Direct and constructive approaches on N-dimensional

Direct and constructive approaches on N-dimensional superintegrable system Fazlul Hoque, YZ Zhang, I Marquette ANZAMP, Newcastle, Australia

December 9-11, 2015

1 / 22 Fazlul Hoque

The University of Queensland

Family of N-dimensional superintegrable double singular ocsillators

H=

p2 ω2 r 2 c1 c2 + + 2 + 2 2 2 x1 + · · · + xn2 xn+1 + · · · + xN2

4D, 8D systems via Hurwitz transformation (Petrosyan 2008, Marquette 2010, 2012) (2,2), (4,4) identified as the duals of 3- and 5-dimensional deformed K-C systems with u(1) and su(2) monopoles respectively not only (2,2), (4,4) but also (1,3) in 4D; (1,7), (2,6), (3,5) in 8D c1 = c2 = 0, the system reduces to isotropic harmonic oscillator

2 / 22 Fazlul Hoque

The University of Queensland

H =

p2 2

+

ω2 r 2

+

2

c1 x12 + · · · + xn2

+

c2 2 2 xn+1 + · · · + xN

Our aim is to Direct approach

Constructive approach

obtain integrals of motion of the system construct quadratic algebra Q(3) with only three generators how the su(N) symmetry algebra of isotropic harmonic oscillator is broken down into Q(3) ⊕ L1 ⊕ L2 , L1 , L2 are certain Lie algebras realize this algebra in terms of deformed oscillator algebra obtain the structure function for the energy spectrum

construct higher order integrals of motion of the system how the integrals close into higher rank cubic algebra C (3) ⊕ L1 ⊕ L2 involving Casimir operators of L1 , L2 Lie algebras realize this cubic algebra C (3) in terms of deformed oscillator algebra construct finite dimensional unitary representation to obtain energy spectrum compare and discuss the novelty of these two approaches

3 / 22 Fazlul Hoque

The University of Queensland

Separation of Variables H is multiseparable separation of variables in double hyper-Eulerian and double hyperspherical coordinates H = H1 + H2 , sum of two singular oscillators of dimensions n and N − n, where H1 =

1 2

2

2

(p1 + ... + pn ) +

ω 2 r12 2

+

c1 r12

,

H2 =

1 2

2

2

(pn+1 + ... + pN ) +

ω 2 r22 2

+

c2 r22

The wave function ψ1 (r1 , Ωn−1 ) in terms of Laguerre functions is proportional to

e

−

ω 0 r12 2

α1 + n 2

r1

α 0 2 Ln 1 (ω r1 )y1 (Ωn−1 ), 1

  E10 n−2 1 n where Lα n (x) the Laguerre polynomial, α1 = 2δ1 + ln + 2 , n1 = 2ω 0 − δ1 + 2 ln + 4 , q  c E ( 21 ln + n−2 )2 + 21 c10 − n−2 − 12 ln , c10 = 12 , ω 0 = ω , E10 = 12 . δ1 = 4 4 ~ ~

~

The energy spectrum of the model H is

 E = 2~ω

p+1+

α1 + α2 2

 ,

q

where α2 = 2δ2 + lN−n + N−n−2 , δ2 = ( 12 lN−n + N−n−2 )2 + 12 c20 − N−n−2 2 4 4   0 E c E2 0 , E c20 = 22 and p = n1 + n2 , n2 = 20 − δ2 + 12 lN−n + N−n = . 2 4 2ω ~ ~2



− 12 lN−n ,

4 / 22 Fazlul Hoque

The University of Queensland

Algebraic derivation in the direct approach Background

Algebraic methods are powerful tools in modern physics N-dimensional hydrogen atom and harmonic oscillator studied using so(N + 1) (Bargmann 1936, Louck, Galbraith 1972) and su(N) (Louck 1965, Hwa, Huyts 1966) Lie algebras respectively Superintegrable systems are an important class of quantum systems which can be solved using algebraic approaches Fris, Smorodinsky, Winternitz(1965, 1966, 1967) inaugurated modern theory and explored multiseparability of superintegrable systems in E2 and E3 Beginning about 2000, Daskaloynnis, Kalnins, Kress, Miller, Pogosyan etc. have largely worked out the structure theory and classification of classical and quantum second order superintegrable systems The quadratic and cubic algebras of the symmetries of these systems and their representation theory has been studied since about 1992 by Zhedanov, Daskaloyannis, Kalnins, Kress, Marquette, Miller, Post, Vinet etc to obtain algebraic derivation of energy spectra Daskaloyannis (2001) studied and applied quadratic algebra with only three generators to 2D superintegrable systems, and later this approach was extended to higher order polynomial algebras with three generators of arbitrary order (Isaac, Marquette 2014); this approach is known as direct approach

5 / 22 Fazlul Hoque

The University of Queensland

Integrals of motion

H =

p2 2

+

ω2 r 2 2

+

c1 x12 + ... + xn2

+

c2 2 2 xn+1 + ... + xN

The second order integrals of motion

A

B

  ( N N N N  X X c1 h2  X 1 X 2 2 2 xi ∂x − xi xj ∂xi ∂xj − (N − 1) xi x i ∂ xi + j  4 i,j=1 2 i=1 x12 + ... + xn2 i,j=1 i=1 ) c2 + , 2 2 xn+1 + ... + xN     n N n N   X X 1 X ω 2 X c1 c2 2 2 2 2 = pi − pi + xi − xi + − ,   2  2  x 2 + ... + x 2 x 2 + ... + x 2 =−

i=1

i=n+1

J(2) =

i=1

X 2 Jij ,

i=n+1

Jij = xi pj − xj pi ,

1

n

n+1

N

i, j = 1, 2, ...., n,

i= n|n, E > and H replaced by E Factorized form

Φ(x; u, E )

=

1 1 (2 + m1 + m2 )][x + u − (2 − m1 + m2 )] 4 4 1 1 ×[x + u − (2 + m1 − m2 )][x + u − (2 − m1 − m2 )] 4 4 −E + ~ω E + ~ω ×[x + u − ][x − ], 2~ω 2~ω −12582912~18 ω 2 [x + u −

where ~2 m12 = 8c1 + 4J(2) + ~2 (n − 2)2 , ~2 m22 = 8c2 + 4K(2) + ~2 (N − n − 2)2 . For the unirreps, we impose the three constraints on the structure function Φ(p + 1, u, E ) = 0,

Φ(0, u, E ) = 0,

Φ(x) > 0,

x = 1, . . . , p,

where p is a positive integer. 12 / 22 Fazlul Hoque

The University of Queensland

Structure functions and Energy spectrum Set − 1 : Φ(x)

−E + ~ω 1 m1 + 2 m2 , E = 2~ω(p + 1 + ), 2~ω 4 18 2 = ηx~ ω [4 + 4p − 4x − (1 − 1 )m1 + (1 + 2 )m2 ][4 + 4p − 4x u=

+(1 + 1 )m1 − (1 − 2 )m2 ][4 + 4p − 4x − (1 − 1 )m1 − (1 − 2 )m2 ] ×[4 + 4p − 4x + (1 + 1 )m1 + (1 + 2 )m2 ][4 + 4p − 2x + 1 m1 + 2 m2 ],

Set − 2 : Φ(x)

1 m1 + 2 m2 E + ~ω , E = 2~ω(p + 1 + ), 2~ω 4 = −ηx[4 + 4p + 4x − (1 − 1 )m1 + (1 + 2 )m2 ][4 + 4p + 4x + (1 + 1 )m1 u=

−(1 − 2 )m2 ][4 + 4p + 4x − (1 − 1 )m1 − (1 − 2 )m2 ][4 + 4p + 4x +(1 + 1 )m1 + (1 + 2 )m2 ][4 + 4p + 2x + 1 m1 + 2 m2 ],

Set − 3 : Φ(x)

1 1 m1 + 2 m2 (2 + 1 m1 + 2 m2 ), E = 2~ω(p + 1 + ), 4 4 = η(p + 1 − x)[4x − (1 − 1 )m1 − (1 − 2 )m2 ][4x + (1 + 1 )m1 − (1 − 2 )m2 ] u=

×[4x − (1 − 1 )m1 + (1 + 2 )m2 ][4x + (1 + 1 )m1 + (1 + 2 )m2 ] ×[2 + 2p + 2x + 1 m1 + 2 m2 ]. For more details visit: J. Phys. A: Math. Theor. 48 (2015) 445207. Fazlul Hoque

The University of Queensland

13 / 22

Constructive approach Background

One important property of superintegrable systems is the existence of non-Abelian symmetry algebras generated by integrals of motion Ladder operators are most known operators to construct integrals of motion for superintegrable systems (Jauch, Hill 1940, Winternitz 1966, Boyer, Miller 1974, Evans 1991, Verrier 2008, Marquette 2010) Junker 1995, Demircioglu et al 2002, Marquette 2009, Ragnisco, Riglioni 2010, Quesne, Marquette 2013 introduced one method which required SUSYQM in combination with ladders and supercharges Another one relies on Darboux-Crum and Krein-Adler approach to SUSYQM combining only with supercharges (Krein 1957, Adler 1994, Marquette 2010, 2011, Marquette, Quesne 2014)[ Note- all these methods apply to superintegrable systems that separable in Cartesian coordinates] Kalnins, Kress and Miller 2011 introduced a recurrence relation approach including ladder operators and applied it to superintegrable systems that are separable in polar or spherical coordinate systems, and established close relation with special functions and orthogonal polynomials. Calzada, Kuru and Negro 2014 explored an operator version of recurrence approach to compute integrals of motion and polynomial algebras. The above approaches are known as constructive approach

14 / 22 Fazlul Hoque

The University of Queensland

Constructive approach Recall the model H = H1 + H2 Consider gauge transformations to Hi 1−n 2

H˜i = µ−1 i Hi µi ,

µ1 = r1

1−N+n 2

,

µ2 = r2

.

The gauge equivalent operators 1 H˜i = − 2

( ∂r2i − ω 02 ri2 −

βi ri2

)

J

(n−1)(n−3)

,

i = 1, 2,

(N−n−1)(N−n−3) 4

+ 2c10 + ~(2) where β1 = 2 and β2 = 4 Denote Z = Xn1 Xn2 y1 (Ωn−1 )y2 (ΩN−n−1 ) with Xni = e −

ω 0 ri2 2

αi + 12

ri

α

Lni i (ω 0 ri2 ),

+ 2c20 +

K(2) ~2

.

i = 1, 2

The wave functions of the gauge transformed H˜i present Z = Z1 Z2 , Zi = Xni yi = µ−1 i ψi (ri , Ω),

i = 1, 2

Alternatively, the gauge transform H˜i back to the initial Hamiltonian ˜ −1 µ−1 , H = µ1 µ2 Hµ 1 2

˜ = H˜1 + H˜2 H 15 / 22

Fazlul Hoque

The University of Queensland

Recurrence formulas The eigenvalue equations H˜i Zi = λri Zi = E˜i0 Zi with λri = ω 0 (2ni + αi + 1),

i = 1, 2

˜ = E˜0 Z is The energy eigenvalue of HZ E˜0 = {2 + 2(n1 + n2 ) + α1 + α2 }ω 0 Define the ladder operators

D˜i± (ω 0 , ri ) = −2H˜i ∓ 2ω 0 ri ∂ri + 2ω 02 ri2 ∓ ω 0 ,

i = 1, 2

The action of the symmetry operators on Zi = Xni yi , i = 1, 2 provide the recurrence formulas D˜i+ (ω 0 , ri )Xni yi = −4ω 0 (ni + 1)Xni +1 yi , D˜i− (ω 0 , ri )Xni yi = −4ω 0 (ni + αi )Xni −1 yi ,

i = 1, 2, i = 1, 2 16 / 22

Fazlul Hoque

The University of Queensland

Integrals of motion Consider the suitable combination of the operators L˜1 = D˜1+ D˜2− ,

L˜2 = D˜1− D˜2+ ,

˜ = H˜1 + H˜2 , H

B˜ = H˜1 − H˜2

The action of the operators on the wave function

L˜1 Z = 16ω 02 (n1 + 1)(n2 + α2 )Xn1 +1 Xn2 −1 y1 y2 ,

L˜2 Z = 16ω 02 (n2 + 1)(n1 + α1 )Xn1 −1 Xn2 +1 y1 y2 ,

L˜1 L˜2 Z = 256ω 04 n1 (n2 + 1)(n1 + α1 )(n2 + α2 + 1)Z ,

L˜2 L˜1 Z = 256ω 04 n2 (n1 + 1)(n2 + α2 )(n1 + α1 + 1)Z 17 / 22 Fazlul Hoque

The University of Queensland

Cubic algebra C (3) The operators form of the cubic algebra C (3),

˜ = 0 = [L˜2 , H], ˜ [L˜1 , H]

L˜1 L˜2

L˜2 L˜1

˜ = −4ω 0 L˜1 , [L˜1 , B]

˜ = 4ω 0 L˜2 , [L˜2 , B]

  02  (n − 2)2 2 0 2 ˜ − 2ω 0 )2 − 4ω J + 2c ~ + ~ = (B˜ + H (2) 1 ~2 4   02  4ω (N − n − 2)2 2 0 2 ˜ − 2ω 0 )2 − × (B˜ − H K + 2c ~ + ~ , (2) 2 ~2 4    02 (n − 2)2 2 ˜ + 2ω 0 )2 − 4ω = (B˜ + H J(2) + 2c10 ~2 + ~ ~2 4    02 (N − n − 2)2 2 0 2 ˜ + 2ω 0 )2 − 4ω ~ × (B˜ − H K + 2c ~ + (2) 2 ~2 4

18 / 22 Fazlul Hoque

The University of Queensland

Full symmetry algebra Jij and Kij generate algebras isomorphic to the so(n) and so(N − n) Lie algebras

[Jij , Jkl ]

= i(δik Jjl + δjl Jik − δil Jjk − δjk Jil )~,

[Kij , Kkl ]

= i(δik Kjl + δjl Kik − δil Kjk − δjk Kil )~,

i, j, k, l = 1, . . . , n, i, j, k, l = n + 1, . . . , N − n

The full symmetry algebra is C (3) ⊕ so(n) ⊕ so(N − n) The su(N) Lie algebra generated by the integrals of motion of the N -dimensional isotropic harmonic oscillators is deformed into higher rank cubic algebra C (3) ⊕ so(n) ⊕ so(N − n) 4th-order integrals of motion Cubic algebra C (3) ⊕ so(n) ⊕ so(N − n) involving Casimir operators of so(n) and so(N − n) Lie algebras The constructive approach presents the cubic algebra C (3) in terms of deformed oscillator algebra

19 / 22 Fazlul Hoque

The University of Queensland

Structure function and Energy Spectra The cubic algebra relations in the form of deformed oscillator by letting

ℵ=

B˜ , 4ω 0

b † = L˜1 ,

b = L˜2

The structure function h i ih ˜ − 2(1 + α1 )ω 0 ˜ ˜ − 2(1 − α1 )ω 0 4ω 0 (x + u) + H Φ(x, u, H) = 4ω 0 (x + u) + H i ih h ˜ − 2(1 + α2 )ω 0 , ˜ − 2(1 − α2 )ω 0 4ω 0 (x + u) − H × 4ω 0 (x + u) − H where u is the arbitrary constant. For the unirreps, we impose the constraints on the structure function Φ(p + 1; u, E˜0 ) = 0,

Φ(0; u, E˜0 ) = 0,

Φ(x) > 0,

x = 1, . . . , p.

All possible structure functions and energy spectra, for ε1 = ±1, ε2 = ±1, u=

−E˜0 + 2ω 0 (1 + ε1 α1 ) , 4ω 0

E˜0 = (2 + 2p + ε1 α1 + ε2 α2 )ω 0 ,

Φ(x) = 256xω 04 (x + ε1 α1 )(1 + p − x)(1 + p − x + ε2 α2 ). For more details visit: arXiv:1511.03331 20 / 22 Fazlul Hoque

The University of Queensland

Comparison Direct approach

Constructive approach

2nd order integrals of motion

4th order integrals of motion

the integrals close to quadratic algebra Q(3)

the integrals close to cubic algebra C (3)

the su(N) symmetry algebra is deformed into Q(3) ⊕ so(n) ⊕ so(N − n)

the su(N) symmetry algebra is deformed into C (3) ⊕ so(n) ⊕ so(N − n)

Q(3) constructed by only three generators

C (3) constructed by only three generators

this algebra need to realize in terms of deformed oscillator algebra

this algebra automatically form of a deformed oscillator algebra

quite complicated to apply this approach

quite easy to apply this approach

21 / 22 Fazlul Hoque

The University of Queensland

Thank You!!!

22 / 22 Fazlul Hoque

The University of Queensland