models for irreducible representations of quadratic algebrasmille003/yerevantalk.pdf1. the quadratic...

Models for irreduciblerepresentations of quadratic

algebrasWillard Miller (Joint with E.G. Kalnins and S. Post.)

[email protected]

University of Minnesota

Montreal talk – p.1/55

Abstract 1

A quantum superintegrable system is an integrable n-

dimensional Hamiltonian system with potential that admits

2n − 1 functionally independent constants of the motion, (fi-

nite order differential operators commuting with the Hamilto-

nian), the maximum number possible. If the operators are all

second order, the system is second order superintegrable.

In 2D and for 3D conformally flat spaces with nondegener-

ate potential, the algebra generated by the constants of the

motion and their commutators has been pr oven to close at

order 3 (the quadratic algebra).


Abstract 2

The representation theory of this algebra gives important in-

formation about the the energy eigenvalues and the spectra

of the symmetries. In the 2D case we study possible re-

alizations of the possible irreducible representations of the

quadratic algebra by differential or difference operators in a

single complex variable t acting on Hilbert spaces of ana-

lytic functions. These models greatly simplify the study of

the representations and are also of considerable interest in

their own right. In particular the Wilson polynomials emerge

naturally in their full generality.


Abstract 3

We demonstrate that models of the classical superintegrable

systems lead directly to models of the quantum systems, so

that, for example, Wilson polynomials emerge directly from

classical mechanics. The examples analyzed provide guid-

ance concerning the models for higher dimensional superin-

tegrable systems and may point the way towards a general

structure theory for representations of quadratic algebras.


2nd order superintegrability (classical)

Classical superintegrable system on an n-dimensional localRiemannian manifold:

H =∑

ij

gij pipj + V (x).

Require that Hamiltonian admits 2n − 1 functionally inde-

pendent 2nd-order symmetries Sk =∑

aij

(k)(x)pipj +W(k)(x),

That is, {H,Sk} = 0 where {f, g} =∑n

j=1(∂xjf∂pj

g−∂pjf∂xj

g)

is the Poisson bracket. Note that 2n−1 is the maximum pos-

sible number of functionally independent symmetries.


Significance

Generically, every trajectory p(t),x(t), i.e., solution of theHamilton equations of motion, is characterized (andparametrized) as a common intersection of the (constantsof the motion) hypersurfaces

Sk(p,x) = ck, k = 0, · · · , 2n− 2.

The trajectories can be obtained without solving the equa-

tions of motion. This is better than integrability.


2nd order superintegrability (quantum)

Schrödinger operator

H = ∆ + V (x)

where ∆ = 1√g

∑

ij ∂xi(√ggij)∂xj

is the Laplace-Beltrami

operator on a Riemannian manifold, expressed in localcoordinates xj and S1, · · · , Sn. Here there are 2n− 1second-order symmetry operators

Sk =1√g

∑

ij

∂xi(√gaij

(k))∂xj

+W(k), k = 1, · · · , 2n− 1

with S1 = H and [H,Sk] ≡ HSk − SkH = 0.


Why second order?

This is the most tractable case due to the association with

separation of variables. Special function theory can be ap-

plied and is relevant for the same reason.


Integrability and superintegrability

1. An integrable system has n functionally independentconstants of the motion in involution. A superintegrablesystem has 2n− 1 functionally independent constants ofthe motion. (Sometimes the definition ofsuperintegrabilty also requires integrability. In this talkwe prove it.)

2. Multiseparable systems yield many examples ofsuperintegrability.

3. Superintegrable systems can be solved explicitly inmultiple ways. It is the information gleaned fromcomparing the distinct solutions and expressing onesolution set in terms of another that is a primary reasonfor their interest.


3D example:

The generalized anisotropic oscillator: Schrödingerequation HΨ = EΨ or

HΨ = −1

2

(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2

)

Ψ + V (x, y, z)Ψ = EΨ.

The 4-parameter “nondegenerate” potential

V (x, y, z) =ω2

2

(

x2 + y2 + 4z2 + ρz)

+1

2

[

λ1

x2+λ2

y2

]


Separable coordinates

The corresponding Schrödinger equation has separable so-

lutions in five coordinate systems: Cartesian coordinates,

cylindrical polar coordinates, cylindrical elliptic coordinates,

cylindrical parabolic coordinates and parabolic coordinates.

The energy eigenstates for this equation are degenerate and

important special function identities arise by expanding one

basis of separable eigenfunctions in terms of another.


Basis for 2nd order symmetries

M1 = ∂2x − ω2x2 +

λ1

x2, M2 = ∂2

y − ω2y2 − λ2

y2,

P = ∂2z − 4ω2(z + ρ)2, L = L2

12 − λ1y2

x2− λ2

x2

y2− 1

2,

S1 = −1

2(∂xL13 + L13∂x) + ρ∂2

x + (z + ρ)(

ω2x2 − λ1x2)

,

S2 = −1

2(∂yL23 + L23∂y) + ρ∂2

y + (z + ρ)(

ω2y2 − λ2y2)

,

where Lij = xi∂xj− xj∂xi

.


Quadratic algebra closing at level 6

The nonzero commutators are

[M1, L] = [L,M2] = Q, [L, S1] = [S2, L] = B,

[Mi, Si] = Ai, [P, Si] = −Ai.

Nonzero commutators of the basis symmetries with Q (4thorder symmetries) are expressible in terms of the secondorder symmetries, e.g.,

[Mi, Q] = [Q,M2] = 4{M1,M2} + 16ω2L,

[S1, Q] = [Q,S2] = 4{M1,M2},[L,Q] = 4{M1, L} − 4{M2, L} − 16λ1M1 + 16λ2M2.


Level 6 closure

The squares of Q, B, Ai and products such as {Q,B}, (all6th order symmetries) are all expressible in terms of 2ndorder symmetries, e.g.,

Q2 =8

3{L,M1,M2} + 8ω2{L,L} + 16λ1M

21 + 16λ2M

22

+64

3{M1,M2} −

128

3ω2L− 128ω2λ1λ2,

{Q,B} = −8

3{M2, L, S1} −

8

3{M1, L, S2} − 16λ1{M2, S2}

−16λ2{M1, S1} −64

3{M1, S2} −

64

3{M2, S1}.

Here {C1, · · · , Cj} is the completely symmetrized product of

operators C1, · · · , Cj . Montreal talk – p.14/55

Important properties-1

1. The algebra generated by the second order symmetriesis closed under commutation in both the classical andoperator cases. This is a remarkable, but typical ofsuperintegrable systems with nondegenerate potentials.

2. Closure is at level 6, since we have to express theproducts of the 3rd order operators in terms of the basisof 2nd order operators.

3. The eigenfunctions of one separable system can beexpanded in terms of the eigenfunctions of another, andthis is the source of nontrivial special functionexpansion theorems.

4. The quadratic algebra identities allow us to relateeigenbases and eigenvalues of one symmetry operatorto those of another.



1. The representation theory of the abstract quadraticalgebra can be used to derive spectral properties of thegenerators Sj , in a manner analogous to the use of Liealgebra representation theory to derive spectralproperties of quantum systems that admit Lie symmetryalgebras,

2. A common feature of quantum superintegrable systemsis that after splitting off a gauge factor, the Schrödingerand symmetry operators are acting on a space ofpolynomials: MULTIVARIABLE ORTHOGONALPOLYNOMIALS.



1. Closely related to the theory of QUASI-EXACTLYSOLVABLE SYSTEMS (QES). In many 2D and 3Dexamples the one-dimensional ODEs are quasi-exactlysolvable and the eigenvalues that give polynomialsolutions are easily obtained from the PDEsuperintegrable systems. Generalizes results ofUshveridze. Leads to new examples.


Basic 2D structure results 1

Theorem 1 Let H be the Hamiltonian of a 2Dsuperintegrable system with nondegenerate (i.e., 3parameter) potential.

1. The space of second order symmetry operators is3-dimensional.

2. The space of third order symmetry operators is1-dimensional.

3. The space of fourth order symmetry operators is6-dimensional and is spanned by symmetric quadraticpolynomials in the second order symmetries.

4. The space of sixth order symmetry operators is10-dimensional and is spanned by symmetric cubicpolynomials in the second order symmetries.



1. Every 2D superintegrable system with 1 or 2-parameterpotential is a restriction of a nondegenerate potential.

2. However, for some 1-parameter potentials the structureof the quadratic algebra changes if the system admits aKilling vector, i.e., a first order symmetry operator. Wewill call these degenerate 1-parameter potentials.



1. In the degenerate 1-parameter potential case there is a1-dimensional space of first order symmetry operatorsand a 4-dimensional space of second order symmetryoperators.

2. The commutator of a first order and a second ordersymmetry operator is always expressible as a linearcombination of second order symmetry operators.

3. The commutator of two second order symmetryoperators is always expressible in terms of symmetricproducts of a first order and a second order symmetry.

4. There is a nontrivial quadratic symmetric polynomialrelating the second order symmetry operators.



1. Thus the quadratic algebra generated by the symmetryoperators always closes, at order 6 for nondegeneratepotentials and at order 4 for degenerate (1-parameter)potentials.

2. Every 2D superintegrable system is Stäckel equivalentto a superintegrable system on a constant curvaturespace, either flat space E2 or the complex 2-sphere S2.

3. All superintegrable systems on E2 and S2 have beenclassified. There are 19 systems on E2, 8 of themdegenerate, and 9 systems on S2, 3 of themdegenerate.

4. Some of these systems are Stäckel equivalent. Thenumber of distinct equivalence classes for all 2Dsuperintegrable systems on possible manifolds is 13, 7nondegenerate and 6 degenerate. Montreal talk – p.21/55


1. The quadratic algebras of two Stäckel equivalentsystems are related by a simple permutation of theparameters in the potential and the energies. ThusThey have the same abstract representation theory.

2. We conclude that the representation theory of quadraticalgebras for second order superintegrable systems in2D reduces to 13 distinct cases, 7 nondegenerate and 6degenerate.


The nondegenerate system E1

1. The potential is

V = ω2(x2 + y2) +1/4 − a2

x2+

1/4 − b2

y2

2. Symmetries: L1 = (x∂y − y∂x)2 +W1 and L2 = ∂2x +W2.

3. The structure equations are (R = [L1, L2]):

[L2, R] = −8L22 + 8HL2 − 16αL1 + 8α,

[L1, R] = −8HL1+8{L2, L1}−8(1+2β)H+16(1+β+γ)L2,

R2 = −8

3{L2, L2, L1}+8H{L2, L1}−4(3+4a)H2−16(a+b−11

3)L

+16(2a+11

3)HL2 +

176

3ωL2 + 16ω(3a+ 3b+ 4ab+

2ω

3)


A model for E1

1. Diagonalize L1: L1 = 4iωt∂t + 2iω(1 − 2m− a)

2. Find raising and lowering operators:

A = L2 −R

4iω+

L21

2ω2− h

2ω2L1 −

1

2= t∂2

t + (1 + b)∂t

A† = L2 +R

4iω+

L21

2ω2− h

2ω2L1 −

1

2= 64t3∂2

t

+(192−64a−128m)t2∂t+(62m2+(64a−128)m+64−64a)t

3. If L1 and L2 are formally self-adjoint, then iω must bereal and A† will be mutually adjoint.


The quantization condition

1. Assume there is a highest weight vector tm−1 for somepositive integer m. Then A†tm−1 = 0.

2. This implies that the energy eigenvalue is given byh = −2iω(2m+ a+ b)

3. Imposing a finite dimensional representation we canrequire At0 = 0.

4. Then the eigenvalue equation L2ψh = λkψk hasspectrum λk = −3/2− 2b− 2a− 4k− 2ba− 4bk− 4ak− 4k2

and the eigenfunctions are hypergeometric polynomials

ψk(t) = lk(8t+ 1)m−1−k2F1

(

−k, −a− k

1 + b| − 8t

)

k = 0, · · · ,m− 1.Montreal talk – p.25/55

The Hilbert space

1. This gives us the energy eigenvalues and the spectraldecompositions for L1, L2 as well as the expansion ofthe L2 eigenbasis in terms of the L1 eigenbasis.

2. Assuming that L1, L2 are self-adjoint we can determinethe orthonormal basis of L1 eigenvectors φn(t) = cnt

n

where cn =√

(−m)n(−m−a)n

n!(b)n

3. Reproducing kernel

∑

φn(t)φn(s) = 2F1

(

−m, −m− a

b|ts)

4. Similarly we can define explicit function space innerproducts to realize the various finite and infinitedimensional irreducible representations of the quadraticalgebra.


A degenerate system on the 2-sphere: S3

1.

H = J21 +J2

2 +J33 +

α

s23, s21+s

22+s

23 = 1, J3 = s1∂s2

−s2∂s1

plus cyclic permutations.

2. Symmetries

L1 = J21 +

αs22s23

, L2 =1

2(J1J2 + J2J1) −

αs1s2s23

, X = J3.


S3 structure equations

1.[L1, X] = 2L2, [L2, X] = −X2 − 2L1 +H − α,

2.

[L1, L2] = −{L1, X} − (2α− 1

2)X,

3.

1

6{L1, X,X}−HL1+L

22+L

21+(α+

11

12)+(α−2

3)L1−

H

6=

5α

6,

where {a, b, c} is the 6 term symmetrizer equal toabc+ acb+ cba+ cab+ bca+ bac.


A recent appearance 1

1. Variable mass Hamiltonians, used in semiconductorresearch , quantum dots, nuclei, and quantum liquids,“Effective mass”.

2. A general variable mass Hamiltonian in 2D:

H = ∂x1

M(x, y)∂x + ∂y

1

M(x, y)∂y + V (x, y)

3. Potential introduced by Quesne (2007).

M(x, y) = − 1cosh2 qx

and V (x, y) = −q2 cosh2 qx+ q2k(k−1)

sinh2 qx

4. The metric must be ds2 = q2 dx2+dy2

cosh2 qx, constant curvature.


A recent appearance 2

1. If we take the coordinates on the sphere to be

s1 =sin qy

cosh qx, s2 =

cos qy

cosh qx, s3 = tanh qx

and perform a gauge transformation we get exactly S3.


The model 1

1. Diagonalize X = i(2t ddt −m)

2. Write h = −(−m+ a− 1)2 + 14 . If µ = −m m ∈ N, then

our model is finite dimensional:

L1 =(

t3 + 2t2 + t) d2

dt2+(

(2 − a−m)t2 + 2(1 −m)t

+a−m)d

dt+m(a− 1)t+ a(m+ 1) −m− 1

2,

L2 = i(

−t3 + t) d2

dt2+i(

(a+m− 2)t2 + a−m) d

dt−im(a−1)t.

3. If m = −µ for arbitrary complex µ then the model isinfinite dimensional bounded below.

4. The finite dimensional representations gives us thequantization for the energy levels. Montreal talk – p.31/55

The model 2

1. We have raising and lowering operators

A† = L1+iL2+1

2(X2−H+α), A = L1−iL2+

1

2(X2−H+α)

2. We can use these to find normalization coefficients forour eigenfunctions, φn = knt

n n = 0, ...,m as well as aweight function and reproducing kernel.


The model 3

1. We can also diagonalize L1ψ − λψ = 0 usinghypergeometric functions. If we are in the finitedimensional model, we have the requirement that ourhypergeometric functions be polynomials of order m;this gives us a quantization conditionλ = −(n− a+ 1

2)2 + a2 − 14 . The eigenfunctions become,

for n = 0, ...,m

ψν(t) = ln(t+ 1)n2F1

(

−m+ n 1 − a+ n

−m+ a| − t

)

where ln is a normalization coefficient.


Classical models 1

1. We can also find models of the classical quadraticalgebras in terms of functions of two canonicallyconjugate variables c, β The analog of the one-variablequantum models.

2. The existence of such models follows easily fromstandard Hamilton-Jacobi theory for integrable systems.

3. Why bother? BECAUSE THE CLASSICAL MODELSTELL US THE FORMS OF THE POSSIBLE QUANTUMMODELS. Sometimes the possible quantum models willbe in terms of differential operators, sometimes in termsof difference operators.


Classical models 2

1. Classical S3 constants of the motion

L1 = J 21 + α

s22s23, L2 = J1J2 − α

s1s2s23

, X = J3

2. Structure relations

{X ,L1} = −2L2, {X ,L2} = 2L1 −H + X 2 + α,

{L1,L2} = −2(L1 + α)X3. Casimir relation

L21 + L2

2 −L1H + L1X 2 + αX 2 + αL1 = 0.


Classical models 3

1. Require X ≡ Xh = c and H = h in the structureequations.

2. Result is

I : L1 =1

2(E−c2−α)+

1

2

√

c4 − 2c2(E + α) + (E − α)2 sin 2β,

3. Factor the term under the square root and set

φ = arctan

(√−4α

c2 − (E + α)2

)

.


Classical models 4

1. Now we let 2β → 2β + φ to obtain

L1 =1

2(E−c2−α)+

1

2

(

(c2 − (E + α)2) sin 2β + 2i√α cos 2β

)

,

L2 =1

2

(

(c2 − (E + α)2) cos 2β − 2i√α sin 2β

)

, X = c.


Classical models 5

1. In this form we can see that the symmetries arepolynomial in c which suggest the substitutionβ → t, c → −∂t leading to a quantum realization of L1, L2

by second order differential operators. This leadsdirectly to the differential operator model of S3 that wehave already exhibited.

2. For a second model we require L1 ≡ (L1)h = c andproceed in a similar fashion. The result is

II : L1 = c, L2 =√

c(E − c− α) sin(2√c+ αβ),

X =

√

c(E − c− α)

c+ αcos(2

√c+ αβ).


Classical models 6

1. This model cannot produce finite order differentialoperator realizations of the quantum quadratic algebra,due to the intertwining of square root dependence for cand exponential dependence for β. However, it willproduce a difference operator realization via Taylor’stheorem: ea∂tf(t) = f(t+ a).


Classical models 7

1. To show this explicitly we make a coordinate changesuch that 2

√c+ α∂c = ∂C which suggests realizations of

the quantum operators in the form

L1f(t) = (t2−α)f(t), Xf(t) = h(t)f(t+ i)+m(t)f(t− i),

L2f(t) = − i

2(i+ 2t)h(t)f(t+ i) +

i

2(−i+ 2t)m(t)f(t− i).

2. A straightforward computation shows that the quantumalgebra structure equations are satisfied if and only if

h(t)m(t+ i) =1

4

(α− t2 − it)(t2 + it− E)

t(t+ i).

We define Tαf(t) = f(t+ α).Montreal talk – p.40/55

S3 difference model 1

1. Some manipulation yields the difference operator model

−iX =(1/2 − a− it)(µ+ a− 1/2 − it)

2tT i

−(1/2 − a+ it)(µ+ a− 1/2 + it))

2tT−i,

L2 = −i(1 − 2it)(1/2 − a− it)(µ+ a− 1/2 − it)

4tT i

−i(1 + 2it)(1/2 − a+ it)(µ+ a− 1/2 + it))

4tT−i,

2. The basis functions are dual Hahn polynomials

fn(t) = 3F2

(

−n 12 − a+ it 1

2 − a− it

µ 1 − a; 1

)

.Montreal talk – p.41/55

S3 difference model 2

1. The orthogonality and normalization are given by

1

2π

∫ ∞

0

∣

∣

∣

∣

Γ(1/2 − a+ it)Γ(µ+ a− 1/2 + it)Γ(1/2 + it)

Γ(2it)

∣

∣

∣

∣

2

×

sn(t2)sn′(t2) dt =Γ(n+ µ)Γ(n+ 1 − a)Γ(n+ µ+ a)n!

(µ)2n|(1 − a)n|2δnn′,

where either 1) µ > 1/2 − a > 0 or 2) µ > 0 anda = ((1 − µ)/2 + iγ is complex.

2. We can also use µ a negative integer to find finitedimensional difference operators representations withbasis vectors of (not continuous) dual Hahn polynomialswith a discrete measure.


The generic system S9 1

1. Potential

V =14 − a2

s21+

14 − b2

s22+

14 − c2

s23

where s21 + s22 + s23 = 1.

2. Hamiltonian

H = J21 + J2

2 + J23 + V (x, y) = H0 + V

where J3 = s1∂s2− s2∂s1

and J2, J3 are obtained bycyclic permutation.

3. Symmetries (symmetric form) L1, L2, L3 where

L1 = J23 +W1, L2 = J2

1 +W2, L3 = J22 +W3,

H = L1 + L2 + L3 + a1 + a2 + a3.Montreal talk – p.43/55

The generic system S9 2

1. Structure equations[Li, R] =4{Li, Lk}−4{Li, Lj}−(8+16aj)Lj+(8+16ak)Lk+8(aj−ak),

R2 =83{L1, L2, L3}−(16a1+12)L2

1−(16a2+12)L22−(16a3+12)L2

3

+523 ({L1, L2} + {L2, L3} + {L3, L1}) + 1

3(16 + 176a1)L1

+13(16 + 176a2)L2 + 1

3(16 + 176a3)L3 + 323 (a1 + a2 + a3)

+48(a1a2 + a2a3 + a3a1) + 64a1a2a3.

Here i, j, k are chosen such that ǫijk = 1 where ǫ is thepure skew-symmetric tensor, and R = [L1, L2].

2. We substitute L3 = H − L1 − L2 − a1 − a2 − a3 into theseequations.


Abstract representation theory for S9 1

1. Assume the existence of a discrete set of eigenvectorsfn for the symmetry operator L1

2. Only possibility is L1fn = λnfn where

λn = −[2n+ B]2 + K, n = 0, 1, · · · ,m.

L1fn = (K − [2n+ B]2)fn, L2fn =∑

ℓ

C(ℓ, n)fℓ.

3. Structure equations give

H = −1

4(−4µ+ 2a+ 2b+ 2c+ 5)(−4µ+ 2a+ 2b+ 2c+ 3).



and

C(n, n) =1

2(2n+ a+ b+ 2)(2n+ a+ b)

−1

2

[

(−2µ+ a+ b+ c+ 2)2 + a2 − b2 − c2 − 1]

+1

2

(a2 − b2)(a+ b− 2µ+ 2)(a+ b+ 2c− 2µ+ 2)

(2n+ a+ b+ 2)(2n+ a+ b),

C(n, n+ 1)C(n+ 1, n) =

16(n+1)(n+µ)(n− c+µ)(n+ b+1)(n+ a+1)(n+ a+ b+1)×(n− µ+ a+ b+ 2)(n− µ+ a+ b+ c+ 2)

(2n+ a+ b+ 3)(2n+ a+ b+ 2)2(2n+ a+ b+ 1).



1. Here µ is an arbitrary complex parameter but ifµ = −m, m ∈ N the representation becomes finitedimensional.

2. Only the coefficients C(n,N) where N = n, n± 1 arenonzero.

3. Only the product C(n, n+ 1)C(n+ 1, n) is determined,the individual factors can be modified via gaugetransformation.


A model for for S9 4

1. The action of L2 on the L1 basis yields the generalthree-term recurrence relation for the Wilsonpolynomials pn:

pn(t2) ≡ pn(t2, α, β, γ, δ) = (α + β)n(α + γ)n(α + δ)n×

4F3

(

−n, α + β + γ + δ − n− 1, α− t, α+ t

α + β, α+ γ, α + δ; 1

)

with the identification α = −a+c+12 + µ, β = a+c+1

2 ,γ = a−c+1

2 , δ = a+c−12 + b− µ+ 2.

2. In the particular case that the representation is finitedimensional, we obtain instead the Racah polynomials.


Classical S9

1. The classical model with H = h and c, β conjugatevariables, gives L1 = c andL2 = 1

2(a1 + 2a2 + E − c) − (a2−a3)(a1+2a2+2a3+E)2(c+a2+a3)

+√

(4a1a2+4a1a3+2c(E+a1+a2+a3)+4ca1−(E+a1+a2+a3)2−c2)(4a2a3−c2)

2(a2+a3+c)

× cos(4β√a2 + a3 + c), which suggests a difference

operator model.

2. We quantize so that β = ∂c and then take a change ofcoordinates so that 4

√a2 + a3 + c∂c = ∂t and obtain an

ansatz difference model. Plugging this into the quantumalgebra relations we recover exactly the recursionrelations for the Wilson and Racah polynomials.


3D nondegenerate systems

1. 2n− 1 = 5 but there are 6 linearly independent secondorder symmetries.

2. The quadratic algebra generated by the second ordersymmetries closes at order 6 again, but there are 4independent commutators R1, · · · , R4 and 10 relationsexpressing RiRj as symmetric cubic polynomials in the6 second order symmetries.

3. The 6 second order symmetries obey a quarticpolynomial relation.

4. The quantum models will be in terms of 2 complexvariables t1, t2.


Example: Singular isotropic oscillator

H = ∂21 + ∂2

2 + ∂23 + a2(x2

1 + x22 + x2

3) +b1x2

1

+b2x2

2

+b3x2

3

∂i ≡ ∂xi.

Basis for 2nd order symmetries:

Mi = ∂2i +a2x2

i +bix2

i

, i = 1, 2, 3, L1 = (x2∂3−x3∂2)2+

b2x23

x22

+b3x

22

x23

,

L2 = (x3∂1−x1∂3)2+

b3x21

x23

+b1x

23

x21

, L3 = (x1∂2−x2∂1)2+

b1x22

x21

+b2x

21

x22

.

HereH = M1 +M2 +M3.


Functional relation

L21M

21 +L2

2M22 +L2

3M23− 1

12{L1, L2,M1,M2}− 112{L1, L3,M1,M3}

− 1

12{L2, L3,M2,M3}−

7

3L1M

21−

7

3L2M

22−

7

3L3M

23 +

2

3a{L1, L2, L3}

− 1

18{L1,M1,M2} −

1

18{L1,M1,M3} −

1

18{L2,M1,M2}

− 1

18{L2,M2,M3} −

1

18{L3,M1,M3} −

1

18{L3,M2,M3}

+1

6(4b1 + 3){L1,M2,M3} +

1

6(4b2 + 3){L2,M1,M3}+

1

6(4b3 + 3){L3,M1,M2} − a2(4b1 + 3)L2

1 − a2(4b2 + 3)L22

−a2(4b3 + 3)L23 + a2

3 ({L1, L2} + {L1, L3}Montreal talk – p.52/55

Functional relation continued

+{L2, L3})−(4b2b3+3b2+3b3+ 43)M2

1 −(4b1b3+3b1+3b3+ 43)M2

2

−(4b1b2 + 3b1 + 3b2 + 43)M2

3 + 23(b3 + 2)M1M2

+23(b2 + 2)M1M3 + 2

3(b1 + 2)M2M3 + 43a

2(7b1 + 4)L1 +43a

2(7b2 + 4)L2 + 43a

2(7b3 + 4)L3

+43a

2(12b1b2b3 + 9b1b2 + 9b1b3 + 9b2b3 + 4b1 + 4b2 + 4b3) = 0.

Here, {A,B,C,D} is the 24 term symmetrizer of 4 operators.


Three 2-variable models in variablest1, t2

1. M1,M3 basis: Differential operator - differential operatormodel. Eigenfunctions are powers of ti.

2. L3,M3 basis: Differential operator - difference operatormodel. Eigenfunctions are powers of t1 and conjugatedual Hahn polynomials in t2.

3. L3, L1 + L2 + L3 basis: Difference operator - differenceoperator model. Eigenfunctions are a restricted class ofWilson polynomials.


Outlook

1. It appears that the 3D nondegenerate case will lead togeneral 2-variable Wilson polynomials and their specialcases. The theory is much more complicated but thequadratic algebra structure is very restricted.

2. Since Wilson polynomials extend naturally toAskey-Wilson polynomials, this suggests the possibilityof a q-theory of superintegrability.

3. The general definition and representation theory forquadratic algebras of all orders is an important futureproject.


models for irreducible representations of quadratic algebrasmille003/yerevantalk.pdf1. the quadratic...

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