научная статья по теме MODELS FOR THE 3D SINGULAR ISOTROPIC OSCILLATOR QUADRATIC ALGEBRA Физика

Текст научной статьи на тему «MODELS FOR THE 3D SINGULAR ISOTROPIC OSCILLATOR QUADRATIC ALGEBRA»

ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 2, с. 380-387

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

MODELS FOR THE 3D SINGULAR ISOTROPIC OSCILLATOR

QUADRATIC ALGEBRA

©2010 E. G. Kalnins1)*, W. Miller, Jr.2), S. Post2)

Received April 17,2009

We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential—differential or differential—difference and difference—difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in n dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.

1. INTRODUCTION

The distinct classical and quantum second-order superintegrable systems on real or complex 3D flat space and with nondegenerate (4-parameter) potentials have been classified [1]. (Recall that a second-order superintegrable system in n dimensions is one that admits 2n — 1 functionally independent constants of the motion quadratic in the momentum variables, the maximum possible, [2—4].) Indeed, the classification for nondegenerate potentials on 3D conformally flat spaces is virtually complete, [5—9]. Characteristic of these systems in all dimensions is that the second-order constants of the motion generate a finite-dimensional algebra, polynomially closed under commutation, the quadratic algebra. In several recent papers [10—12] for the 2D cases, the authors have launched a study of the irreducible representations of these algebras and their applications via models of the representations, in terms of differential and difference operators. (Some earlier work on this subject can be found in [13—18].) For the 3D case where 2n — 1 = 5, we have shown that in fact there are always six linearly independent second-order symmetries and that these generate a quadratic algebra closing at order six in the momenta. The second-order generators are always functionally dependent via a polynomial relation at order eight. In this paper we present, for the first time, the details for a nontrivial quadratic algebra in 3D: the singular isotropic oscillator. Two-variable models for irreducible representations of the quantum system follow

!)Department of Mathematics, University of Waikato, Hamilton, New Zealand.

2)School of Mathematics, University of Minnesota, Minneapolis, USA.

E-mail: math0236@waikato.ac.nz

directly from models for the classical system. There are three possible models expressed in differential or difference operators, corresponding to separation of the eigenvalue equation for the Schrodinger operator in Cartesian, cylindrical or spherical coordinates. The models have important connections with the theory of dual Hahn and Wilson polynomials. This is a prototype for the general study of the representations of 3D quadratic algebras.

2. THE QUANTUM SUPERINTEGRABLE SYSTEM

The Hamiltonian operator is H = 6% + 6% + df + a2(xf + + x23)+ (1)

+ h h h Q.=d

\ 9 \ 9 \ 95 — uXi'

ry>2 ry>2 ry>2 b

i 2 3

A basis for the second-order constants of the motion is (with H = + M2 + M3)

Mi = dj + a2xj + %, i = 1,2,3, (2)

X 0

U = (Xjdk - xkdj)2 +

bj xl bk xj

+

where i, j, k are pairwise distinct and run from 1 to 3. There are four linearly independent commutators of the second-order symmetries:

Si = [Li,M2] = [M3,LI], (3)

52 = — [M3,L2] = [Mi,L2 ],

53 = — [Mi,L3] = [M2,L3 ],

R =[Li,L2] = [L2, L3] = [L3, Li],

2

x

k

j

[Mt,Mj] = [Mi,Li]=0, 1 < i,j < 3.

Here we define the commutator of linear operators F, G by [F, G]= FG - GF. (Thus a second-order constant of the motion is a second-order partial differential operator K in the variables Xj such that [K, H] = 0, where 0 is the zero operator.)

The fourth-order structure equations are [Mi,Si] = 0, i = 1,2,3, and

j[Mi, Sj] = 8MiMk - 16a2Lj + 8a2, (4) eijk[Mi, R] = 8(MjLj - MkLk) + 4(Mk - Mj), eijk[Si, Lj] = 8MiLi - 8MkLk + 4(Mk - Mi), j[Li, Si] = 4{Li, Mk - Mj} + 16bjMk -- 16bkMj + 8(Mk - Mj), eijk [Li, R] = 4{Li, Lk - Lj } - 16bjLj + + 16bkLk + 8(Lk - Lj + bj - bk).

Here, {F, G} = FG + GF and eijk is the completely antisymmetric tensor.

The fifth-order structure equations are obtainable directly from the fourth order equations and the Jacobi

identity. The sixth-order structure equations are

8

S2 - - {Lj, Mj, Mk} + 16a%2 + (5)

+ (16bk + 12)M2 + (16bj + 12)M| - ^MjMk -

3

176 16

- — a2U - —a2(2 + 9bj + 9bk + 12bjbk) = 0,

1 4 4

si) + + 3 ~

- 8LkMl - 8a2{Li, Lj} - (16bk + 12)MiMj + + 4Mk2 - 4Mk(Mi + Mj) + al(32bk + 24)Lk +

+ 8a2(Li + Lj) - 16a2(bk + 1) = 0,

R} - 8L2Mt + ^{Lk, Li} Mk} +

4

+ -{Lt,LJ,MJ}~ (8bk + 6){Lk,MJ}~

- (8bj + 6){Lj,Mk} - 2{Li, Mk + Mj} +

88 52

+ —LiMi + —{LkMk + LjMj) +

+ (32bkbj + 24bk + 246,- + ^ ] Mi +

3

+ I Щ - - ) Mk + ( 8bk - - ) Mj = 0,

52

+ (1662 + 12)L2 + (1663 + 12)L2 - — {LUL2} -

52 52 16

- —{L\, L3} --{L2, L3} - y (llbi + l)Li -

16 16 - y (H62 + 1)^2 " у (ПЬз + 1)Ьз "

32 / 9

у i 6616263 + 3(6162 + 6163 + 6263) +

+ bi + b2 + Ьз

Here, {A, B, C} = ABC + ACB + BAC + BCA + + CAB + CBA and i, j, k are pairwise distinct. The eighth-order functional relation is

L?M2 + L2M22 + l2M32 - (6)

--L{L1,L2,m1,M2}--^{L1,L3,M1,M3}~

1 7 7

- — {L2,L3,M2,M3} - -LXM2 - -L2M2 -

7

2

1

- -L3M2 + -a{L\,L2, L3} - -{LbMbM2} -

^-{L3,M2,M3} + l(4b1 + 3){L1,M2,M3} + 18 6

+ ^(462 + 3){L2, Mb M3} + i(4b3 + 3) x 66

x [L3, Mi,M2} - a2(4bi + 3)L2 -

a2

- a2(462 + 3)L\ - a2(46з + 3)L2 + y ({Lb L2} + + {Li, L3} + {L2, L3}) - ( 4b2b3 + 3b2 +

4

4

R2 - -{L\, L2, L3} + (166! + 12)L2 +

+ З63 + - M{ - ^46i63 + З61 + З63 + - ) Mi -

- (аЬхЬ2 + 36i + 362 + ^ M| +

22 + -(63 + 2 )MiM2 + -(62 + 2)M\M3 +

24 + -(61 + 2)M2M3 + -a (76i + 4)L1 +

+ ^a2(762 + 4)L2 + ^a2(763 + 4)L3 +

+ ^a2(126i6263 + 96i62 + 96i63 + 96263 + 3

382

KALNINS et al.

+ 4bi + 4bi + 4Ьз ) = 0.

Here, {A, B,C, D} is the 24-term symmetrizer of four operators.

2.1. Cartesian Case: A Quantum Model with Mir M2 Diagonal

For the model we choose variables u, v in which the eigenfunctions are polynomials, and write the parameters as bj = 1/4 — kj. Then we have

Mi = 2ia(2udu + ki + 1), (7)

M2 = 2ia(2vdv + k2 + 1),

Mi + M2 + M3 = E,

Li = 4 v( u2dl + 2uvdu dv + (v2 + 1)2 — (8)

- (¿-^-^-4)9^+4(1 + k2)dv +

+ ¿M2Ms + i, Li = 4 J v292 + dv + (u2 + 1)d£ - (9)

- (J^ " h - k2 - udy^j + 4(1 + h)du +

L3 = 4(uvdU + uvdf + (ki + 1) vdu + (10)

+ (k2 + 1)«0„) + +

In the model, the monomials fN,j = ujvN-j are simultaneous eigenfunctions of the operators Mj:

MifN,j = 2ia(2j + ki + 1)fN,j, (11)

MfNj = 2m(2W - 2j + k2 + 1)/n Further, we have the expansion formulas

j

L1 fN,j =

- fcl) fN+ij + 4(N - j)(N - j + ki)fN-1 j +

i-

\2ia

+ ( 2( -2N - 2ki -2k2-2\ x

x (2ЛГ - 2J + fc2 + 1) + - )/Wj,

S

= + 3 - — + h + k2J - (13)

- k3^J fN+1 j+i + 4j(j + ki)fN-i,j-1 + + [2 — 2N — 2k\ — 2k2 — 2 j x

x (2j + k1 + l) + - )fNJ,

L3fNj = 4(N - j)(N - j + kifj+i + (14) + 4j(j + ki)fNj-i + (2(2j + ki + 1) x

x (2N-2j + k2 + l) + -)fN,3.

From the model we can find a family of finite-dimensional irreducible representations, labeled by the nonnegative integer M. A basis of eigenfunctions is given by {¡N,j = ujvN-j }, such that the N and j are nonnegative integers satisfying 0 < j < N < M. The energy satisfies

E = 2ia(2M + ki + k2 + k3 + 3). (15)

The dimension of the representation is (M + 2)(M + + 1)/2. Now we introduce an inner product such that the operators Mj ,Lj are formally self-adjoint for j = = 1,2,3. This forces a to be pure imaginary.

Normalization coefficients: Let fNj = KN juj x xvN-j suchthat \\fN> j|| = 1. If we assume K0, 0 = 1, then the coefficients become

(-M)n(-M - k3)N

KN j =

(N - j)!j!(ki + 1)n-j(ki + 1)j

E

2N + 3 — ——\-k\+k2) - (12) 2ia

2.2. Recurrence Relations for Wilson Polynomials

The spherical case is intimately bound up with recurrence relations for Wilson polynomials and the cylindrical case with the dual Hahn polynomials. To see this we modify some of the results of [19]. The unnormalized Wilson polynomials are

2

Wn(y2) = Wn(y2,a, 3,7,5) = (a + 3)n(a + 7)n(a + 5)n x (16)

x a + 3 + 7 + 5 + n - a - y a + = (a + 3)n(a + 7)n(a + ^ • ' • - • <V),

\ a + 3, a + 7, a + 5 J

where (a)n is the Pochhammer symbol and 4F3(1) is a generalized hypergeometric function of unit argument. For fixed a, 3,7,5 > 0 the Wilson polynomials are orthogonal with respect to the inner product

~ 2

{wn,wn>) = ^ J wn(-y2)wn>(-y2)

Г(а + 1у)Г(в + iy)r(7 + iy)r(S + iy)

r(2iy)

dy = (17)

= 5nn>n!(a + (3 + 7 + 5 + n — 1)„ x r(q + (3 + n)T(a + 7 + n)T(a + S + n)T((3 + 7 + n)T((3 + S + n)T(-y + 6 + n) X T(a + /3 + 7 + 5 + 2n) '

The Wilson polynomials $n(y2) = ' 1'&\y2), satisfy the three-term recurrence formula

y2$n(y2) = K(n + 1,n)$n+i(y2) + K(n, n)$n(y2) + K(n — 1,n)$n-i(y2), (18)

where

+ = (a+ l3 + 7 + 5 + 2+n -+l)(a + pl , + 6 + 2n)^ + -)(a + , + n)(a + S + n), (19)

_ n(/3 + 7 + rc-l)(/3-M + n-l)(7 + a + n-l)

1 ' (a + /3 + 7 + ^ + 2n-2)(a + /3 + 7 + 5 + 2n-l)' (ZUj

0

K (n,n) = a2 — K (n + 1, n) — K (n — 1, n). (21)

Moreover, they satisfy the following parameter-changing recurrence relations when acting on the basis polynomials $n = '^ 'Y ';here TT f (y) = f (y + + r): 1.

2 y[ J'

= n(n + q + /3 + 7 + ^-l)

(a + /3)(a + 7)(a + S)

^(«+i/2 ,^+i/2,y+1/2,¿+i/2),

x ^n-i >

ЬФп = (a + в - 1)(a + 7 - 1)(a + S - 1) x

Ф(а-1/2 ,0-1/2,Y-1/2 ,¿-1/2). x Фп+1 ;

La/3 — —

2y

" (a--+y) x

f3-\ + y)jT1/2 +[a-^-y\ x LaeФп = -(a + в - 1) x

Ф(а-1/2 ,в-1/2,y+1/2 ,¿+1/2). x ^ n )

L

1

x(7-l+y)(i-l+y)Tl/2-

x Y

1

Ra13 = — 2 У

~[7-l+y)(ï-l + y)Tl/2 +

+ (7-±-tf)(i-±-tf)^

ла/зф = (n + 7 + ^-l)(n + a + /3)

"" a; + (3

x ф(а+1/2,в+1/2,7-1/2,5-1/2) _

1

X

384

KALNINS

The operators LaY, Las, RaY, Ras are obtained by obvious substitutions

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