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Separation of variables, superintegrability andBôcher contractions.

Willard Miller, [Joint with E.G. Kalnins (Waikato), Sarah Post (Hawaii),Eyal Subag (Technion) and Robin Heinonen (Minnesota]

University of Minnesota

W. Miller (University of Minnesota) Bôcher contractions Torino 1 / 48

Abstract

Quantum superintegrable systems are exactly solvable quantum eigenvalueproblems. Their solvability is due to symmetry, but the symmetry isoften“hidden”. The symmetry generators of 2nd order superintegrablesystems in 2 dimensions close under commutation to define quadraticalgebras, a generalization of Lie algebras. The irreducible representations ofthese algebras yields important information about the eigenvalues andeigenspaces of the quantum systems. Distinct superintegrable systems andtheir quadratic algebras are related by geometric contractions, induced bygeneralized Inönü-Wigner Lie algebra contractions which have importantphysical and geometric implications, such as the Askey scheme for obtainingall hypergeometric orthogonal polynomials as limits of Racah/Wilsonpolynomials. This can all be unified by ideas first introduced in the 1894 thesisof Bôcher to study R- separable solutions of the wave equation.

Outline

1 Introduction

2 Representatives of Nondegenerate Systems

3 Representatives of Degenerate Systems

4 Models of Superintegrable Systems

5 The S9 Difference Operator Model

6 Lie algebra contractions

7 Contractions of Superintegrable Systems

8 The Bôcher Method

9 Discussion and Conclusions

Introduction

Superintegrable Systems: HΨ = EΨ

A quantum superintegrable system is an integrable Hamiltonian systemon an n-dimensional Riemannian/pseudo-Riemannian manifold withpotential:

H = ∆n + V

that admits 2n − 1 algebraically independent partial differential operatorscommuting with H, the maximum possible.

[H,Lj ] = 0, L2n−1 = H, n = 1,2, · · · ,2n − 1.

Superintegrability captures the properties of quantum Hamiltoniansystems that allow the Schrödinger eigenvalue problem HΨ = EΨ to besolved exactly, analytically and algebraically.A system is of order K if the maximum order of the symmetry operators,other than H, is K . For n = 2, K = 1,2 all systems are known.

Introduction

H = ∆n + V

[H,Lj ] = 0, L2n−1 = H, n = 1,2, · · · ,2n − 1.

Introduction

H = ∆n + V

[H,Lj ] = 0, L2n−1 = H, n = 1,2, · · · ,2n − 1.

Introduction

Free 2nd order superintegrable systems, (no potential,K = 2)We apply these ideas to 2nd order systems in 2D (2n − 1 = 3). The complexspaces with Laplace-Beltrami operators admitting at least three 2nd ordersymmetries were classified by Koenigs (1896). They are:

The two constant curvature spaces, six linearly independent 2nd ordersymmetries and three 1st order symmetries,The four Darboux spaces, four 2nd order symmetries and one 1st ordersymmetry,

ds2 = 4x(dx2 + dy2), ds2 =x2 + 1

x2 (dx2 + dy2),

ds2 =ex + 1

e2x (dx2 + dy2), ds2 =2 cos 2x + b

sin2 2x(dx2 + dy2),

Eleven 4-parameter Koenigs spaces. No 1st order symmetries. Example

ds2 = (c1

x2 + y2 +c2

x2 +c3

y2 + c4)(dx2 + dy2).

Introduction

2nd order systems with potential, K = 2

The symmetry operators of each system close under commutation togenerate a quadratic algebra, and the irreducible representations of thisalgebra determine the eigenvalues of H and their multiplicityAll the 2nd order superintegrable systems are limiting cases of a singlesystem: the generic 3-parameter potential on the 2-sphere, S9 in ourlisting. Analogously all quadratic symmetry algebras of these systemsare contractions of S9.

S9 : H = ∆2 +a1

s23, s2

1 + s22 + s2

3 = 1,

L1 = (s2∂s3 − s3∂s2 )2 +a3s2

s22, L2, L3,

H = L1 + L2 + L3 + a1 + a2 + a3

Introduction

2nd order systems with potential, K = 2

The symmetry operators of each system close under commutation togenerate a quadratic algebra, and the irreducible representations of thisalgebra determine the eigenvalues of H and their multiplicityAll the 2nd order superintegrable systems are limiting cases of a singlesystem: the generic 3-parameter potential on the 2-sphere, S9 in ourlisting. Analogously all quadratic symmetry algebras of these systemsare contractions of S9.

S9 : H = ∆2 +a1

s23, s2

1 + s22 + s2

3 = 1,

L1 = (s2∂s3 − s3∂s2 )2 +a3s2

s22, L2, L3,

H = L1 + L2 + L3 + a1 + a2 + a3

Introduction

3 types of 2nd order superintegrable systems:

1 Nondegenerate:

V (x) = a1V(1)(x) + a2V(2)(x) + a3V(3)(x) + a4

2 Degenerate:V (x) = a1V(1)(x) + a2

3 Free:V = a1, no potential

Introduction

Nondegenerate systems (2n − 1 = 3 generators)

The symmetry algebra generated by H,L1,L2 always closes undercommutation. Define 3rd order commutator R by R = [L1,L2]. Then

[Lj ,R] = A(j)1 L2

1 + A(j)2 L2

2 + A(j)3 H2 + A(j)

4 L1, L2+ A(j)5 HL1 + A(j)

+A(j)7 L1 + A(j)

8 L2 + A(j)9 H + A(j)

10, L1, L2 = L1L2 + L2L1,

R2 = b1L31 + b2L3

2 + b3H3 + b4L21, L2+ b5L1, L2

2+ b6L1L2L1 + b7L2L1L2

+b8HL1, L2+ b9HL21 + b10HL2

2 + b11H2L1 + b12H2L2 + b13L21 + b14L2

2 + b15L1, L2

+b16HL1 + b17HL2 + b18H2 + b19L1 + b20L2 + b21H + b22,

This structure is an example of a quadratic algebra. If we know the expansionfor R2 we can compute the other structure relations.

Introduction

Nondegenerate systems (2n − 1 = 3 generators)

The symmetry algebra generated by H,L1,L2 always closes undercommutation. Define 3rd order commutator R by R = [L1,L2]. Then

[Lj ,R] = A(j)1 L2

1 + A(j)2 L2

2 + A(j)3 H2 + A(j)

4 L1, L2+ A(j)5 HL1 + A(j)

+A(j)7 L1 + A(j)

8 L2 + A(j)9 H + A(j)

10, L1, L2 = L1L2 + L2L1,

R2 = b1L31 + b2L3

2 + b3H3 + b4L21, L2+ b5L1, L2

2+ b6L1L2L1 + b7L2L1L2

+b8HL1, L2+ b9HL21 + b10HL2

2 + b11H2L1 + b12H2L2 + b13L21 + b14L2

2 + b15L1, L2

+b16HL1 + b17HL2 + b18H2 + b19L1 + b20L2 + b21H + b22,

This structure is an example of a quadratic algebra. If we know the expansionfor R2 we can compute the other structure relations.

Introduction

Degenerate systems (2n − 1 = 3)

There are 4 generators: one 1st order X and 3 second order H,L1,L2.

[X , Lj ] = C(j)1 L1 + C(j)

2 L2 + C(j)3 H + C(j)

4 X 2 + C(j)5 , j = 1, 2,

[L1, L2] = E1L1,X+ E2L2,X+ E3HX + E4X 3 + E5X ,

Since 2n − 1 = 3 there must be an identity satisfied by the 4 generators. It isof 4th order:

c1L21 + c2L2

2 + c3H2 + c4L1, L2+ c5HL1 + c6HL2 + c7X 4 + c8X 2, L1+ c9X 2, L2

+c10HX 2 + c11XL1X + c12XL2X + c13L1 + c14L2 + c15H + c16X 2 + c17 = 0

If we know the 4th order identity, we can compute the other structure relationsto within an overall scale factor.

Introduction

Stäckel Equivalence Classes

All 2nd order 2d superintegrable systems with potential are known. There are59 types, on a variety of manifolds, but under the Stäckel transform, aninvertible structure preserving mapping, they divide into 12 equivalenceclasses with representatives on flat space and the 2-sphere, 6 withnondegenerate 3-parameter potentials

S9,E1,E2,E3′,E8,E10

and 6 with degenerate 1-parameter potentials

S3,E3,E4,E5,E6,E14

Representatives of Nondegenerate Systems

Example: S9

H = J21 + J2

2 + J23 +

where J3 = s1∂s2 − s2∂s1 and J2, J3 are obtained by cyclic permutations of indices.

Basis symmetries: (J3 = s2∂s1 − s1∂s2 , · · · )

L1 = J21 +

, L2 = J22 +

, L3 = J23 +

Structure equations:

[Li ,R] = 4Li , Lk − 4Li , Lj − (8 + 16aj)Lj + (8 + 16ak )Lk + 8(aj − ak ),

R2 =83L1, L2, L3 − (16a1 + 12)L2

1 − (16a2 + 12)L22 − (16a3 + 12)L2

523(L1, L2+L2, L3+L3, L1)+

13(16+176a1)L1+

13(16+176a2)L2+

13(16+176a3)L3

+323(a1 + a2 + a3) + 48(a1a2 + a2a3 + a3a1) + 64a1a2a3, R = [L1, L2].

Interesting 2 variable O.P.’s: Prorial-Karlin-MacGregor, (right triangle)W. Miller (University of Minnesota) Bôcher contractions Torino 11 / 48

Example: S9

H = J21 + J2

2 + J23 +

where J3 = s1∂s2 − s2∂s1 and J2, J3 are obtained by cyclic permutations of indices.

Basis symmetries: (J3 = s2∂s1 − s1∂s2 , · · · )

L1 = J21 +

, L2 = J22 +

, L3 = J23 +

[Li ,R] = 4Li , Lk − 4Li , Lj − (8 + 16aj)Lj + (8 + 16ak )Lk + 8(aj − ak ),

R2 =83L1, L2, L3 − (16a1 + 12)L2

1 − (16a2 + 12)L22 − (16a3 + 12)L2

523(L1, L2+L2, L3+L3, L1)+

13(16+176a1)L1+

13(16+176a2)L2+

13(16+176a3)L3

+323(a1 + a2 + a3) + 48(a1a2 + a2a3 + a3a1) + 64a1a2a3, R = [L1, L2].

Interesting 2 variable O.P.’s: Prorial-Karlin-MacGregor, (right triangle)W. Miller (University of Minnesota) Bôcher contractions Torino 11 / 48

Example: E1

H = ∂2x + ∂2

y − ω2(x2 + y2) +b1

x2 +b2

Generators:

L1 = ∂2x − ω2x2 +

x2 , L2 = ∂2y − ω2y2 +

y2 , L3 = (x∂y − y∂x)2 + y2 b1

x2 + x2 b2

Structure relations:[R, L1] = 8L2

1 − 8HL1 − 16ω2L3 + 8ω2,

[R, L3] = 8HL3 − 8L1, L3+ (16b1 + 8)H − 16(b1 + b2 + 1)L1,

R2 +83L1, L1, L3−8HL1, L3+(16b1 +16b2 +

)L21−16ω2L2

3− (32b1 +176

+(16b1 + 12)H2 +1763

ω2L3 + 16ω2(3b1 + 3b2 + 4b1b2 +23) = 0

Example: E1

H = ∂2x + ∂2

y − ω2(x2 + y2) +b1

x2 +b2

Generators:

L1 = ∂2x − ω2x2 +

x2 , L2 = ∂2y − ω2y2 +

y2 , L3 = (x∂y − y∂x)2 + y2 b1

x2 + x2 b2

Structure relations:[R, L1] = 8L2

1 − 8HL1 − 16ω2L3 + 8ω2,

[R, L3] = 8HL3 − 8L1, L3+ (16b1 + 8)H − 16(b1 + b2 + 1)L1,

R2 +83L1, L1, L3−8HL1, L3+(16b1 +16b2 +

)L21−16ω2L2

3− (32b1 +176

+(16b1 + 12)H2 +1763

ω2L3 + 16ω2(3b1 + 3b2 + 4b1b2 +23) = 0

Representatives of Degenerate Systems

Relations between nondegenerate/degeneratesystems

Every 1-parameter potential can be obtained from some 3-parameterpotential by parameter restriction.It is not simply a restriction, however, because the structure of thesymmetry algebra changes.A formally skew-adjoint 1st order symmetry appears and this induces anew 2nd order symmetry.Thus the restricted potential has a strictly larger symmetry algebra than isinitially apparent.

Example: S3 (Higgs oscillator)

H = J21 + J2

2 + J23 +

The system is the same as S9 with a1 = a2 = 0, a3 = a with the former L2 replaced by

L2 =12(J1J2 + J2J1)−

andX = J3 = s2∂s3 − s3∂s2 .

Structure relations:

[L1,X ] = 2L2, [L2,X ] = −X 2 − 2L1 + H − a, [L1, L2] = −(L1X + XL1)− (12+ 2a)X ,

(X 2L1 + XL1X + L1X 2

1 + L22 −HL1 + (a +

)X 2 − 16

H + (a− 23)L1 −

Interesting 2-variable O.P.’s: Koschmieder, Zerneke, (disk)

Example: E3 (Harmonic oscillator)

H = ∂2x + ∂2

y − ω2(x2 + y2)

Basis symmetries:

L1 = ∂2x − ω2x2, L3 = ∂xy − ω2xy , X = x∂y − y∂x .

Also we set L2 = ∂2y − ω2y2 = H − L1.

[L1,X ] = 2L3, [L3,X ] = H − 2L1, [L1, L3] = 2ω2X ,

L21 + L2

3 − L1H − ω2X 2 + ω2 = 0

Closure Theorem (Kalnins-Miller, 2014)

TheoremThere is a 1-1 relationship between quadratic algebras generated by 2ndorder elements in the envelping algebra of so(3,C), called free, and 2nd ordersuperintegrable systems on the complex 2-sphere. There is a 1-1 relationshipbetween quadratic algebras generated by 2nd order elements in theenveloping algebra of e(2,C) and 2nd order superintegrable systems on the2D complex flat space.

Remark :This theorem is constructive. Given a free quadratic algebra Q onecan compute the potential V and the symmetries of the quadratic algebra Q.

What are models of quadratic algebras?

A representation of a quadratic algebra Q is a homomorphism of Q intothe associative algebra of linear operators on some vector space:products go to products, commutators to commutators, etc.A model M is a faithful representation of Q in which the vector space is aspace of polynomials in one complex variable and the action is viadifferential/difference operators acting on that space.The dimensions of the finite dimensional irreducible representationsdetermine the multiplicities of the quantum energy eigenspaces and therepresentations determine the eigenvalues.

What is the significance of models?

They are easier to work with than the original abstract algebras.Eigenspaces in the models can be interpreted as expansion coefficientsof one separable eigenbasis of the original quantum system with potentialin terms of another eigenbasis.Racah and Wilson polynomials are related to S9 in this way.

Models of Superintegrable Systems

The big picture: Special functions and contractions

Connection with special functions

Special functions arise in two distinct ways:As separable eigenfunctions of the quantum Hamiltonian.As eigenfunctions in the model. Often solutions of difference equations.

Connection with contractionsTaking coordinate limits starting from quantum system S9 we can obtainother superintegrable systems.These limits induce limit relations between the special functionsassociated with the superintegrable systems.The limits induce contractions of the associated quadratic algebras, andvia the models, limit relations between the associated special functions.For constant curvature systems the required limits are all induced byWigner-type Lie algebra contractions of o(3.C) and e(2,C).The Askey scheme for orthogonal functions of hypergeometric type fitsnicely into this picture. (Kalnins-Miller-Post, 2014)

Model interplay

Figure :

The S9 Difference Operator Model

The S9 difference operator model. 1

L2 fn,m = (−4t2 −1

1 + B23 )fn,m,

L3 fn,m = (−4τ∗τ − 2[B1 + 1][B2 + 1] +1

2)fn,m,

H = L1 + L2 + L3 +3

4− (B2

1 + B22 + B2

3 ) = −4(m + 1)(B1 + B2 + B3 + m + 1) − 2(B1B2 + B1B3 + B2B3) +3

4− (B2

1 + B22 + B2

Here n = 0, 1, · · · ,m if m is a nonnegative integer and n = 0, 1, · · · otherwise. Also

4− B2

j , α = −(B1 + B3 + 1)/2 − m, β = (B1 + B3 + 1)/2, γ = (B1 − B3 + 1)/2, δ = (B1 + B3 − 1)/2 + B2 + m + 2,

EAF (t) = F (t + A), τ =1

2t(E1/2 − E−1/2), τ

∗ =1

[(α + t)(β + t)(γ + t)(δ + t)E1/2 − (α − t)(β − t)(γ − t)(δ − t)E−1/2

wn(t2) = (α + β)n(α + γ)n(α + δ)n4F3

(−n, α + β + γ + δ + n − 1, α − t, α + tα + β, α + γ, α + δ

= (α + β)n(α + γ)n(α + δ)nΦ(α,β,γ,δ)n (t2), Φn ≡ fn,m,

τ∗τΦn = n(n + α + β + γ + δ − 1)Φn,

where (a)n is the Pochhammer symbol and 4F3(1) is a hypergeometric function of unit argument. The polynomial wn(t2) is symmetric in α, β, γ, δ. For

the finite dimensional representations the spectrum of t2 is (α + k)2, k = 0, 1, · · · ,m and the orthogonal basis eigenfunctions are Racah

polynomials. In the infinite dimensional case they are Wilson polynomials.

L2 fn,m = (−4t2 −1

1 + B23 )fn,m,

L3 fn,m = (−4τ∗τ − 2[B1 + 1][B2 + 1] +1

2)fn,m,

H = L1 + L2 + L3 +3

4− (B2

1 + B22 + B2

3 ) = −4(m + 1)(B1 + B2 + B3 + m + 1) − 2(B1B2 + B1B3 + B2B3) +3

4− (B2

1 + B22 + B2

Here n = 0, 1, · · · ,m if m is a nonnegative integer and n = 0, 1, · · · otherwise. Also

4− B2

j , α = −(B1 + B3 + 1)/2 − m, β = (B1 + B3 + 1)/2, γ = (B1 − B3 + 1)/2, δ = (B1 + B3 − 1)/2 + B2 + m + 2,

EAF (t) = F (t + A), τ =1

2t(E1/2 − E−1/2), τ

∗ =1

[(α + t)(β + t)(γ + t)(δ + t)E1/2 − (α − t)(β − t)(γ − t)(δ − t)E−1/2

wn(t2) = (α + β)n(α + γ)n(α + δ)n4F3

(−n, α + β + γ + δ + n − 1, α − t, α + tα + β, α + γ, α + δ

= (α + β)n(α + γ)n(α + δ)nΦ(α,β,γ,δ)n (t2), Φn ≡ fn,m,

τ∗τΦn = n(n + α + β + γ + δ − 1)Φn,

where (a)n is the Pochhammer symbol and 4F3(1) is a hypergeometric function of unit argument. The polynomial wn(t2) is symmetric in α, β, γ, δ. For

the finite dimensional representations the spectrum of t2 is (α + k)2, k = 0, 1, · · · ,m and the orthogonal basis eigenfunctions are Racah

polynomials. In the infinite dimensional case they are Wilson polynomials.

The action of L2 and L3 on an L3 eigenbasis is

L2fn,m = −4K (n + 1, n)fn+1,m − 4K (n, n)fn,m − 4K (n − 1, n)fn−1,m + (B21 + B2

3 −12)fn,m,

L3fn,m = −(4n2 + 4n[B1 + B2 + 1] + 2[B1 + 1][B2 + 1]− 12)fn,m,

K (n + 1, n) =(B1 + B2 + n + 1)(n −m)(−B3 −m + n)(B2 + n + 1)

(B1 + B2 + 2n + 1)(B1 + B2 + 2n + 2),

K (n − 1, n) =n(B1 + n)(B1 + B2 + B3 + m + n + 1)(B1 + B2 + m + n + 1)

(B1 + B2 + 2n)(B1 + B2 + 2n + 1),

K (n, n) = [B1 + B2 + 2m + 1

2]2 − K (n + 1, n)− K (n − 1, n),

Lie algebra contractions

Let (A; [; ]A), (B; [; ]B) be two complex Lie algebras. We say B is a contractionof A if for every ε ∈ (0; 1] there exists a linear invertible map tε : B → A suchthat for every X ,Y ∈ B,

limε→0

t−1ε [tεX , tεY ]A = [X ,Y ]B.

Thus, as ε→ 0 the 1-parameter family of basis transformations can becomenonsingular but the structure constants go to a finite limit.

Lie algebra contractions

Contractions of e(2,C) and o(3,C)

These are the symmetry algebras of free systems on constant curvaturespaces. Their contractions have long since been classified. The are 6nontrivial contractions of e(2,C) and 4 of o(3,C).

Example: An Inönü-Wigner- contraction of o(3,C). We use the classicalrealization for o(3,C) acting on the 2-sphere, with basisJ1 = s2p3 − s3p2, J2 = s3p1 − s1p3, J3 = s1p2 − s2p1, commutation relations

[J2, J1] = J3, [J3, J2] = J1, [J1, J3] = J2,

and Hamiltonian H = J21 + J2

2 + J23 . Here s2

1 + s22 + s2

3 = 1.

Basis change : J ′1, J ′2, J ′3 = εJ1, εJ2, J3, o < ε ≤ 1

coordinate implementation x =s1

ε, y =

ε, s3 ≈ 1, J = J3

New structure relations : [J ′2, J′1] = ε2J ′3, [J ′3, J

′2] = J ′1, [J ′1, J

′3] = J ′2,

Let ε→ 0 : [J ′2, J′1] = 0, [J ′3, J

′2] = J ′1, [J ′1, J

′3] = J ′2, get e(2,C)

Contractions of Superintegrable Systems

Contractions of nondegenerate systems. 1

Suppose we have a nondegenerate superintegrable system with generatorsH,L1,L2, R = [L1,L2] and the usual structure equations, defining a quadraticalgebra Q. If we make a change of basis to new generators H, L1, L2 andparameters a1, a2, a3 such that L1

A1,1 A1,2 A1,3

A2,1 A2,2 A2,3

0 0 A3,3

B1,1 B1,2 B1,3

B2,1 B2,2 B2,3

B3,1 B3,2 B3,3

C1,1 C1,2 C1,3

C2,1 C2,2 C2,3

C3,1 C3,2 C3,3

for some 3× 3 constant matrices A = (Ai,j ),B,C such that det A · det C 6= 0,we will have the same system with new structure equations of the same formfor R = [L1, L2], [Lj , R], R2, but with transformed structure constants.

Choose a continuous 1-parameter family of basis transformation matricesA(ε),B(ε),C(ε, 0 < ε ≤ 1 such that A(1) = C(1) is the identity matrix,B(1) = 0 and det A(ε) 6= 0, det C(ε) 6= 0.Now suppose as ε→ 0 the basis change becomes singular, (i.e., thelimits of A,B,C either do not exist or, if they exist do not satisfydet A(0) det C(0) 6= 0) but the structure equations involvingA(ε),B(ε),C(ε), go to a limit, defining a new quadratic algebra Q′.We call Q′ a contraction of Q in analogy with Lie algebra contractions.

There is a similar definition of a contraction of a degenerate superintegrablesystem.

Lie algebra contractions⇒ quadratic algebracontractions

Constant curvature spaces:

Theorem

(Kalnins-Miller, 2014) Every Lie algebra contraction of A = e(2,C) orA = o(3,C) induces uniquely a contraction of a free quadratic algebra Qbased on A, which in turn induces uniquely a contraction of the quadraticalgebra Q with potential. This is true for both classical and quantum algebras.

A complication: Darboux spaces: The Lie symmetry algebra is only1-dimensional so Wigner contractions don’t apply and geometrical freequadratic algebra contractions must be determined on a case-by-case basis.

Figure : The Askey scheme and contractions of superintegrable systems

Figure : The Askey contraction scheme

Example of contraction hierarchy, S=sphere, E=flatspace, D=Darboux space

The Bôcher Method

2D conformal superintegrability of the 2nd order

Systems of Laplace type are of the form

HΨ ≡ ∆nΨ + V Ψ = 0.

Here ∆n is the Laplace-Beltrami operator on a conformally flat nDRiemannian or pseudo-Riemannian manifold.

A conformal symmetry of this equation is a partial differential operator S in thevariables x = (x1, · · · , xn) such that [S,H] ≡ SH − HS = RSH for somedifferential operator RS.

The system is conformally superintegrable for n > 2 if there are 2n − 1functionally independent conformal symmetries, S1, · · · ,S2n−1 with S1 = H. Itis second order conformally superintegrable if each symmetry Si can bechosen to be a differential operator of at most second order.

The Bôcher Method

The Bôcher approach

In his 1894 thesis Bôcher developed a geometrical method for finding andclassifying the R-separable orthogonal coordinate systems for the flat spaceLaplace equation ∆nΨ = 0 in n dimensions. It was based on the conformalsymmetry of these equations. The conformal symmetry algebra in thecomplex case is so(n + 2,C). We will use his ideas for n = 2 , but applied tothe Laplace equation with potential

HΨ ≡ (∂2x + ∂2

y + V )Ψ = 0.

The so(4,C) conformal symmetry algebra in the case n = 2 has the basis

P1 = ∂x , P2 = ∂y , J = x∂y − y∂x , D = x∂x + y∂y ,

K1 = (x2 − y2)∂x + 2xy∂y , K2 = (y2 − x2)∂y + 2xy∂x .

Bôcher linearizes this action by introducing tetraspherical coordinates. Theseare 4 projective coordinates (x1, x2, x3, x4) confined to the nullconex2

1 + x22 + x2

3 + x24 = 0.

The Bôcher Method

Tetraspherical coordinates

They are complex projective coordinates on the null cone

x21 + x2

2 + x23 + x2

4 = 0.

Relation to Cartesian coordinates (x , y):

x = − x1

x3 + ix4, y = − x2

x3 + ix4,

H = ∂xx + ∂yy + V = (x3 + ix4)2

∂2xk

where V = (x3 + ix4)2V .

The Bôcher Method

Relation to flat space 1st order conformal symmetries

We defineLjk = xj∂xk − xk∂xj , 1 ≤ j , k ≤ 4, j 6= k ,

where Ljk = −Lkj . The generators for flat space conformal symmetries arerelated to these via

P1 = ∂x = L13 + iL14, P2 = ∂y = L23 + iL24, D = iL34,

J = L12, K1 = L13 − iL14, K2 = L23 − iL24.

HereD = x∂x + y∂y , J = x∂y − y∂x , K1 = 2xD − (x2 + y2)∂x ,

The Bôcher Method

Relation to separation of variables and Bôcher’s limits

Bôcher uses symbols of the form [n1,n2, ..,np] where n1 + ...+ np = 4, todefine coordinate surfaces as follows. Consider the quadratic forms

Ω = x21 + x2

2 + x23 + x2

4 = 0, Φ =x2

1λ− e1

2λ− e2

λ− e3+

λ− e4.

If the parameters ej are pairwise distinct, the elementary divisors of these twoforms are denoted by [1,1,1,1].

Given a point in 2D flat space with Cartesian coordinates (x0, y0, z0), therecorresponds a set of tetraspherical coordinates (x0

1 , x02 , x

03 , x

04 ), unique up to

multiplcation by a nonzero constant. If we substitute into Φ we see that thereare exactly 2 roots λ = ρ, µ such that Φ = 0. These are elliptic coordinates.They are orthogonal with respect to the metric ds2 = dx2 + dy2 and areR-separable for the Laplace equations (∂2

x + ∂2y )Θ = 0 or (

∑4j−1 ∂

)Θ = 0.

The Bôcher Method

The potential V[1,1,1,1]

Consider the potential

V[1,1,1,1] =a1

It is the only potential V such that equation

∂2xj

+ V )Θ = 0

is R-separable in elliptic coordinates for all choices of the parameters ej . Theseparation is characterized by 2nd order conformal symmetry operators thatare linear in the parameters ej . In particular the symmetries span a3-dimensional subspace of symmetries, so the systemHΘ = (

∑4j−1 ∂

+ V[1,1,1,1])Θ = 0 must be conformally superintegrable.

The Bôcher Method

Contraction [1,1,1,1]→ [2,1,1], 1

If some of the ei become equal then Bôcher shows that the process of makinge1 → e2 together with suitable transformations of the a′i s produces aconformally equivalent H. This corresponds to the choice of coordinate curvesobtained by the Bôcher limiting process

e1 = e2 + ε2, x1 →iy1

ε, x2 →

ε+ εy2, xj → yj , j = 3,4,

which results in the pair of quadratic forms

Ω = 2y1y2 + y23 + y2

4 = 0, Φ =y2

1(λ− e2)2 +

(λ− e2)+

(λ− e3)+

(λ− e4)= 0.

The Bôcher Method

Contraction [1,1,1,1]→ [2,1,1], 2

The coordinate curves with e4 =∞ correspond to cyclides with elementarydivisors [2,1,

∞1 ], i.e.

Φ =y2

1(λ− e2)2 +

(λ− e2)+

(λ− e3)= 0.

The λ roots of this form yield planar elliptic coordinates In order to identify“Cartesian" coordinates on the cone we can choose

y1 =1√2

(x ,1 + ix ,2), y2 =i√2

(x ,1 − ix ,2), y3 = x ,3, y4 = x ,4.

The Bôcher Method

Contraction [1,1,1,1]→ [2,1,1], 3

The composite linear coordinate mapping

x1+ix2 =i√

(x ′1+ix ′2)+iε√2

(x ′1−ix ′2), x1−ix2 = − iε√2

(x ′1−ix ′2), x3 = x ′3, x4 = x ′4,

satisfies limε→0∑4

j=1 x2j =

∑4j=1 x ′2j = 0, and induces a contraction of so(4,C)

to itself:

L′12 = L12, L′13 = − i√2 ε

(L13− iL23)− i ε√2

L13, L′23 = − i√2 ε

(L13− iL23)− ε√2

L′34 = L34, L′14 = − i√2 ε

(L14−iL24)− i ε√2

L14, L′24 = − i√2 ε

(L14−iL24)− ε√2

This is the Bôcher contraction [1,1,1,1]→ [2,1,1].

The Bôcher Method

Bôcher contractions

This is family of contractions of so(4,C) to itself that we call Bôchercontractions. All these contractions are implemented via coordinatetransformations.

The Bôcher Method

Every 2D Riemannian manifold is conformally flat, so we can always find aCartesian-like coordinate system with coordinates (x , y) ≡ (x1, x2) such thatthe Laplace equation takes the form

(∗) H =1

λ(x , y)(∂2

x + ∂2y ) + V (x) = 0.

However, this equation is equivalent to the flat space equation

(∗∗) H ≡ ∂2x + ∂2

y + V (x) = 0, V (x) = λ(x)V (x).

In particular, the conformal symmetries of (*) are identical with the conformalsymmetries of (**). Thus without loss of generality we can assume themanifold is flat space with λ ≡ 1.

The Bôcher Method

The conformal Stäckel transform

Suppose we have a second order conformal superintegrable system

λ(x , y)(∂xx + ∂yy ) + V (x , y) = 0, H = H0 + V

where V (x , y) = a1W (x , y) + a2U(x , y) for arbitrary parameters a1,a2.

Theorem: The transformed (Helmholtz) system

H = E , H =1λ

(∂xx + ∂yy ) + V

with potential V (x , y) is truly superintegrable, where

λ = λU, V =VU.

The Bôcher Method

Thus any second order conformal Laplace superintegrable system admitting anonconstant potential U can be Stäckel transformed to a Helmholtzsuperintegrable system.

By choosing all possible special potentials U associated with the fixedLaplace system we generate the equivalence class of all Helmholtzsuperintegrable systems obtainable through this process.

Theorem: There is a one-to-one relationship between flat space conformallysuperintegrable Laplace systems with nondegenerate potential and Stäckelequivalence classes of superintegrable Helmholtz systems withnondegenerate potential.

The Bôcher Method

The 6 nondegenerate conformally superintegrablesystemsHere, systems are

∑4j=1 ∂

V[1,1,1,1] =4∑

V[2,1,1] =a1

+a3(x3 − ix4)

(x3 + ix4)3 +a4

(x3 + ix4)2 ,

V[2,2] =a1

(x1 + ix2)2 +a2(x1 − ix2)

(x1 + ix2)3 +a3

(x3 + ix4)2 +a4(x3 − ix4)

(x3 + ix4)3 ,

V[3,1] =a1

(x3 + ix4)2 +a2x1

(x3 + ix4)3 +a3(4x1

2 + x22)

(x3 + ix4)4 +a4

V[4] =a1

(x3 + ix4)2 + a2x1 + ix2

(x3 + ix4)3 + a33(x1 + ix2)2 − 2(x3 + ix4)(x1 − ix2)

(x3 + ix4)4 ,

V[0] =a1

(x3 + ix4)2 +a2x1 + a3x2

(x3 + ix4)3 + a4x2

1 + x22

(x3 + ix4)4 .

The Bôcher Method

Helmholtz contractions from Bôcher contractions, 1We describe how Bôcher contractions of conformal superintegrable systemsinduce contractions of Helmholtz superintegrable systems.

The basic idea is that the procedure of taking a conformal Stäckel transform ofa conformal superintegrable system, followed by a Helmholtz contractionyields the same result as taking a Bôcher contraction followed by an ordinaryStäckel transform: The diagrams commute.

Example: Consider the conformal Stäckel transforms of system [1,1,1,1] withpotential V[1,1,1,1]. Let H be the initial Hamiltonian. In terms of tetrasphericalcoordinates the conformal Stäckel transformed potential will take the form

+ a3x2

+ A3x2

=V[1,1,1,1]

F (x,A), F (x,A) =

and the transformed Hamiltonian will be

F (x,A)H,

where the transform is determined by the fixed vector (A1,A2,A3,A4).W. Miller (University of Minnesota) Bôcher contractions Torino 44 / 48

The Bôcher Method

Helmholtz contractions from Bôcher contractions, 2

We apply the Bôcher contraction [1,1,1,1]→ [2,1,1] to this system. In thelimit as ε→ 0 the potential V[1,1,1,1] → V[2,1,1], (??), and H → H ′ the [2,1,1]system. Now consider

F (x(ε),A) = V ′(x′,A)εα + O(εα+1),

where the the integer exponent α depends upon our choice of A. We willprovide the theory to show that the system defined by Hamiltonian

H ′ = limε→0

εαH(ε) =1

V ′(x′,A)H ′

is a superintegrable system that arises from the system [2,1,1] by aconformal Stäckel transform induced by the potential V ′(x′,A). Thus theHelmholtz superintegrable system with potential V = V1,1,1,1/F contracts tothe Helmholtz superintegrable system with potential V[2,1,1]/V ′. Thecontraction is induced by a generalized Inönü-Wigner Lie algebra contractionof the conformal algebra so(4,C).

Discussion and Conclusions

Figure : Relationship between conformal Stäckel transforms and Bôcher contractions

Wrap-up. 1

Free quadratic algebras uniquely determine associated superintegrablesystems with potential.A contraction of a free quadratic algebra to another uniquely determinesa contraction of the associated superintegrable systems.For a 2D superintegrable systems on a constant curvature space thesecontractions can be induced by Lie algebra contractions of the underlyingLie symmetry algebra.For all 2D nondegenerate superintegrable systems these contrations areinduced by Bôcher contractions of the conformal algebra to itself.Every 2D superintegrable system is obtained either as a sequence ofcontractions from S9 or is Stäckel equivalent to a system that is soobtained.

Wrap-up. 2

Taking contractions step-by-step from the S9 model we can recover theAskey Scheme. However, the contraction method is more general. Itapplies to all special functions that arise from the quantum systems viaseparation of variables, not just polynomials of hypergeometric type, andit extends to higher dimensions.The special functions arising from the models can be described as thecoefficients in the expansion of one separable eigenbasis for the originalquantum system in terms of another separable eigenbasis.The functions in the Askey Scheme are just those hypergeometricpolynomials that arise as the expansion coefficients relating twoseparable eigenbases that are both of hypergeometric type. Thus, thereare some contractions which do not fit in the Askey scheme since thephysical system fails to have such a pair of separable eigenbases.

Wrap-up. 3

Even though 2nd order 2D nondegenerate superintegrable systemsadmitno group symmetry, their structure is determined completely by theunderlying conformal symmetry of flat space Laplace equations(∆ + V )Ψ = 0.

separation of variables, superintegrability and bôcher ...miller/torinotalk1.pdfgeneralized...

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