 β 1) (H 2 4αL 2) (β 1 L 2 L 3) 2 4L 2L 3 Everything apart from L 1 is known to be a constant of the motion and hence ( ) L 1 = H 2L2 sin(4θ1) 2(L2 cos(4θ1) + L3 β1) r 2 p θ1 p r + r r 2 pr (H2 αl 2) cos(4θ 1) is a constant quartic in he momenta.

31 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example sinh(8 L 2(N 1 M 1)) = sinh(a 1 2B 1) = 2 cosh A 1 sinh B 1 cosh B sinh A 1 cosh 2 B 1 sinh A 1 = (( ) ) 4iL 2 H 2L2 sin(4θ1) 2(L2 cos(4θ1) + L3 β1) r 2 p θ1 p r + r r 2 pr 2 (H 2 4αL 2) (β 1 L 2 L 3) 2 4L 2L 3 i (H 2 4αL 2)(L 2 cos(4θ 1) + L 3 β 1) (H 2 4αL 2) (β 1 L 2 L 3) 2 4L 2L 3 = 4iL 2L 1 ih 2 (L 3 β 1) (H 2 4αL 2) (β 1 L 2 L 3) 2 4L 2L 3 Everything apart from L 1 is known to be a constant of the motion and hence ( ) L 1 = H 2L2 sin(4θ1) 2(L2 cos(4θ1) + L3 β1) r 2 p θ1 p r + r r 2 pr (H2 αl 2) cos(4θ 1) is a constant quartic in he momenta.

32 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example Next M 2(θ 1, H, L 2, L 3, L 4) and N 2(θ 2, H, L 2, L 3, L 4) must satisfy and so {H, M 2} = M 2 = N 2 = 1 r 2 sin 2 (k 1θ 1) 2 sin 2 (k 1θ 1) L 2 2 L 3 and {H, N 2} = dθ 2 dθ 1 1 r 2 sin 2 (k 1θ 1). β1 cos 2 (k 1 θ1) L 3 sin 2 (k 1 θ1) β2 cos 2 (k 2 θ2) L 4 sin 2 (k 2 θ2), where M 2 = B 2 4k 1 L3 and N 2 = A 2 4k 2 L3 sinh B 2 = 2L3cosec2 (k 1θ 1) + β 1 L 2 L 3 4β1L2 (L 3 L 2 β 1) 2 2i L 3 cot(k 1θ 1)p θ1 cosh B 2 = 4β1L 2 (L 3 L 2 β 1) 2 sinh A 2 = i L 3 cos(2k 2θ 2) + β 3 β 2 L3 sin(2k 2θ 2)p θ2 cosh A 2 = (β2. (β2 β 3 L 3) 2 4β 3L 3 β 3 L 3) 2 4β 3L 3

33 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example From this we can nd another quartic constant L 2 = 2(L 3 cos(2θ 2) + L 4 β 2) cot(2θ 1) sin(2θ 2)p θ1 p θ2 + ((β 2 L 3 L 4) 2 4L 3L 4)cosec 2 (2θ 1) sin 2 (2θ 2)(2L 3cosec 2 (2θ 1) + β 1 L 2 L 3)p 2 θ2 and then M 3(θ 2, H, L 2, L 3, L 4) and N 3(θ 3, H, L 2, L 3, L 4) lead to a cubic constant L 3 = 2(L 4 cos(2θ 3) + β 4 β 3) cot(θ 2)p θ2 (2L 4cosec 2 (θ 2) + β 2 L 3 L 4) sin(2θ 3)p θ3 So we have 7 functionally independent polynomial constants in total and hence the system is superintegrable. The same process works for any rational k 1, k 2 and k 3.

34 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example From this we can nd another quartic constant L 2 = 2(L 3 cos(2θ 2) + L 4 β 2) cot(2θ 1) sin(2θ 2)p θ1 p θ2 + ((β 2 L 3 L 4) 2 4L 3L 4)cosec 2 (2θ 1) sin 2 (2θ 2)(2L 3cosec 2 (2θ 1) + β 1 L 2 L 3)p 2 θ2 and then M 3(θ 2, H, L 2, L 3, L 4) and N 3(θ 3, H, L 2, L 3, L 4) lead to a cubic constant L 3 = 2(L 4 cos(2θ 3) + β 4 β 3) cot(θ 2)p θ2 (2L 4cosec 2 (θ 2) + β 2 L 3 L 4) sin(2θ 3)p θ3 So we have 7 functionally independent polynomial constants in total and hence the system is superintegrable. The same process works for any rational k 1, k 2 and k 3.

35 Four-dimensional TTW (k 1, k 2, k 3 = 2, 1, 1) Generalized TTW Subgroup separable classical Hamiltonians A 4D example From this we can nd another quartic constant L 2 = 2(L 3 cos(2θ 2) + L 4 β 2) cot(2θ 1) sin(2θ 2)p θ1 p θ2 + ((β 2 L 3 L 4) 2 4L 3L 4)cosec 2 (2θ 1) sin 2 (2θ 2)(2L 3cosec 2 (2θ 1) + β 1 L 2 L 3)p 2 θ2 and then M 3(θ 2, H, L 2, L 3, L 4) and N 3(θ 3, H, L 2, L 3, L 4) lead to a cubic constant L 3 = 2(L 4 cos(2θ 3) + β 4 β 3) cot(θ 2)p θ2 (2L 4cosec 2 (θ 2) + β 2 L 3 L 4) sin(2θ 3)p θ3 So we have 7 functionally independent polynomial constants in total and hence the system is superintegrable. The same process works for any rational k 1, k 2 and k 3.

36 Four-dimensional Quantum TTW The quantum version has four mutually commuting dierential operators, H = 2 r + 3 r r ω2 r 2 + L1 r 2 L 1 = 2 θ1 + 2k1 cot(k1θ1) θ1 + L 2 = 2 θ2 + k2 cot(k2θ2) θ2 + L 3 = 2 θ3 + β 3 cos 2 (k 3θ 3) + β 4 sin 2 (k 3θ 3) β 1 cos 2 (k + L 2 1θ 1) sin 2 (k 1θ 1) β 2 cos 2 (k + L 3 2θ 2) sin 2 (k 2θ 2) The Hamiltonian is of the form H = 2 + V 0 where 2 is the Laplacian on a four-dimensional manifold with metric g = e 0 e 0 + e 1 e 1 + e 2 e 2 + e 3 e 3, e 0 = dr, e 1 = rdθ 1, e 2 = r sin(k 1θ 1)dθ 2, e 3 = r sin(k 1θ 1) sin(k 2θ 2)dθ 3.

37 Four-dimensional Quantum TTW The equation HΨ = EΨ is separable in the coordinate r, θ 1, θ 2, θ 3 but for reasons that will be explained, we add some extra terms. H = L 0 = r r r ω2 r 2 + L1 r k2 1 r 2 L 1 = 2 θ1 + 2k1 cot(k1θ1) β 1 θ1 + cos 2 (k + L 2 1θ 1) sin 2 (k + k2 1 k2 2 1θ 1) 4 sin 2 (k 1θ 1) L 2 = 2 θ2 + k2 cot(k2θ2) β 2 θ2 + cos 2 (k + L 3 2θ 2) sin 2 (k 2θ 2) L 3 = 2 θ3 + β 3 cos 2 (k 3θ 3) + β 4 sin 2 (k 3θ 3) HΨ = EΨ remains separable with the additional terms.

38 Four-dimensional Quantum TTW For the metric g, the scalar curvature is and if we dene R = 6 r 2 + k2 1 ( 6 r 2 2 r 2 sin 2 (k 1θ 1) ) + W = 3W abcd W abcd = 2(k2 1 k 2 2 ) r 2 sin 2 (k 1θ 1). 2k 2 2 r 2 sin 2 (k 1θ 1) then H = 2 + V R 1 24 W The metric g is conformally at if and only if k 2 1 = k 2 2.

39 Four-dimensional Quantum TTW We look for solutions of the form HΨ = EΨ, Ψ = Ψ 0(r)Ψ 1(θ 1)Ψ 2(θ 2)Ψ 3(θ 3), and L 3Ψ 3 = l 3Ψ 3, L 2Ψ 2Ψ 3 = l 2Ψ 2Ψ 3, L 1Ψ 1Ψ 2Ψ 3 = l 1Ψ 1Ψ 2Ψ 3. For the angular equations we will nd the equation ( 1 u 4 + α2 1 sin 2 y + 4 ) β2 cos 2 + (2n + α + β + 1)2 u = 0. y which has solution u(y) = (sin y) α+ 1 2 (cos y) β+ 1 2 P (α,β) n (cos 2y) where P n (α,β) (x) is a Jacobi function.

40 Four-dimensional Quantum TTW We look for solutions of the form HΨ = EΨ, Ψ = Ψ 0(r)Ψ 1(θ 1)Ψ 2(θ 2)Ψ 3(θ 3), and L 3Ψ 3 = l 3Ψ 3, L 2Ψ 2Ψ 3 = l 2Ψ 2Ψ 3, L 1Ψ 1Ψ 2Ψ 3 = l 1Ψ 1Ψ 2Ψ 3. For the angular equations we will nd the equation ( 1 u 4 + α2 1 sin 2 y + 4 ) β2 cos 2 + (2n + α + β + 1)2 u = 0. y which has solution u(y) = (sin y) α+ 1 2 (cos y) β+ 1 2 P (α,β) n (cos 2y) where P n (α,β) (x) is a Jacobi function.

41 Four-dimensional Quantum TTW With the replacements β 3 = k 2 3 ( ) ( ) 1 4 a2 4, β 4 = k a2 3, the separated θ 3 equation is ( ( k 2 1 L 3Ψ 3(θ 3) = Ψ (θ 3) + ( 1 4) a2 cos 2 (k + k2 3 4 ) 3) a2 3θ 3) sin 2 Ψ 3(θ 3) = l 3Ψ 3(θ 3) (k 3θ 3) which has solutions Ψ a 3,a 4 3,n 3 (θ 3) = (sin(k 3θ 3)) a (cos(k 3θ 3)) a P (a 3,a 4 ) n 3 (cos(2k 3θ 3)) and eigenvalues L 3Ψ a 3,a 4 3,n 3 (θ 3) = l 3Ψ a 3,a 4 3,n 3 (θ 3), l 3 = k 2 3 (2n 3 + a 3 + a 4 + 1) 2.

42 Four-dimensional Quantum TTW The separated θ 2 equation L 2Ψ 2(θ 2) = l 2Ψ 2(θ 2) is ( Ψ 2 (θ 2) + k 2 cot(k 2θ 2)Ψ 2(θ 2) + which we transform with β 2 cos 2 (k 2θ 2) + l 3 sin 2 (k 2θ 2) Ψ 2(θ 2) = (sin(k 2θ 2)) 1 2 ψ(θ 2) to absorb the rst derivative term to give ( ψ 2 (θ 2) + β 2 cos 2 (k 2θ 2) + l k2 2 sin 2 (k 2θ 2) k2 2 l 2 ) Ψ 2(θ 2) = l 2Ψ 2(θ 2) ) ψ 2(θ 2) = 0 and we make the replacements ( ) β 2 = k a2 2, l ( ) 4 k2 2 = k A2 2 A 2 = k3 k 2 (2n 3 + a 3 + a 4 + 1)

43 Four-dimensional Quantum TTW The separated θ 2 equation becomes ( 1 k2 2 ψ 2 4 (θ 2) + a2 1 2 cos 2 (k + 4 ) A2 2 2θ 2) sin 2 (k + 1 2θ 2) 4 l2 k2 2 ψ 2(θ 2) = 0 where 1 4 l2 k 2 2 = (2n 2+a 2+A 2+1) 2 l 2 = k 2 2 Λ 2(Λ 2+1), Λ 2 = 2n 2+a 2+A and has solution Ψ A 2,a 2 2,n 2 (θ 2) = (sin(k 2θ 2)) A2 (cos(k 2θ 2)) a P (A 2,a 2 ) n 2 (cos(2k 2θ 2)).

44 Four-dimensional Quantum TTW The separated θ 1 equation L 1Ψ 1(θ 1) = l 1Ψ 1(θ 1) is ( Ψ 1 (θ 1)+2k 1 cot(k 1θ 1)Ψ 1(θ 1)+ which we transform with to absorb the rst order term to give k 2 1 ψ 1 (θ 1) + and we make the replacements ( β 1 = k a2 1 β 1 cos 2 (k + l (k2 1 k2 2 ) 1θ 1) sin 2 (k 1θ 1) Ψ 1(θ 1) = (sin(k 1θ 1)) 1 ψ(θ 1) ( 1 4 a2 1 cos 2 (k + l (k2 1 k2 2 ) 1θ 1) sin l1 (k 1θ 1) k1 2 ), l (k2 1 k 2 2 ) = k 2 1 A 1 = k2 k 1 (2n 2 + A 2 + a 2 + 1) ) ) Ψ 1(θ 1) = l 1Ψ 1(θ 1), ψ 1(θ 1) = 0 ( ) 1 4 A2 1

45 Four-dimensional Quantum TTW The separate θ 1 equation becomes where 1 l1 k 2 1 ( 1 k1 2 2 θ1 ψ1(θ1) + 4 a2 1 1 cos 2 (k + 4 A2 1 1θ 1) sin 2 (k + 1 l1 1θ 1) k 2 1 ) ψ 1(θ 1) = 0 = (2n 1 + A 1 + a 1 + 1) 2 l 1 = k 2 1 Λ 1(Λ 1 + 2), Λ 1 = 2n 1 + a 1 + A 1 and has solutions Ψ A 1,a 1 1,n 1 (θ 1) = (sin(k 1θ 1)) A (cos(k 1θ 1)) a P (A 1,a 1 ) n 1 (cos(2k 1θ 1))

46 Four-dimensional Quantum TTW Finally, the separated radial equation is HΨ 0(r) = r 2 Ψ 0(r) + 3 ( r r Ψ0(r) + ω 2 r 2 + l1 ) k r 2 Ψ 0(r) = EΨ 0(r). which has solution Ψ A 0 0,n 0 (r) = e ωr2 2 r A 0 1 L (A 0 ) n 0 (ωr 2 ) where L A n (x) is a Laguerre functions and A 2 0 = k 2 1 l 1, A 0 = k 1(2n 1 + a 1 + A 1 + 1), E = ω(4n 0 + 2A 0 + 2).

47 Four-dimensional Quantum TTW Now, putting these together we nd E = ω(4n 0 + 2A 0 + 2) A 0 = k 1(2n 1 + a 1 + A 1 + 1) A 1 = k2 k 1 (2n 2 + A 2 + a 2 + 1) A 2 = k3 k 2 (2n 3 + a 3 + a 4 + 1) E = 2ω(2n 0+2k 1n 1+2k 2n 2+2k 3n 3+k 1a 1+k 2a 2+k 3a 3+k 3a 4+k 1+k 2+k 3+1) for a solution of the form Ψ n0,n 1,n 2,n 3 = Ψ A 0 0,n 0 (r)ψ A 1,a 1 1,n 1 (θ 1)Ψ A 2,a 2 2,n 2 (θ 2)Ψ a 3,a 4 3,n 3 (θ 3). Our aim is to use special function identities to raise and lower the n i while preserving E and produce new operators commuting with H.

48 Raising and lowering operators for TTW solutions Some examples... 0 n 0 = 1 A0 r + (2n 0 + A 0 + 1)ω + 1 A2 0 r r 2 K + A 0 0 n 0 = 1 + A0 r + (2n 0 + A 0 + 1)ω + 1 A2 0 r r 2 K A 0 These raising and lower n 0 and A 0 simultaneously. K + A 0 0 n 0 Ψ A 0 n 0 = 2(n 0 + 1)(n 0 + A 0)Ψ A 0 2 n 0 +1 K A 0 0 n 0 Ψ A 0 n 0 = 2ω 2 Ψ A 0+2 n 0 1

49 Raising and lowering operators for TTW solutions For the angular functions, we can use Jacobi function identities to make operators that raising and lower n alone. J n + (N + 1) sin(2kθ) = θ 1 ( (N + 1)(N + 1 c d) cos(2kθ) 2k 2 (N + 1)(c d) + a 2 b 2) Jn (N 1) sin(2kθ) = θ 1 ( (N 1)(N 1 + c + d) cos(2kθ) 2k 2 + (N 1)(c d) + a 2 b 2) N = 2n + a + b + 1, Θ (a,b) n = sin a+c (kθ) cos b+d (kθ)p n (a,b) cos(2kθ) J + n Θ (a,b) n = 2(n + 1)(n + a + b + 1)Θ (a,b) n+1 J n Θ (a,b) n = 2(n + a)(n + b)θ (a,b) n 1

50 Raising and lowering operators for TTW solutions We can also use Jacobi function identities to make operators that raising and lower n and a simultaneously. Again, for we have K n + a (1 a) cos(kθ) = θ 2(n(n + a + b + 1) + a(a + b)) k sin(kθ) (1 a)(a c) (1 a)(a + c + b + d) sin 2 (kθ) Kn a (1 + a) cos(kθ) = θ 2n(n + a + b + 1) k sin(kθ) (1 + a)(a + c) (1 + a)(a + c + b + d) + sin 2 (kθ) Θ (a,b) n = sin a+c (kθ) cos b+d (kθ)p n (a,b) cos(2kθ) K n + a Θ (a,b) n = 2(n + 1)(n + a)θn+1 a 2,b Kn a Θ (a,b) n = 2(n + a + b + 1)(n + b)θ a+2,b n 1

51 Raising and lowering operators for TTW solutions We now consider k 1 = p 1/q 1 for relatively prime p 1 and q 1 and form the operator Ξ + 01 = K A 0+2(p 1 1) 0 n 0 (p 1 1) K A 0 0 n 0 J 1 + n 1 +q 1 1 J 1 + n 1 which has the eect on a basis function of and so n 0 n 0 p 1, n 1 n 1 + q 1, A 0 A 0 + 2p 1. E = 2ω(2n 0 + 2k 1n 1 + ) 2ω(2(n 0 p 1) + 2k 1(n 1 + q 1) + ) = 2ω(2n 0 + 2k 1n 1 + ), that is, E is unchanged. A similar lowering operator is Ξ 01 = K + A 0 2(p 1 1) 0 n 0 +(p 1 1) K + A 0 0 n 0 J 1 n 1 (q 1 1) J 1 n 1.

52 Additional symmetries of generalized TTW The transformation n 1 n 1 A 1 a 1 1 while holding E constant has the eect of changing the sign of A 0. It is straightforward to check from the explicit expressions for the operators that L + 01 = Ξ Ξ 01 and L 01 = Ξ+ 01 Ξ 01 A 0 are polynomials in E, A 2 0 and A 2 1. Since A 2 0 = k 2 1 l 1 and A 2 1 = 1 4 k2 2 l 2 k 2 1 we can replace E, A 2 0 and A 2 1 with a second order dierential operators where ever they appear in these expressions. E.g. for k 1, k 2, k 3 = 2, 1, 1 these are operators that commute with H of orders 5 and 6 that are algebraically independent of the second order operators.

53 Additional symmetries of generalized TTW For the case k 1, k 2, k 3 = 2, 1, 1, L + 01 is a 5th order operator. ( L + 01 = 2 r 3 r + 6 r 4 ( + + ) ( A 2 0 A2 1 + cos(4θ 1) 2r 3 r + sin(4θ 1) 4r 4 θ cos(4θ ) 1) 2r 4 A0 4 ( ) ( 1 r + 2 r r 2 E A1 2 sin(4θ1 ) + θ1 + cos(4θ ) 1) + 1 E r + sin(4θ 1) 4r 2 θ1 + 3 cos(4θ ) ( ) 1) + 1 2r 2 E A r 3 r + 4 r 2 2 r 6 r 4 A1 2 cos(4θ 1) 4r ( sin(4θ1 ) r θ1 + 3 cos(4θ 1) + 2 a1 2 r r ( + sin(4θ 1) 4r 2 r 2 θ sin(4θ 1) r 5 sin(4θ 1) 4r 2 θ1 (6 cos(4θ 1) + 5 4a 2 1 ) ) 2r 2 E 4r 3 r θ1 (4 cos(4θ 1) + 1) 2r 2 r cos(4θ 1) 4a 2 1 2r 3 r 5 sin(4θ 1) r 4 θ1 25 cos(4θ 1) a 2 ) 1 2r 4 A sin(4θ 1) 4r 2 r 2 (11 8a 2 θ cos(4θ 1)) 2r 2 r 2 23 sin(4θ 1) 4r 3 r θ1 26 cos(4θ 1) a1 2 2r 3 r 21 sin(4θ 1) 4r 4 θ1 3(10 cos(4θ 1) + 7 4a 2 1 ) 2r 4 with the replacements E H, A 2 0 k2 1 L 1, A 2 1 (k2 2 4L 2)/(4k 2 1 ).

54 Additional symmetries of generalized TTW The operators L ± 01 are not of the lowest possible order. We can attempt to nd operators M 01 ± such that [L 1, M 01] ± = L ± 01, [H, M ± 01] = 0, [L 2, M ± 01] = 0, [L 3, M ± 01] = 0. For M01 the solution (when acting on a basis function) is M01 = 1 ( L ) 01 4q 1 A 0(A 0 + p + L + 01 S1(H, L2) + 1) A 0(A 0 p 1) A 2 0 p2 1 where k 1 = p 1/q 1 with p 1 and q 1 relatively prime and S 1(H, L 2) is an polynomial in H and L 2 to be determined.

55 Additional symmetries of generalized TTW From M01 = 1 ( 4q 1 L 01 A 0(A 0 + p 1) + L + 01 A 0(A 0 p 1) ) + S1(H, L2) A 2 0, p2 1 multiplying both sides by A 2 0 p 2 1 gives 4q 1(A 2 0 p1)m 2 01 = L 01 (A0 p1) + L+ 01(A 0 + p 1) + S 1(H, L 2) A 0 A 0 and so S 1(H, L 2) can be determined by setting A 0 = ±p 1. For example for p 1 = 2 and q 1 = 1, S 1(H, L 2) = (H2 4ω)(A 2 1 a 2 1) 16 and M 01 + turns out to be a polynomial in E and even powers of A0 and A1.

56 Further work Background The commutator algebra of the symmetry operators can be derived using similar methods (see e.g. SIGMA 7 (2011) 031). This provides examples of irreducible Killing tensors that naturally lead to symmetries of a modied Laplacian. Why does this work?

### Structure relations for the symmetry algebras of classical and quantum superintegrable systems

UNAM talk p. 1/4 Structure relations for the symmetry algebras of classical and quantum superintegrable systems Willard Miller miller@ima.umn.edu University of Minnesota UNAM talk p. 2/4 Abstract 1 A quantum

### Superintegrability and exactly solvable problems in classical and quantum mechanics

Superintegrability and exactly solvable problems in classical and quantum mechanics Willard Miller Jr. University of Minnesota W. Miller (University of Minnesota) Superintegrability Penn State Talk 1 /

### Equivalence of superintegrable systems in two dimensions

Equivalence of superintegrable systems in two dimensions J. M. Kress 1, 1 School of Mathematics, The University of New South Wales, Sydney 058, Australia. In two dimensions, all nondegenerate superintegrable

### Warped product of Hamiltonians and extensions of Hamiltonian systems

Journal of Physics: Conference Series PAPER OPEN ACCESS Warped product of Hamiltonians and extensions of Hamiltonian systems To cite this article: Claudia Maria Chanu et al 205 J. Phys.: Conf. Ser. 597

### Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties

Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties E. G. Kalnins Department of Mathematics, University of Waikato, Hamilton, New Zealand. J. M. Kress School of Mathematics,

### arxiv: v1 [math-ph] 8 May 2016

Superintegrable systems with a position dependent mass : Kepler-related and Oscillator-related systems arxiv:1605.02336v1 [math-ph] 8 May 2016 Manuel F. Rañada Dep. de Física Teórica and IUMA Universidad

### Variable separation and second order superintegrability

Variable separation and second order superintegrability Willard Miller (Joint with E.G.Kalnins) miller@ima.umn.edu University of Minnesota IMA Talk p.1/59 Abstract In this talk we shall first describe

### Models of quadratic quantum algebras and their relation to classical superintegrable systems

Models of quadratic quantum algebras and their relation to classical superintegrable systems E. G, Kalnins, 1 W. Miller, Jr., 2 and S. Post 2 1 Department of Mathematics, University of Waikato, Hamilton,

### Superintegrable 3D systems in a magnetic field and Cartesian separation of variables

Superintegrable 3D systems in a magnetic field and Cartesian separation of variables in collaboration with L. Šnobl Czech Technical University in Prague GSD 2017, June 5-10, S. Marinella (Roma), Italy

### arxiv: v1 [math-ph] 31 Jan 2015

Symmetry, Integrability and Geometry: Methods and Applications SIGMA? (00?), 00?,?? pages Structure relations and Darboux contractions for D nd order superintegrable systems R. Heinonen, E. G. Kalnins,

### 20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R

20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian

### Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

JOURNAL OF MATHEMATICAL PHYSICS 48, 113518 2007 Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties E. G. Kalnins Department of Mathematics, University of

### Superintegrable three body systems in one dimension and generalizations

arxiv:0907.5288v1 [math-ph] 30 Jul 2009 Superintegrable three body systems in one dimension and generalizations Claudia Chanu Luca Degiovanni Giovanni Rastelli Dipartimento di Matematica, Università di

### Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties

Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties E. G. Kalnins Department of Mathematics, University of Waikato, Hamilton, New Zealand. J. M. Kress School of Mathematics,

### Models for the 3D singular isotropic oscillator quadratic algebra

Models for the 3D singular isotropic oscillator quadratic algebra E. G. Kalnins, 1 W. Miller, Jr., and S. Post 1 Department of Mathematics, University of Waikato, Hamilton, New Zealand. School of Mathematics,

### arxiv: v1 [math-ph] 21 Oct 2013

Extensions of Hamiltonian systems dependent on a rational parameter Claudia M. Chanu, Luca Degiovanni, Giovanni Rastelli arxiv:30.5690v [math-ph] 2 Oct 203 Dipartimento di Matematica G. Peano, Università

### arxiv: v1 [math-ph] 22 Apr 2018

CARTER CONSTANT AND SUPERINTEGRABILITY K Rajesh Nayak, and Payel Mukhopadhyay 2,, Center of Excellence in Space Sciences, India and Department of Physical Sciences, Indian Institute of Science Education

### Complete sets of invariants for dynamical systems that admit a separation of variables

Complete sets of invariants for dynamical systems that admit a separation of variables. G. Kalnins and J.. Kress Department of athematics, University of Waikato, Hamilton, New Zealand, e.kalnins@waikato.ac.nz

### Superintegrability of Calogero model with oscillator and Coulomb potentials and their generalizations to (pseudo)spheres: Observation

Superintegrability of Calogero model with oscillator and Coulomb potentials and their generalizations to (pseudo)spheres: Observation Armen Nersessian Yerevan State University Hannover 013 Whole non-triviality

### Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere

Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere Willard Miller, [Joint with E.G. Kalnins (Waikato) and Sarah Post (CRM)] University of Minnesota Special Functions

### Complete integrability of geodesic motion in Sasaki-Einstein toric spaces

Complete integrability of geodesic motion in Sasaki-Einstein toric spaces Mihai Visinescu Department of Theoretical Physics National Institute for Physics and Nuclear Engineering Horia Hulubei Bucharest,

### Physics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I

Physics 342 Lecture 26 Angular Momentum Lecture 26 Physics 342 Quantum Mechanics I Friday, April 2nd, 2010 We know how to obtain the energy of Hydrogen using the Hamiltonian operator but given a particular

### COULOMB SYSTEMS WITH CALOGERO INTERACTION

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 016, 3, p. 15 19 COULOMB SYSTEMS WITH CALOGERO INTERACTION P h y s i c s T. S. HAKOBYAN, A. P. NERSESSIAN Academician G. Sahakyan

### Angular momentum. Quantum mechanics. Orbital angular momentum

Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular

### Vectors, metric and the connection

Vectors, metric and the connection 1 Contravariant and covariant vectors 1.1 Contravariant vectors Imagine a particle moving along some path in the 2-dimensional flat x y plane. Let its trajectory be given

### Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

### Lecture 5: Orbital angular momentum, spin and rotation

Lecture 5: Orbital angular momentum, spin and rotation 1 Orbital angular momentum operator According to the classic expression of orbital angular momentum L = r p, we define the quantum operator L x =

### d 1 µ 2 Θ = 0. (4.1) consider first the case of m = 0 where there is no azimuthal dependence on the angle φ.

4 Legendre Functions In order to investigate the solutions of Legendre s differential equation d ( µ ) dθ ] ] + l(l + ) m dµ dµ µ Θ = 0. (4.) consider first the case of m = 0 where there is no azimuthal

### 2 Canonical quantization

Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.

### The 3 dimensional Schrödinger Equation

Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum

### Introduction to Matrix Algebra

Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

### Symmetries for fun and profit

Symmetries for fun and profit Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 August 28, 2008 Sourendu Gupta (TIFR Graduate School) Symmetries for fun and profit QM I 1 / 20 Outline 1 The isotropic

### arxiv: v1 [math-ph] 28 Apr 2014

Polynomial symmetries of spherical Lissajous systems J.A. Calzada 1, Ş. Kuru 2, and J. Negro 3 arxiv:1404.7066v1 [math-ph] 28 Apr 2014 1 Departamento Matemática Aplicada, Universidad de Valladolid, 47011

### arxiv: v4 [math-ph] 3 Nov 2015

Symmetry Integrability and Geometry: Methods and Applications Examples of Complete Solvability of D Classical Superintegrable Systems Yuxuan CHEN Ernie G KALNINS Qiushi LI and Willard MILLER Jr SIGMA 5)

### UNIVERSITY OF DUBLIN

UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429

### Quantum Mechanics in 3-Dimensions

Quantum Mechanics in 3-Dimensions Pavithran S Iyer, 2nd yr BSc Physics, Chennai Mathematical Institute Email: pavithra@cmi.ac.in August 28 th, 2009 1 Schrodinger equation in Spherical Coordinates 1.1 Transforming

### arxiv: v1 [math-ph] 15 Sep 2009

On the superintegrability of the rational Ruijsenaars-Schneider model V. Ayadi a and L. Fehér a,b arxiv:0909.2753v1 [math-ph] 15 Sep 2009 a Department of Theoretical Physics, University of Szeged Tisza

### More On Carbon Monoxide

More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations of CO 1 / 26 More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations

### which implies that we can take solutions which are simultaneous eigen functions of

Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated,

### 16.1. PROBLEM SET I 197

6.. PROBLEM SET I 97 Answers: Problem set I. a In one dimension, the current operator is specified by ĵ = m ψ ˆpψ + ψˆpψ. Applied to the left hand side of the system outside the region of the potential,

### Irreducible Killing Tensors from Conformal Killing Vectors

Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 708 714 Irreducible Killing Tensors from Conformal Killing Vectors S. Brian EDGAR, Rafaelle RANI and Alan BARNES Department

### Physics 70007, Fall 2009 Answers to Final Exam

Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,

### LECTURE 1: LINEAR SYMPLECTIC GEOMETRY

LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 *********************************************************************************

### Second quantization: where quantization and particles come from?

110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian

### Sample Quantum Chemistry Exam 2 Solutions

Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)

### Brief course of lectures at 18th APCTP Winter School on Fundamental Physics

Brief course of lectures at 18th APCTP Winter School on Fundamental Physics Pohang, January 20 -- January 28, 2014 Motivations : (1) Extra-dimensions and string theory (2) Brane-world models (3) Black

### Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a

### arxiv: v2 [math-ph] 9 May 2018

Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials Adrian M. Escobar-Ruiz, 1,a J. C. López Vieyra,,b P. Winternitz, 1,c and İ. Yurduşen 1,,d 1 Centre de recherches

### arxiv: v2 [math-ph] 14 Apr 2008

Exact Solution for the Stokes Problem of an Infinite Cylinder in a Fluid with Harmonic Boundary Conditions at Infinity Andreas N. Vollmayr, Jan-Moritz P. Franosch, and J. Leo van Hemmen arxiv:84.23v2 math-ph]

### GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical

### General N-th Order Integrals of Motion

General N-th Order Integrals of Motion Pavel Winternitz E-mail address: wintern@crm.umontreal.ca Centre de recherches mathématiques et Département de Mathématiques et de Statistique, Université de Montréal,

Atoms 2012 update -- start with single electron: H-atom x z φ θ e -1 y 3-D problem - free move in x, y, z - easier if change coord. systems: Cartesian Spherical Coordinate (x, y, z) (r, θ, φ) Reason: V(r)

### GLASGOW Paolo Lorenzoni

GLASGOW 2018 Bi-flat F-manifolds, complex reflection groups and integrable systems of conservation laws. Paolo Lorenzoni Based on joint works with Alessandro Arsie Plan of the talk 1. Flat and bi-flat

### 26 Group Theory Basics

26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.

### Dierential geometry for Physicists

Dierential geometry for Physicists (What we discussed in the course of lectures) Marián Fecko Comenius University, Bratislava Syllabus of lectures delivered at University of Regensburg in June and July

### On local normal forms of completely integrable systems

On local normal forms of completely integrable systems ENCUENTRO DE Răzvan M. Tudoran West University of Timişoara, România Supported by CNCS-UEFISCDI project PN-II-RU-TE-2011-3-0103 GEOMETRÍA DIFERENCIAL,

### Introduction and Vectors Lecture 1

1 Introduction Introduction and Vectors Lecture 1 This is a course on classical Electromagnetism. It is the foundation for more advanced courses in modern physics. All physics of the modern era, from quantum

### Quantum Theory and Group Representations

Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)

### Collection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators

Basic Formulas 17-1-1 Division of Material Science Hans Weer The de Broglie wave length λ = h p The Schrödinger equation Hψr,t = i h t ψr,t Stationary states Hψr,t = Eψr,t Collection of formulae Quantum

### Liouville integrability of Hamiltonian systems and spacetime symmetry

Seminar, Kobe U., April 22, 2015 Liouville integrability of Hamiltonian systems and spacetime symmetry Tsuyoshi Houri with D. Kubiznak (Perimeter Inst.), C. Warnick (Warwick U.) Y. Yasui (OCU Setsunan

### E 2 = p 2 + m 2. [J i, J j ] = iɛ ijk J k

3.1. KLEIN GORDON April 17, 2015 Lecture XXXI Relativsitic Quantum Mechanics 3.1 Klein Gordon Before we get to the Dirac equation, let s consider the most straightforward derivation of a relativistically

### PHYS 404 Lecture 1: Legendre Functions

PHYS 404 Lecture 1: Legendre Functions Dr. Vasileios Lempesis PHYS 404 - LECTURE 1 DR. V. LEMPESIS 1 Legendre Functions physical justification Legendre functions or Legendre polynomials are the solutions

### Massive Scalar Field in Anti-deSitter Space: a Superpotential Approach

P R A Y A S Students Journal of Physics c Indian Association of Physics Teachers Massive Scalar Field in Anti-deSitter Space: a Superpotential Approach M. Sc., Physics Department, Utkal University, Bhubaneswar-751

### Transformation Matrices; Geometric and Otherwise As examples, consider the transformation matrices of the C 3v

Transformation Matrices; Geometric and Otherwise As examples, consider the transformation matrices of the v group. The form of these matrices depends on the basis we choose. Examples: Cartesian vectors:

### Intertwining Symmetry Algebras of Quantum Superintegrable Systems

Symmetry, Integrability and Geometry: Methods and Applications Intertwining Symmetry Algebras of Quantum Superintegrable Systems SIGMA 5 (9), 39, 3 pages Juan A. CALZADA, Javier NEGRO and Mariano A. DEL

### Donoghue, Golowich, Holstein Chapter 4, 6

1 Week 7: Non linear sigma models and pion lagrangians Reading material from the books Burgess-Moore, Chapter 9.3 Donoghue, Golowich, Holstein Chapter 4, 6 Weinberg, Chap. 19 1 Goldstone boson lagrangians

### Quantum Computing Lecture 2. Review of Linear Algebra

Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces

### STEP Support Programme. Pure STEP 3 Solutions

STEP Support Programme Pure STEP 3 Solutions S3 Q6 Preparation Completing the square on gives + + y, so the centre is at, and the radius is. First draw a sketch of y 4 3. This has roots at and, and you

### Massachusetts Institute of Technology Physics Department

Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,

### Left-invariant Einstein metrics

on Lie groups August 28, 2012 Differential Geometry seminar Department of Mathematics The Ohio State University these notes are posted at http://www.math.ohio-state.edu/ derdzinski.1/beamer/linv.pdf LEFT-INVARIANT

### 1 Commutators (10 pts)

Final Exam Solutions 37A Fall 0 I. Siddiqi / E. Dodds Commutators 0 pts) ) Consider the operator Â = Ĵx Ĵ y + ĴyĴx where J i represents the total angular momentum in the ith direction. a) Express both

### Symmetries and particle physics Exercises

Symmetries and particle physics Exercises Stefan Flörchinger SS 017 1 Lecture From the lecture we know that the dihedral group of order has the presentation D = a, b a = e, b = e, bab 1 = a 1. Moreover

### [L 2, L z ] = 0 It means we can find the common set of eigen function for L 2 and L z. Suppose we have eigen function α, m > such that,

Angular Momentum For any vector operator V = {v x, v y, v z } For infinitesimal rotation, [V i, L z ] = i hɛ ijk V k For rotation about any axis, [V i, L z ]δφ j = i hɛ ijk V k δφ j We know, [V i, n.l]dφ

### Chapter 8B - Trigonometric Functions (the first part)

Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of

### arxiv:solv-int/ v1 4 Sep 1995

Hidden symmetry of the quantum Calogero-Moser system Vadim B. Kuznetsov Institute of Mathematical Modelling 1,2,3 Technical University of Denmark, DK-2800 Lyngby, Denmark arxiv:solv-int/9509001v1 4 Sep

### Existence of Antiparticles as an Indication of Finiteness of Nature. Felix M. Lev

Existence of Antiparticles as an Indication of Finiteness of Nature Felix M. Lev Artwork Conversion Software Inc., 1201 Morningside Drive, Manhattan Beach, CA 90266, USA (Email: felixlev314@gmail.com)

### arxiv: v1 [math-ph] 7 Apr 2014

7/04/2014 Zoll and Tannery metrics from a superintegrable geodesic flow arxiv:1404.1793v1 [math-ph] 7 Apr 2014 Galliano VALENT 1 Abstract We prove that for Matveev and Shevchishin superintegrable system,

### Rotation Eigenvectors and Spin 1/2

Rotation Eigenvectors and Spin 1/2 Richard Shurtleff March 28, 1999 Abstract It is an easily deduced fact that any four-component spin 1/2 state for a massive particle is a linear combination of pairs

### Stress-energy tensor is the most important object in a field theory and have been studied

Chapter 1 Introduction Stress-energy tensor is the most important object in a field theory and have been studied extensively [1-6]. In particular, the finiteness of stress-energy tensor has received great

### Solutions to chapter 4 problems

Chapter 9 Solutions to chapter 4 problems Solution to Exercise 47 For example, the x component of the angular momentum is defined as ˆL x ŷˆp z ẑ ˆp y The position and momentum observables are Hermitian;

### 2-Form Gravity of the Lorentzian Signature

2-Form Gravity of the Lorentzian Signature Jerzy Lewandowski 1 and Andrzej Oko lów 2 Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoża 69, 00-681 Warszawa, Poland arxiv:gr-qc/9911121v1 30

### E.G. KALNINS AND WILLARD MILLER, JR. The notation used for -series and -integrals in this paper follows that of Gasper and Rahman [3].. A generalizati

A NOTE ON TENSOR PRODUCTS OF -ALGEBRA REPRESENTATIONS AND ORTHOGONAL POLYNOMIALS E.G. KALNINSy AND WILLARD MILLER, Jr.z Abstract. We work out examples of tensor products of distinct generalized s`) algebras

### 1 r 2 sin 2 θ. This must be the case as we can see by the following argument + L2

PHYS 4 3. The momentum operator in three dimensions is p = i Therefore the momentum-squared operator is [ p 2 = 2 2 = 2 r 2 ) + r 2 r r r 2 sin θ We notice that this can be written as sin θ ) + θ θ r 2

### Quantum Theory of Angular Momentum and Atomic Structure

Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the

### Quantization of scalar fields

Quantization of scalar fields March 8, 06 We have introduced several distinct types of fields, with actions that give their field equations. These include scalar fields, S α ϕ α ϕ m ϕ d 4 x and complex

### As usual, these notes are intended for use by class participants only, and are not for circulation. Week 6: Lectures 11, 12

As usual, these notes are intended for use by class participants only, and are not for circulation Week 6: Lectures, The Dirac equation and algebra March 5, 0 The Lagrange density for the Dirac equation

### Lecture I: Constrained Hamiltonian systems

Lecture I: Constrained Hamiltonian systems (Courses in canonical gravity) Yaser Tavakoli December 15, 2014 1 Introduction In canonical formulation of general relativity, geometry of space-time is given

### Introduction to Modern Quantum Field Theory

Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical

### Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p

Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the

### arxiv: v2 [math-ph] 10 Dec 2015

A superintegrable discrete harmonic oscillator based on bivariate Charlier polynomials Vincent X. Genest, 1, Hiroshi Miki,, Luc Vinet, 3,4, and Guofu Yu 4, 1 Department of Mathematics, Massachusetts Institute

### ( ) = 9φ 1, ( ) = 4φ 2.

Chemistry 46 Dr Jean M Standard Homework Problem Set 6 Solutions The Hermitian operator A ˆ is associated with the physical observable A Two of the eigenfunctions of A ˆ are and These eigenfunctions are

### Very quick introduction to the conformal group and cft

CHAPTER 1 Very quick introduction to the conformal group and cft The world of Conformal field theory is big and, like many theories in physics, it can be studied in many ways which may seem very confusing

### arxiv: v1 [gr-qc] 16 Jul 2014

An extension of the Newman-Janis algorithm arxiv:1407.4478v1 [gr-qc] 16 Jul 014 1. Introduction Aidan J Keane 4 Woodside Place, Glasgow G3 7QF, Scotland, UK. E-mail: aidan@worldmachine.org Abstract. The

### Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics

Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics We now consider the spatial degrees of freedom of a particle moving in 3-dimensional space, which of course is an important

### REPRESENTATION OF THE LORENTZ TRANSFORMATIONS IN 6-DIMENSIONAL SPACE-TIME

Kragujevac Journal of Mathematics Volume 35 Number 2 (2011), Pages 327 340. EPESENTATION OF THE LOENTZ TANSFOMATIONS IN 6-DIMENSIONAL SPACE-TIME KOSTADIN TENČEVSKI Abstract. In this paper we consider first

### 2. Matrix Algebra and Random Vectors

2. Matrix Algebra and Random Vectors 2.1 Introduction Multivariate data can be conveniently display as array of numbers. In general, a rectangular array of numbers with, for instance, n rows and p columns

### Konstantin E. Osetrin. Tomsk State Pedagogical University

Space-time models with dust and cosmological constant, that allow integrating the Hamilton-Jacobi test particle equation by separation of variables method. Konstantin E. Osetrin Tomsk State Pedagogical