Bôcher Contractions of Conformally Superintegrable Laplace Equations
The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often ''hidden''. The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebra...
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2016 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2016
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147737 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Bôcher Contractions of Conformally Superintegrable Laplace Equations / E.G. Kalnins, Willard Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 38 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-147737 |
|---|---|
| record_format |
dspace |
| spelling |
Kalnins, E.G. Miller Jr., Willard Subag, E. 2019-02-15T18:56:59Z 2019-02-15T18:56:59Z 2016 Bôcher Contractions of Conformally Superintegrable Laplace Equations / E.G. Kalnins, Willard Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 38 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R05; 81R12; 33C45 DOI:10.3842/SIGMA.2016.038 https://nasplib.isofts.kiev.ua/handle/123456789/147737 The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often ''hidden''. The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems on constant curvature spaces are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group SO(4,C), and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that Bôcher's prescription for coalescing roots of these forms induces contractions of the conformal algebra so(4,C) to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html. This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller Jr). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bôcher Contractions of Conformally Superintegrable Laplace Equations Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| spellingShingle |
Bôcher Contractions of Conformally Superintegrable Laplace Equations Kalnins, E.G. Miller Jr., Willard Subag, E. |
| title_short |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_full |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_fullStr |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_full_unstemmed |
Bôcher Contractions of Conformally Superintegrable Laplace Equations |
| title_sort |
bôcher contractions of conformally superintegrable laplace equations |
| author |
Kalnins, E.G. Miller Jr., Willard Subag, E. |
| author_facet |
Kalnins, E.G. Miller Jr., Willard Subag, E. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often ''hidden''. The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems on constant curvature spaces are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group SO(4,C), and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that Bôcher's prescription for coalescing roots of these forms induces contractions of the conformal algebra so(4,C) to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147737 |
| citation_txt |
Bôcher Contractions of Conformally Superintegrable Laplace Equations / E.G. Kalnins, Willard Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 38 назв. — англ. |
| work_keys_str_mv |
AT kalninseg bochercontractionsofconformallysuperintegrablelaplaceequations AT millerjrwillard bochercontractionsofconformallysuperintegrablelaplaceequations AT subage bochercontractionsofconformallysuperintegrablelaplaceequations |
| first_indexed |
2025-11-24T15:49:13Z |
| last_indexed |
2025-11-24T15:49:13Z |
| _version_ |
1850848944281616384 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 038, 31 pages
Bôcher Contractions of Conformally Superintegrable
Laplace Equations?
Ernest G. KALNINS †, Willard MILLER Jr. ‡ and Eyal SUBAG §
† Department of Mathematics, University of Waikato, Hamilton, New Zealand
E-mail: math0236@math.waikato.ac.nz
URL: http://www.math.waikato.ac.nz
‡ School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
E-mail: miller@ima.umn.edu
URL: http://www.ima.umn.edu/ miller/
§ Department of Mathematics, Pennsylvania State University, State College,
Pennsylvania, 16802 USA
E-mail: eus25@psu.edu
Received January 24, 2016, in final form April 11, 2016; Published online April 19, 2016
http://dx.doi.org/10.3842/SIGMA.2016.038
Abstract. The explicit solvability of quantum superintegrable systems is due to symmetry,
but the symmetry is often “hidden”. The symmetry generators of 2nd order superintegrable
systems in 2 dimensions close under commutation to define quadratic algebras, a generaliza-
tion of Lie algebras. Distinct systems on constant curvature spaces are related by geometric
limits, induced by generalized Inönü–Wigner Lie algebra contractions of the symmetry al-
gebras of the underlying spaces. These have physical/geometric implications, such as the
Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satis-
factory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional
Lie algebras. We show that these systems can be best understood by transforming them
to Laplace conformally superintegrable systems, with flat space conformal symmetry group
SO(4,C), and using ideas introduced in the 1894 thesis of Bôcher to study separable so-
lutions of the wave equation in terms of roots of quadratic forms. We show that Bôcher’s
prescription for coalescing roots of these forms induces contractions of the conformal alge-
bra so(4,C) to itself and yields a mechanism for classifying all Helmholtz superintegrable
systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we
announced our main findings. This paper provides the proofs and more details.
Key words: conformal superintegrability; contractions; Laplace equations
2010 Mathematics Subject Classification: 81R05; 81R12; 33C45
1 Introduction
A quantum (or Helmholtz) superintegrable system is an integrable Hamiltonian system on an
n-dimensional Riemannian/pseudo-Riemannian manifold with potential: H = ∆n + V that
admits 2n−1 algebraically independent partial differential operators Lj commuting with H, the
maximum possible. [H,Lj ] = 0, j = 1, 2, . . . , 2n− 1. Superintegrability captures the properties
of quantum Hamiltonian systems that allow the Schrödinger eigenvalue problem (or Helmholtz
equation) HΨ = EΨ to be solved exactly, analytically and algebraically [6, 7, 32, 36, 37].
A system is of order K if the maximum order of the symmetry operators, other than H, is K.
?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
mailto:math0236@math.waikato.ac.nz
mailto:miller@ima.umn.edu
mailto:eus25@psu.edu
http://dx.doi.org/10.3842/SIGMA.2016.038
http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 E.G. Kalnins, W. Miller Jr. and E. Subag
For n = 2, K = 1, 2 all systems are known, e.g., [5, 12, 13, 14, 15, 16, 17]. For K = 1 the
symmetry algebras are just Lie algebras.
We review quickly the facts for free 2nd order superintegrable systems (i.e., no potential,
K = 2) in the case n = 2, 2n − 1 = 3. The complex spaces with Laplace–Beltrami operators
admitting at least three 2nd order symmetries were classified by Koenigs [29]. They are:
• the two constant curvature spaces (flat space and the complex sphere), six linearly inde-
pendent 2nd order symmetries and three 1st order symmetries,
• the four Darboux spaces (one of which, D4, contains a parameter), four 2nd order sym-
metries and one 1st order symmetry, see Section 1.1 and [19],
• 6 families of 4-parameter Koenigs spaces. No 1st order symmetries, see Section 1.1.
For 2nd order systems with non-constant potential, K = 2, the generating symmetry operators
of each system close under commutation to determine a quadratic algebra, and the irreducible
representations of this algebra determine the eigenvalues of H and their multiplicities. Here we
consider only the nondegenerate superintegrable systems. Those with 4-parameter potentials
(including the additive constant) (the maximum possible):
V (x) = a1V(1)(x) + a2V(2)(x) + a3V(3)(x) + a4,
where {V(1)(x), V(2)(x), V(3)(x), 1} is a linearly independent set. For these the symmetry algebra
generated by H, L1, L2 always closes under commutation and gives the following quadratic
algebra structure: Define the 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)
6 HL2
+A
(j)
7 L1 +A
(j)
8 L2 +A
(j)
9 H +A
(j)
10 ,
R2 = b1L
3
1 + b2L
3
2 + b3H
3 + b4
{
L2
1, L2
}
+ b5
{
L1, L
2
2
}
+ b6L1L2L1 + b7L2L1L2
+ b8H{L1, L2}+ b9HL
2
1 + b10HL
2
2 + b11H
2L1 + b12H
2L2 + b13L
2
1 + b14L
2
2
+ b15{L1, L2}+ b16HL1 + b17HL2 + b18H
2 + b19L1 + b20L2 + b21H + b22,
where {L1, L2} = L1L2 + L2L1 and the A
(j)
i , bk are constants, different for each algebra.
All 2nd order 2D superintegrable systems with potential and their quadratic algebras are
known. There are 33 nondegenerate systems, on a variety of manifolds, see Section 1.1 (just
manifolds classified by Koenigs), where the numbering for constant curvature systems is taken
from [20] (the numbers are not always consecutive because the lists in [20] also include dege-
nerate systems). Under the Stäckel transform (we discuss this in Section 2.1) these systems
divide into 8 equivalence classes with representatives on flat space and the 2-sphere, see [30] and
Section 3.16.
1.1 The Helmholtz nondegenerate superintegrable systems
Flat space systems: HΨ = (∂2
x + ∂2
y + V )Ψ = EΨ.
1. E1: V = α
(
x2 + y2
)
+ β
x2
+ γ
y2
,
2. E2: V = α
(
4x2 + y2
)
+ βx+ γ
y2
,
3. E3′: V = α(x2 + y2) + βx+ γy,
4. E7: V = α(x+iy)√
(x+iy)2−b
+ β(x−iy)√
(x+iy)2−b
(
x+iy+
√
(x+iy)2−b
)2 + γ
(
x2 + y2
)
,
5. E8: V = α(x−iy)
(x+iy)3
+ β
(x+iy)2
+ γ
(
x2 + y2
)
,
Bôcher Contractions of Conformally Superintegrable Laplace Equations 3
6. E9: V = α√
x+iy
+ βy + γ(x+2iy)√
x+iy
,
7. E10: V = α(x− iy) + β
(
x+ iy − 3
2(x− iy)2
)
+ γ
(
x2 + y2 − 1
2(x− iy)3
)
,
8. E11: V = α(x− iy) + β(x−iy)√
x+iy
+ γ√
x+iy
,
9. E15: V = f(x− iy), where f is arbitrary (the exceptional case, characterized by the fact
that the symmetry generators are functionally linearly dependent [12, 13, 14, 15, 16, 17,
20]),
10. E16: V = 1√
x2+y2
(
α+ β
y+
√
x2+y2
+ γ
y−
√
x2+y2
)
,
11. E17: V = α√
x2+y2
+ β
(x+iy)2
+ γ
(x+iy)
√
x2+y2
,
12. E19: V = α(x+iy)√
(x+iy)2−4
+ β√
(x−iy)(x+iy+2)
+ γ√
(x−iy)(x+iy−2)
,
13. E20: V = 1√
x2+y2
(
α+ β
√
x+
√
x2 + y2 + γ
√
x−
√
x2 + y2
)
.
Systems on the complex 2-sphere: HΨ = (J2
23 + J2
13 + J2
12 + V )Ψ = EΨ. Here, Jk` =
sk∂s` − s`∂sk and s2
1 + s2
2 + s2
3 = 1.
1. S1: V = α
(s1+is2)2
+ βs3
(s1+is2)2
+
γ(1−4s23)
(s1+is2)4
,
2. S2: V = α
s23
+ β
(s1+is2)2
+ γ(s1−is2)
(s1+is2)3
,
3. S4: V = α
(s1+is2)2
+ βs3√
s21+s22
+ γ
(s1+is2)
√
s21+s22
,
4. S7: V = αs3√
s21+s22
+ βs1
s22
√
s21+s22
+ γ
s22
,
5. S8: V = αs2√
s21+s23
+ β(s2+is1+s3)√
(s2+is1)(s3+is1)
+ γ(s2+is1−s3)√
(s2+is1)(s3−is1)
,
6. S9: V = α
s21
+ β
s22
+ γ
s23
.
Darboux 1 systems: HΨ =
(
1
4x(∂2
x + ∂2
y) + V
)
Ψ = EΨ.
1. D1A: V = b1(2x−2b+iy)
x
√
x−b+iy + b2
x
√
x−b+iy + b3
x + b4,
2. D1B: V = b1(4x2+y2)
x + b2
x + b3
xy2
+ b4,
3. D1C: V = b1(x2+y2)
x + b2
x + b3y
x + b4.
Darboux 2 systems: HΨ =
(
x2
x2+1
(∂2
x + ∂2
y) + V
)
Ψ = EΨ.
1. D2A: V = x2
x2+1
(
b1(x2 + 4y2) + b2
x2
+ b3y
)
+ b4,
2. D2B: V = x2
x2+1
(
b1(x2 + y2) + b2
x2
+ b3
y2
)
+ b4,
3. D2C: V = x2√
x2+y2(x2+1)
(
b1 + b2
y+
√
x2+y2
+ b3
y−
√
x2+y2
)
+ b4.
Darboux 3 systems: HΨ =
(
1
2
e2x
ex+1(∂2
x + ∂2
y) + V
)
Ψ = EΨ.
1. D3A: V = b1
1+ex + b2ex√
1+2ex+iy(1+ex)
+ b3ex+iy
√
1+2ex+iy(1+ex)
+ b4,
4 E.G. Kalnins, W. Miller Jr. and E. Subag
2. D3B: V = ex
ex+1
(
b1 + e−
x
2 (b2 cos y2 + b3 sin y
2 )
)
+ b4,
3. D3C: V (= ex
ex+1
(
b1 + ex( b2
cos2 y
2
+ b3
sin2 y
2
)
)
+ b4,
4. D3D: V = e2x
1+ex
(
b1e
−iy + b2e
−2iy
)
+ b3
1+ex + b4.
Darboux 4 systems: HΨ =
(
− sin2 2x
2 cos 2x+b(∂
2
x + ∂2
y) + V
)
Ψ = EΨ.
1. D4(b)A: V = sin2 2x
2 cos 2x+b
(
b1
sinh2 y
+ b2
sinh2 2y
)
+ b3
2 cos 2x+b + b4,
2. D4(b)B: V = sin2 2x
2 cos 2x+b
(
b1
sin2 2x
+ b2e
4y + b3e
2y
)
+ b4,
3. D4(b)C: V = e2y
b+2
sin2 x
+ b−2
cos2 x
(
b1
Z+(1−e2y)
√
Z
+ b2
Z+(1+e2y)
√
Z
+ b3e−2y
cos2 x
)
+ b4,
Z = (1− e2y)2 + 4e2y cos2 x.
Generic Koenigs spaces:
1. K[1, 1, 1, 1]: HΨ = 1
V (b1,b2,b3,b4)
(
∂2
x + ∂2
y + V (a1, a2, a3, a4)
)
Ψ = EΨ,
V (a1, a2, a3, a4) = a1
x2
+ a2
y2
+ 4a3
(x2+y2−1)2
− 4a4
(x2+y2+1)2
,
2. K[2, 1, 1]: HΨ = 1
V (b1,b2,b3,b4)
(
∂2
x + ∂2
y + V (a1, a2, a3, a4)
)
Ψ = EΨ,
V (a1, a2, a3, a4) = a1
x2
+ a2
y2
− a3
(
x2 + y2
)
+ a4,
3. K[2, 2]: HΨ = 1
V (b1,b2,b3,b4)
(
∂2
x + ∂2
y + V (a1, a2, a3, a4)
)
Ψ = EΨ,
V (a1, a2, a3, a4) = a1
(x+iy)2
+ a2(x−iy)
(x+iy)3
+ a3 − a4
(
x2 + y2
)
,
4. K[3, 1]: HΨ = 1
V (b1,b2,b3,b4)
(
∂2
x + ∂2
y + V (a1, a2, a3, a4)
)
Ψ = EΨ,
V (a1, a2, a3, a4) = a1 − a2x+ a3
(
4x2 + y2
)
+ a4
y2
,
5. K[4]: HΨ = 1
V (b1,b2,b3,b4)
(
∂2
x + ∂2
y + V (a1, a2, a3, a4)
)
Ψ = EΨ,
V (a1, a2, a3, a4) = a1−a2(x+ iy)+a3
(
3(x+ iy)2 +2(x− iy)
)
−a4
(
4(x2 +y2)+2(x+ iy)3
)
,
6. K[0]: HΨ = 1
V (b1,b2,b3,b4)
(
∂2
x + ∂2
y + V (a1, a2, a3, a4)
)
Ψ = EΨ,
V (a1, a2, a3, a4) = a1 − (a2x+ a3y) + a4
(
x2 + y2
)
.
1.2 Lie algebras and quadratic algebras
Important for 2nd order superintegrable systems are the Lie algebras e(2,C) and o(3,C). These
are the (K = 1) symmetry Lie algebras of free (zero potential) systems on constant curvature
spaces. Every 2nd order symmetry operator on a constant curvature space takes the form
L = K +W (x), where K is a 2nd order element in the enveloping algebra of o(3,C) or e(2,C).
An important example is S9:
H = J2
1 + J2
2 + J2
3 +
a1
s2
1
+
a2
s2
2
+
a3
s2
3
,
where J3 = s1∂s2 − s2∂s1 and J2, J3 are obtained by cyclic permutations of indices. Basis
symmetries are
L1 = J2
1 +
a3s
2
2
s2
3
+
a2s
2
3
s2
2
, L2 = J2
2 +
a1s
2
3
s2
1
+
a3s
2
1
s2
3
, L3 = J2
3 +
a2s
2
1
s2
2
+
a1s
2
2
s2
1
.
Bôcher Contractions of Conformally Superintegrable Laplace Equations 5
Theorem 1.1. There is a bijection between quadratic algebras generated by 2nd order elements
in the enveloping algebra of o(3,C), called free, and 2nd order nondegenerate superintegrable
systems on the complex 2-sphere. Similarly, there is a bijection between quadratic algebras
generated by 2nd order elements in the enveloping algebra of e(2,C) and 2nd order nondegenerate
superintegrable systems on the 2D complex flat space.
The proof of this theorem is constructive [21]. Given a free quadratic algebra Q̃ one can
compute the potential V and the symmetries of the quadratic algebra Q of the nondegenerate
superintegrable system. These systems are closely related to the special functions of mathemati-
cal physics and their properties. The special functions arise in two distinct ways: 1) As separable
eigenfunctions of the quantum Hamiltonian. Second order superintegrable systems are multi-
separable, (with one exception) [12, 13, 14, 15, 16, 17]. 2) As interbasis expansion coefficients
relating distinct separable coordinate eigenbases [22, 23, 31, 35]. Most of the classical special
functions in the Digital Library of Mathematical Functions, as well as Wilson polynomials,
appear in these ways [34].
In [21] it has been shown that all the 2nd order superintegrable systems are obtained by taking
coordinate limits of the generic system S9 [20], or are obtained from these limits by a Stäckel
transform (an invertible structure preserving mapping of superintegrable systems [12, 13, 14,
15, 16, 17]). Analogously all quadratic symmetry algebras of these systems are limits of that
of S9. These coordinate limits induce limit relations between the special functions associated
as eigenfunctions of the superintegrable systems. The limits also induce contractions of the
associated quadratic algebras, and via the models of the irreducible representations of these
algebras, limit relations between the associated special functions. The Askey scheme for orthogo-
nal functions of hypergeometric type is an example of this [25]. For constant curvature systems
the required limits are all induced by Inönü–Wigner-type Lie algebra contractions of o(3,C)
and e(2,C) [9, 33, 38]. Inönü–Wigner-type Lie algebra contractions have long been applied to
relate separable coordinate systems and their associated special functions, see, e.g., [10, 11] for
some more recent examples, but the application to quadratic algebras is due to the authors and
their collaborators.
Recall the definition of (natural) Lie algebra contractions: Let (A; [ ; ]A), (B; [ ; ]B) be two
complex Lie algebras. We say that B is a contraction of A if for every ε ∈ (0; 1] there exists
a linear invertible map tε : B → A such that for every X,Y ∈ B, lim
ε→0
t−1
ε [tεX, tεY ]A = [X,Y ]B.
Thus, as ε → 0 the 1-parameter family of basis transformations can become nonsingular but
the structure constants of the Lie algebra go to a finite limit, necessarily that of another Lie
algebra.
The contractions of the symmetry algebras of constant curvature spaces have long since been
classified [21]. There are 6 nontrivial contractions of e(2,C) and 4 of o(3,C). They are each
induced by coordinate limits.
Contractions of quadratic algebras: Just as for Lie algebras we can define a contraction of
a quadratic algebra in terms of 1-parameter families of basis changes in the algebra: As ε → 0
the 1-parameter family of basis transformations becomes singular but the structure constants
go to a finite limit [21].
Let H = H(0) +V , S1 = S
(0)
1 +W1, S2 = S
(0)
2 +W2 be a superintegrable system on a constant
curvature space with quadratic algebra Q and free quadratic algebra Q̃ of H(0), S
(0)
1 , S
(0)
2 .
Motivating idea: Lie algebra contractions induce quadratic algebra contractions. For constant
curvature spaces we have
Theorem 1.2 ([21]). Every Lie algebra contraction of A = e(2,C) or A = o(3,C) induces
a contraction of a free (zero potential) quadratic algebra Q̃ based on A, which in turn induces
a contraction of the quadratic algebra Q with potential. This is true for both classical and
quantum algebras.
6 E.G. Kalnins, W. Miller Jr. and E. Subag
Similarly the coordinate limit associated with each contraction takes H to a new superin-
tegrable system with the contracted quadratic algebra. This relationship between coordinate
limits, Lie algebra contractions and quadratic algebra contractions for superintegrable systems
on constant curvature spaces breaks down for Darboux and Koenigs spaces. For Darboux spaces
the Lie symmetry algebra is only 1-dimensional, and there is no Lie symmetry algebra at all for
Koenigs spaces. Furthermore, there is the issue of finding a more systematic way of classifying
the 44 distinct Helmholtz superintegrable eigenvalue systems on different manifolds, and their re-
lations. These issues can be clarified by considering the Helmholtz systems as Laplace equations
(with potential) on flat space. This point of view was introduced in the paper [18] and applied
in [3] to solve important classification problems in the case n = 3. As announced in [28], the
proper object to study is the conformal symmetry algebra so(4, C) of the flat space Laplacian and
its contractions. The basic idea is that families of (Stäckel-equivalent) Helmholtz superintegrable
systems on a variety of manifolds correspond to a single conformally superintegrable Laplace
equation on flat space. We exploit this relation in the case n = 2, but it generalizes easily to all
dimensions n ≥ 2. The conformal symmetry algebra for Laplace equations with constant poten-
tial on flat space is the conformal algebra so(n+ 2,C). We review these concepts in Section 2.
In his 1894 thesis [1] Bôcher introduced a limit procedure based on the roots of quadratic
forms to find families of R-separable solutions of the ordinary (zero potential) flat space Laplace
equation in n dimensions. An important feature of his work was the introduction of special
projective coordinates in which the action of the conformal group so(n + 2,C) on solutions of
the Laplace equation can be linearized. For n = 2 these are tetraspherical coordinates. In
Section 4 we describe in detail the Laplace equation mechanism and how it can be applied
to systematize the classification of Helmholtz superintegrable systems and their relations via
limits. We show that Bôcher’s limit procedure can be interpreted as constructing generalized
Inönü–Wigner Lie algebra contractions of so(4,C) to itself. We call these Bôcher contractions
and show that they induce contractions of the conformal quadratic algebras associated with
Laplace superintegrable systems. All of the limits of the Helmholtz systems classified before for
n = 2 [21] are induced by the larger class of Bôcher contractions.
2 2D conformal superintegrability of the 2nd order
Systems of Laplace type are of the form
HΨ ≡ ∆nΨ + VΨ = 0. (2.1)
Here ∆n is the Laplace–Beltrami operator on a real or complex conformally flat nD Riemannian
or pseudo-Riemannian manifold. We assume that all functions occurring in this paper are locally
analytic, real or complex.) A conformal symmetry of this equation is a partial differential opera-
tor S in the variables x = (x1, . . . , xn) such that [S,H] ≡ SH−HS = RSH for some differential
operator RS . A conformal symmetry maps any solution Ψ of (2.1) to another solution. Two
conformal symmetries S, S′ are identified if S = S′ + RH for some differential operator R,
since they agree on the solution space of (2.1). (For short we will say that S = S′, mod(H) and
that S is a symmetry if [S,H] = 0, mod(H).) The system is conformally superintegrable for
n > 2 if there are 2n − 1 functionally independent conformal symmetries, S1, . . . , S2n−1 with
S1 = H. It is second order conformally superintegrable if each symmetry Si can be chosen to be
a differential operator of at most second order.
For n = 2 the definition must be restricted, since for a potential V = 0 there will be an
infinite-dimensional space of conformal symmetries when n = 2; every analytic function induces
such symmetries. We assume V 6= 0, possibly a constant.
Every 2D Riemannian manifold is conformally flat, so we can always find a Cartesian-like
coordinate system with coordinates x ≡ (x, y) ≡ (x1, x2) such that the Laplace equation takes
Bôcher Contractions of Conformally Superintegrable Laplace Equations 7
the form
H̃ =
1
λ(x, y)
(
∂2
x + ∂2
y
)
+ Ṽ (x) = 0. (2.2)
However, this equation is equivalent to the flat space equation
H ≡ ∂2
x + ∂2
y + V (x) = 0, V (x) = λ(x)Ṽ (x). (2.3)
In particular, the conformal symmetries of (2.2) are identical with the conformal symmetries
of (2.3). Indeed, denoting by Λ the operator of multiplication by the function λ(x, y) and using
the operator identity [A,BC] = B[A,C] + [A,B]C we have
[S,H] = [S,ΛH̃] = Λ[S, H̃] + [S,Λ]H̃ = ΛRH̃ + [S,Λ]H̃ =
(
ΛRΛ−1 + [S,Λ]Λ−1
)
H.
Thus without loss of generality we can assume the manifold is flat space with λ ≡ 1.
Since the Hamiltonians are formally self-adjoint, without loss of generality we can always
assume that a 2nd order conformal symmetry S is formally self-adjoint:
S =
1
λ
2∑
k,j=1
∂k ·
(
λakj(x)
)
∂j +W (x) ≡ S0 +W, ajk = akj .
Equating coefficients of the partial derivatives on both sides of
[S,H] =
(
R(1)(x)∂x +R(2)(x)∂y
)
H,
we can derive the conditions
aiii = 2aijj + ajji , i 6= j, Wj =
2∑
s=1
asjVs + ajjj V, k, j = 1, 2.
Here a subscript j on a`m, V or W denotes differentiation with respect to xj . The requirement
that ∂xW2 = ∂yW1 leads to the second order (conformal) Bertrand–Darboux partial differential
equation for the potential
a12(V11 − V22) +
(
a22 − a11
)
V12 +
(
a12
1 + a22
2 − a11
2
)
V1 +
(
a22
1 − a11
1 − a12
2
)
V2 + 2a12
12V = 0.
The following results are easy modifications of results for 3D conformal superintegrable systems
proved in [18]. For a conformally superintegrable system there are 3 functionally independent
symmetries, each leading to a Bertrand–Darboux equation. The result is that the potential
function V must satisfy a canonical system of equations of the form
V22 = V11 +A22(x)V1 +B22(x)V2 + C22(x)V,
V12 = A12(x)V1 +B12(x)V2 + C12(x)V. (2.4)
If the integrability conditions for this system (2.4) are satisfied identically, the vector space of
solutions V is four-dimensional and we say that the potential is nondegenerate. Otherwise the
potential is degenerate and the potential involves ≤ 3 parameters. In this paper we consider
only systems with nondegenerate potentials. Since we can always add the trivial conformal
symmetry ρ(x)H to S we could assume that, say a11 = 0.
In general the space of 2nd order conformal symmetries could be infinite-dimensional. How-
ever, the requirement that H have a multiparameter potential reduces the possible symmetries
to a finite-dimensional space. Indeed the conformal Bertrand–Darboux conditions for a 2nd
8 E.G. Kalnins, W. Miller Jr. and E. Subag
order symmetry yields the requirement ∂xy(a
11−a22) = 0. The result is that the pure derivative
terms S0 belong to the space spanned by symmetrized products of the conformal Killing vectors
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. (2.5)
and terms g(x)
(
∂2
x+∂2
y
)
, where g is an arbitrary function. For a given multiparameter potential
only a subspace of these conformal tensors occurs. Note that on the hypersurface H = 0 in phase
space all symmetries g(x)H vanish, so any two symmetries differing by g(x)H can be identified.
2.1 The conformal Stäckel transform
We review quickly the concept of the Stäckel transform [24] and extend it to conformally super-
integrable systems. Suppose we have a second order conformal superintegrable system
H =
1
λ(x, y)
(∂xx + ∂yy) + V (x, y) = 0, H = H0 + V (2.6)
with V the general solution for this system, and suppose U(x, y) is a particular potential solution,
nonzero in an open set. The Stäckel transform induced by U is the system
H̃ = E, H̃ =
1
λ̃
(∂xx + ∂yy) + Ṽ , where λ̃ = λU, Ṽ =
V
U
. (2.7)
In [18, 28] we proved
Theorem 2.1. The transformed (Helmholtz) system H̃ is truly superintegrable.
Note that if HΨ = 0 then S̃Ψ = SΨ and H(SΨ) = 0 so S and S̃ agree on the null space of H
and they preserve this null space. This result shows that any second order conformal Laplace
superintegrable system admitting a nonconstant potential U can be Stäckel transformed to
a Helmholtz superintegrable system. This operation is invertible, but the inverse is not a Stäckel
transform. By choosing all possible special potentials U associated with the fixed Laplace
system (2.6) we generate the equivalence class of all Helmholtz superintegrable systems (2.7)
obtainable through this process. As is easy to check, any two Helmholtz superintegrable systems
lie in the same equivalence class if and only if they are Stäckel equivalent in the standard sense,
see [28, Theorem 4]. All Helmholtz superintegrable systems are related to conformal Laplace
systems in this way, so the study of all Helmholtz superintegrability on conformally flat manifolds
can be reduced to the study of all conformal Laplace superintegrable systems on flat space.
In [12, 13, 14, 15, 16, 17] it is demonstrated that for the 3-parameter Helmholtz system H ′
and the Stäckel transform H̃ ′,
H ′ = H0 + V ′ = H0 + U (1)α1 + U (2)α2 + U (3)α3,
H̃ ′ =
1
U (1)
H0 +
−U (1)E + U (2)α2 + U (3)α3
U (1)
,
if H ′Ψ = EΨ then H̃ ′Ψ = −α1Ψ. The effect of the Stäckel transform is to replace α1 by −E
and E by −α1. Further, a 2nd order symmetry S of H ′ transforms to the 2nd order symmetry S̃
of H̃ ′ such that S and S̃ agree on eigenspaces of H ′.
We know that the symmetry operators of all 2nd order nondegenerate superintegrable systems
in 2D generate a quadratic algebra of the form
[R,S1] = f (1)(S1, S2, α1, α2, α3, H
′), [R,S2] = f (2)(S1, S2, α1, α2, α3, H
′),
Bôcher Contractions of Conformally Superintegrable Laplace Equations 9
R2 = f (3)(S1, S2, α1, α2, α3, H
′), R ≡ [S1, S2], (2.8)
where {S1, S2, H} is a basis for the 2nd order symmetries and α1, α2, α3 are the parameters for
the potential [12, 13, 14, 15, 16, 17, 32]. We see that the effect of a Stäckel transform generated
by the potential function U (1) is to determine a new superintegrable system with structure
[R̃, S̃1] = f (1)
(
S̃1, S̃2,−H̃ ′, α2, α3,−α1
)
, [R̃, S̃2] = f (2)
(
S̃1, S̃2,−H̃ ′, α2, α3,−α1
)
,
R̃2 = f (3)
(
S̃1, S̃2,−H̃ ′, α2, α3,−α1
)
, R̃ ≡ [S̃1, S̃2]. (2.9)
Of course, the switch of α1 and H ′ is only for illustration; there is a Stäckel transform that
replaces any αj by −H ′ and H ′ by −αj .
Formulas (2.8) and (2.9) are just instances of the quadratic algebras of the superintegrable
systems belonging to the equivalence class of a single nondegenerate conformally superintegrable
Hamiltonian Ĥ = ∂xx + ∂yy +
4∑
j=1
αjV
(j)(x, y). Let Ŝ1, Ŝ2, Ĥ be a basis of 2nd order conformal
symmetries of Ĥ. From the above discussion we can conclude the following.
Theorem 2.2. The symmetries of the 2D nondegenerate conformal superintegrable Hamilto-
nian Ĥ generate a quadratic algebra
[R̂, Ŝ1] = f (1)
(
Ŝ1, Ŝ2, α1, α2, α3, α4
)
, [R̂, Ŝ2] = f (2)
(
Ŝ1, Ŝ2, α1, α2, α3, α4
)
,
R̂2 = f (3)
(
Ŝ1, Ŝ2, α1, α2, α3, α4
)
, (2.10)
where R̂ = [Ŝ1, Ŝ2] and all identities hold mod(Ĥ). A conformal Stäckel transform generated
by the potential V (j)(x, y) yields a nondegenerate Helmholtz superintegrable Hamiltonian H̃ with
quadratic algebra relations identical to (2.10), except that we make the replacements Ŝ` → S̃` for
` = 1, 2 and αj → −H̃. These modified relations (2.10) are now true identities, not mod(Ĥ).
3 Tetraspherical coordinates
The tetraspherical coordinates (x1, . . . , x4) satisfy x2
1 + x2
2 + x2
3 + x2
4 = 0 (the null cone) and
4∑
k=1
xk∂xk = 0. They are projective coordinates on the null cone and have 3 degrees of free-
dom. Their principal advantage over flat space Cartesian coordinates is that the action of the
conformal algebra (2.5) and of the conformal group ∼ SO(4,C) is linearized in tetraspherical
coordinates.
Relation to Cartesian coordinates (x, y) and coordinates on the 2-sphere (s1, s2, s3):
x1 = 2XT, x2 = 2Y T, x3 = X2 + Y 2 − T 2, x4 = i
(
X2 + Y 2 + T 2
)
,
x =
X
T
= − x1
x3 + ix4
, y =
Y
T
= − x2
x3 + ix4
, x =
s1
1 + s3
, y =
s2
1 + s3
,
s1 =
2x
x2 + y2 + 1
, s2 =
2y
x2 + y2 + 1
, s3 =
1− x2 − y2
x2 + y2 + 1
,
H = ∂xx + ∂yy + Ṽ = (x3 + ix4)2
(
4∑
k=1
∂2
xk
+ V
)
= (1 + s3)2
3∑
j=1
p2
sj + V
,
Ṽ = (x3 + ix4)2V, (1 + s3) = −i(x3 + ix4)
x4
,
s1 =
ix1
x4
, s2 =
ix2
x4
, s3 =
−ix3
x4
.
10 E.G. Kalnins, W. Miller Jr. and E. Subag
Relation to flat space and 2-sphere 1st order conformal constants of the motion:
We define
Ljk = xj∂xk − xk∂xj , 1 ≤ j, k ≤ 4, j 6= k,
where Ljk = −Lkj . The generators for flat space conformal symmetries are related to these via
P1 = ∂x = L13 + iL14, P2 = ∂y = L23 + iL24, D = iL34, J = L12,
Kj = Lj3 − iLj4, j = 1, 2, D = x∂x + y∂y, J = x∂y − y∂x,
K1 = 2xD −
(
x2 + y2
)
∂x, . . . .
The generators for 2-sphere conformal constants of the motion are related to the Ljk via
L12 = J12 = s1∂s2 − s2∂s1 , L13 = J13, L23 = J23,
L14 = −i∂s1 , L24 = −i∂s2 , L34 = −i∂s3 .
Note that in identifying tetraspherical coordinates we can always permute the parameters 1–4.
Also, we can apply an arbitrary SO(4,C) transformation to the tetraspherical coordinates, so
the above relations between Euclidean and tetraspherical coordinates are far from unique.
2nd order conformal symmetries ∼ H: The 11-dimensional space of conformal symmet-
ries ∼ H has basis
L2
12 − L2
34, L2
13 − L2
24, L2
23 − L2
14, L2
12 + L2
13 + L2
23, L12L34 + L23L14 − L13L24,
{L13, L14}+ {L23, L24}, {L13, L23}+ {L14, L24}, {L12, L13}+ {L34, L24},
{L12, L14} − {L34, L23}, {L12, L23} − {L34, L14}, {L12, L24}+ {L34, L13}.
All of this becomes much clearer if we make use of the decomposition so(4,C) ≡ so(3,C) ⊕
so(3,C) and the functional realization of the Lie algebra. Setting
J1 = 1
2(L23 − L14), J2 = 1
2(L13 + L24), J3 = 1
2(L12 − L34),
K1 = 1
2(L23 + L14), K2 = 1
2(L13 − L24), K3 = 1
2(L12 + L34),
we have [Ji, Jj ] = εijkJk, [Ki,Kj ] = εijkKk, [Ji,Kj ] = 0. In the variables z = x+ iy, z̄ = x− iy:
J1 = 1
2
(
i∂z − iz2∂z
)
, J2 = 1
2
(
∂z + z2∂z
)
, J3 = iz∂z,
K1 = 1
2
(
−i∂z̄ + iz̄2∂z̄
)
, K2 = 1
2
(
∂z̄ + z̄2∂z̄
)
, K3 = −iz̄∂z̄,
so the Ji operators depend only on z and the Kj operators depend only on z̄. Also J2
1 +J2
2 +J2
3 ≡
0, K2
1 + K2
2 + K2
3 ≡ 0. The space of 2nd order elements in the enveloping algebra is thus
21-dimensional and decomposes as Az ⊕ Az̄ ⊕ Azz̄, where Az is 5-dimensional with basis J2
1 ,
J2
3 , {J1, J2}, {J1, J3}, {J2, J3}, Az̄ is 5-dimensional with basis K2
1 , K2
3 , {K1,K2}, {K1,K3},
{K2,K3}, and Azz̄ is 9-dimensional with basis JiKj , 1 ≤ i, j ≤ 3. Note that all of the elements
of Azz̄ are ∼ H, whereas none of the nonzero elements of Az, Az̄ have this property. Here, the
transposition Ji ↔ Ki is a conformal equivalence.
3.1 Classif ication of nondegenerate conformally superintegrable systems
With this simplification it becomes feasible to classify all conformally 2nd order superinte-
grable systems with nondegenerate potential. Since every such system has generators S(1) =
S
(1)
0 + W1(z, z̄), S(2) = S
(2)
0 + W2(z, z̄), it is sufficient to classify, up to SO(4,C) conjugacy,
all free conformal quadratic algebras with generators S
(1)
0 , S
(2)
0 , modH0 (H0 = ∂zz̄) and then
to determine for which of these free conformal algebras the integrability conditions for equa-
tions (2.4) hold identically, so that the system admits a nondegenerate potential Ṽ (z, z̄) which
can be computed. The classification breaks up into the following possible cases:
Bôcher Contractions of Conformally Superintegrable Laplace Equations 11
• Case 1: S
(1)
0 , S
(2)
0 ∈ Az. (This is conformally equivalent to S
(1)
0 , S
(2)
0 ∈ Az̄.) The possible
free conformal quadratic algebras of this type, classified up to SO(3,C) conjugacy mod J2
1 +
J2
2 + J2
3 can easily be obtained from the computations in [21]. They are the pairs
1) J2
3 , J
2
1 , 2) J2
3 , {J1 + iJ2, J3}, 3) J2
3 , {J1, J3},
4) {J2, J2 + iJ1}, {J2, J3}, 5) J2
3 , (J1 + iJ2)2,
6) {J1 + iJ2, J3}, (J1 + iJ2)2.
Checking pairs 1)–5) we find that they do not admit a nonzero potential, so they do not
correspond to nondegenerate conformal superintegrable systems. This is in dramatic dis-
tinction to the results of [21], where for Helmholtz systems on constant curvature spaces
there was a 1-1 relationship between free quadratic algebras and nondegenerate superin-
tegrable systems. Pair 6), does correspond to a superintegrable system, the exceptional
case Ṽ = f(z), where f(z) is arbitrary. (This system is conformally Stäckel equivalent
to the singular Euclidean system E15.) Equivalently, the system in Az̄ with analogous
K-operators yields the potential Ṽ = f(z̄), see (3.7) in Section 3.2.
• Case 2: S
(1)
0 = S
(1)
J + S
(1)
K , S
(2)
0 = S
(2)
J , where S
(1)
J , S
(2)
J are selected from one of the pairs
1)–6) above and S
(1)
K is a nonzero element of Az̄. Again there is a conformally equivalent
case, where the roles of Ji and Ki are switched. To determine the possibilities for S
(1)
K
we classify the 2nd order elements in the enveloping algebra of so(3,C) up to SO(3,C)
conjugacy, modK2
1 +K2
2 +K2
3 . From the computations in [21] we see easily that there are
the following representatives for the equivalence classes:
a) K2
3 , b) K2
1 + aK2
2 , a 6= 0, 1, c) (K1 + iK2)2,
d) K2
3 + (K1 + iK2)2, e) {K3,K1 + iK2}.
For pairs 1), 3), 4), 5) above and all choices a)–e) we find that the integrability conditions
are never satisfied, so there are no corresponding nondegenerate superintegrable systems.
For pair 2), however, we find that any choice a)–e) leads to the same nondegenerate
superintegrable system [2, 2], see (3.3) in Section 3.2. While it appears that there are
multiple generators for this one system, each set of generators maps to any other set by
a conformal Stäckel transformation and a change of variable. For pair 6), we find that
any choice a)–e) leads to the same nondegenerate superintegrable system [4], see (3.5) in
Section 3.2. Again each set of generators maps to any other set by a conformal Stäckel
transformation and a change of variable.
• Case 3: S
(1)
0 = S
(1)
J , S
(2)
0 = S
(2)
J + S
(2)
K , where S
(1)
J , S
(2)
J are selected from one of the pairs
1)–6) above and S
(2)
K is a nonzero element of Az̄. Again there is a conformally equivalent
case, where the roles of Ji and Ki are switched. To determine the possibilities for S
(2)
K
we classify the 2nd order elements in the enveloping algebra of so(3,C) up to SO(3,C)
conjugacy, modK2
1 + K2
2 + K2
3 . They are a)–e) above. For pairs 1)–4), 6) above and all
choices a)–e) the integrability conditions are never satisfied, so there are no corresponding
nondegenerate superintegrable systems. For pair 5), however, we find that any choice a)–e)
leads to the same nondegenerate superintegrable system [2, 2], see (3.3) in Section 3.2.
Again each set of generators maps to any other set (and to any [2, 2] generators in Case 2)
by a conformal Stäckel transformation and a change of variable.
• Case 4: S
(1)
0 = S
(1)
J , S
(2)
0 = S
(2)
K , where S
(1)
J is selected from one of the representatives a)–e)
above and S
(2)
K is selected from one of the analogous representatives a)–e) expressed as
K-operators. We find that each of the 25 sets of generators leads to the single conformally
12 E.G. Kalnins, W. Miller Jr. and E. Subag
superintegrable system [0], see (3.6) in Section 3.2, and each set of generators maps to any
other set by a conformal Stäckel transformation and a change of variable.
• Case 5: S
(1)
0 = S
(1)
J + S
(1)
K , S
(2)
0 = S
(2)
J + S
(2)
K , where S
(1)
J , S
(2)
J are selected from one of
the pairs 1)–6) above and S
(1)
K , S
(2)
K are obtained from S
(1)
J , S
(2)
J , respectively, by replacing
each Ji by Ki. We find the following possibilities:
i) S
(1)
0 = J2
1 + K2
1 , S
(2)
0 = J2
3 + K2
3 . This extends to the system [1, 1, 1, 1], see (3.1) in
Section 3.2.
ii) S
(1)
0 = J2
3 +K2
3 , S
(2)
0 = {J3, J1 + iJ2}+ {K3,K1 + iK2}. This extends to the system
[2, 1, 1], see (3.2) in Section 3.2.
iii) S
(1)
0 = J2
3 + K2
3 , S
(2)
0 = {J1, J3} + {K1,K3}. This extends to the system [1, 1, 1, 1],
see (3.1) in Section 3.2, again, equivalent to the generators i) by a conformal Stäckel
transformation and a change of variable.
iv) S
(1)
0 = {J1, J2 + iJ1} + {K1,K2 + iK1}, S(2)
0 = {J2, J3} + {K2,K3}. This does not
extend to a conformal superintegrable system.
v) S
(1)
0 = (J1 + iJ2)2 +(K1 + iK2)2, S
(2)
0 = J2
3 +K2
3 . This extends to the system [2, 1, 1],
see (3.2) in Section 3.2, again, equivalent to the generators ii) by a conformal Stäckel
transformation and a change of variable.
vi) S
(1)
0 = {J3, J1 + iJ2} + {K3,K1 + iK2}, S(2)
0 = (J1 + iJ2)2 + (K1 + iK2)2, which
extends to the system with potential [3, 1], see (3.4) in Section 3.2.
Example 3.1. We describe how apparently distinct superintegrable systems of a fixed type
are actually the same. In Case 2 consider the system with generators {J1 + iJ2, J3} + (K1 +
iK2)2, (J1 + iJ2)2. This extends to the conformally superintegrable system [4] with flat space
Hamiltonian operator H1 = ∂zz̄ + V (1), where V (1) = 2k3zz̄ + 2k4z + k3z̄
3 + 3k4z̄
2 + k1z̄ + k2.
The system with generators {J1 + iJ2, J3}+K2
3 + (K1 + iK2)2, (J1 + iJ2)2 again extends to the
conformally superintegrable system [4]. Indeed, replacing z, z̄ by Z, Z̄ to distinguish the two
systems, we find the 2nd flat space Hamiltonian operator H2 = ∂ZZ̄ + V (2), where
V (2) =
c3 arcsinh3(Z̄) + 3c4 arcsinh2(Z̄) + (2c3Z + c1) arcsinh(Z̄) + 2c4Z + c2√
1− Z̄2
.
Now we perform a conformal Stäckel transform on H2 to obtain the new flat space system
H̃2 =
√
1− Z̄2∂ZZ̄ + c3 arcsinh3(Z̄) + 3c4 arcsinh2(Z̄)
+ (2c3Z + c1) arcsinh(Z̄) + 2c4Z + c2.
Making the change of variable Z̄ = sinhW , we find
H̃2 = ∂ZW + c3W
3 + 3c4W
2 + (2c3Z + c1)W + 2c4Z + c2.
Thus, with the identifications Z = z, W = z̄, ci = ki, we see that H1 ≡ H̃2.
This completes the classification. The results are summarized in the next section.
3.2 The 8 Laplace superintegrable systems with nondegenerate potentials
The systems are all of the form 4∑
j=1
∂2
xj + V (x)
Ψ = 0, or
(
∂2
x + ∂2
y + Ṽ
)
Ψ = 0
Bôcher Contractions of Conformally Superintegrable Laplace Equations 13
as a flat space system in Cartesian coordinates. The potentials are
V[1,1,1,1] =
a1
x2
1
+
a2
x2
2
+
a3
x2
3
+
a4
x2
4
,
Ṽ[1,1,1,1] =
a1
x2
+
a2
y2
+
4a3
(x2 + y2 − 1)2
− 4a4
(x2 + y2 + 1)2
, (3.1)
V[2,1,1] =
a1
x2
1
+
a2
x2
2
+
a3(x3 − ix4)
(x3 + ix4)3
+
a4
(x3 + ix4)2
,
Ṽ[2,1,1] =
a1
x2
+
a2
y2
− a3
(
x2 + y2
)
+ a4, (3.2)
V[2,2] =
a1
(x1 + ix2)2
+
a2(x1 − ix2)
(x1 + ix2)3
+
a3
(x3 + ix4)2
+
a4(x3 − ix4)
(x3 + ix4)3
,
Ṽ[2,2] =
a1
(x+ iy)2
+
a2(x− iy)
(x+ iy)3
+ a3 − a4
(
x2 + y2
)
, (3.3)
V[3,1] =
a1
(x3 + ix4)2
+
a2x1
(x3 + ix4)3
+
a3(4x1
2 + x2
2)
(x3 + ix4)4
+
a4
x2
2
,
Ṽ[3,1] = a1 − a2x+ a3
(
4x2 + y2
)
+
a4
y2
, (3.4)
V[4] =
a1
(x3 + ix4)2
+ a2
x1 + ix2
(x3 + ix4)3
+ a3
3(x1 + ix2)2 − 2(x3 + ix4)(x1 − ix2)
(x3 + ix4)4
+ a4
4(x3 + ix4)(x2
3 + x2
4) + 2(x1 + ix2)3
(x3 + ix4)5
,
Ṽ[4] = a1 − a2(x+ iy) + a3
(
3(x+ iy)2 + 2(x− iy)
)
− a4
(
4
(
x2 + y2
)
+ 2(x+ iy)3
)
, (3.5)
V[0] =
a1
(x3 + ix4)2
+
a2x1 + a3x2
(x3 + ix4)3
+ a4
x2
1 + x2
2
(x3 + ix4)4
,
Ṽ[0] = a1 − (a2x+ a3y) + a4
(
x2 + y2
)
, (3.6)
Varb =
1
(x3 + ix4)2
f
(
−x1 − ix2
x3 + ix4
)
,
Ṽarb = f(x+ iy), f is arbitrary, (3.7)
V (1) = a1
1
(x1 + ix2)2
+ a2
1
(x3 + ix4)2
+ a3
(x3 + ix4)
(x1 + ix2)3
+ a4
(x3 + ix4)2
(x1 + ix2)4
,
Ṽ (1) =
a1
(x+ iy)2
+ a2 −
a3
(x+ iy)3
+
a4
(x+ iy)4
(a special case of (3.7)), (3.8)
V (2)′ = a1
1
(x3 + ix4)2
+ a2
(x1 + ix2)
(x3 + ix4)3
+ a3
(x1 + ix2)2
(x3 + ix4)4
+ a4
(x1 + ix2)3
(x3 + ix4)5
,
Ṽ (2)′ = a1 + a2(x+ iy) + a3(x+ iy)2 + a4(x+ iy)3 (a special case of (3.7)). (3.9)
We note that systems (3.8), (3.9) are not the fundamental Bôcher classes; they are merely special
cases of the singular system (3.7). We list them because they, and not the general (3.7), appear
as contractions of the fundamental systems.
3.3 Contractions of conformal superintegrable systems
with potential induced by generalized Inönü–Wigner contractions
The basis symmetries S(j) = S(j)
0 +W (j), H = H0 +V of a nondegenerate 2nd order conformally
superintegrable system determine a conformal quadratic algebra (2.10), and if the parameters
of the potential are set equal to 0, the free system S(j)
0 , H0, j = 1, 2 also determines a conformal
quadratic algebra without parameters, which we call a free conformal quadratic algebra. The
14 E.G. Kalnins, W. Miller Jr. and E. Subag
elements of this free algebra belong to the enveloping algebra of so(4,C) with basis (2.5). Since
the system is nondegenerate the integrability conditions for the potential are satisfied identically
and the full quadratic algebra can be computed from the free algebra, modulo a choice of basis
for the 4-dimensional potential space. Once we choose a basis for so(4,C), its enveloping algebra
is uniquely determined by the structure constants. Structure relations in the enveloping algebra
are continuous functions of the structure constants, so a contraction of one so(4,C) to itself
induces a contraction of the enveloping algebras. Then the free conformal quadratic algebra
constructed in the enveloping algebra will contract to another free quadratic algebra. (In [21]
essentially the same argument was given in more detail for Helmholtz superintegrable systems
on constant curvature spaces.)
In this paper we consider a family of contractions of so(4,C) to itself that we call Bôcher
contractions. All these contractions are implemented via coordinate transformations. Suppose
we have a conformal nondegenerate superintegrable system with free generators H0, S(1)
0 , S(2)
0
that determines the conformal and free conformal quadratic algebras Q and Q(0) and has struc-
ture functions Aij(x), Bij(x), Cij(x) in Cartesian coordinates x = (x, y). Further, suppose this
system contracts to another nondegenerate system H′0, S ′(1)
0 , S ′(2)
0 with conformal quadratic
algebra Q′(0). We show here that this contraction induces a contraction of the associated non-
degenerate superintegrable system H = H0 + V , S(1) = L(1)
0 + W (1), S(2) = S(2)
0 + W (2), Q to
H′ = H′0 + V ′, S ′(1) = S ′(1)
0 +W (1)′, S ′(2) = S ′(2)
0 +W (2)′, Q′. The point is that in the contrac-
tion process the symmetries H′0(ε), S ′(1)
0 (ε), S ′(2)
0 (ε) remain continuous functions of ε, linearly
independent as quadratic forms, and lim
ε→0
H′0(ε) = H′0, lim
ε→0
S ′(j)0 (ε) = S ′(j)0 . Thus the associa-
ted functions Aij(ε), Bij(ε), C(ij) will also be continuous functions of ε and lim
ε→0
Aij(ε) = A′ij ,
lim
ε→0
Bij(ε) = B′ij , lim
ε→0
Cij(ε) = C ′ij . Similarly, the integrability conditions for the potential
equations
V
(ε)
22 = V
(ε)
11 +A22(ε)V
(ε)
1 +B22(ε)V
(ε)
2 + C22(ε)V (ε),
V
(ε)
12 = A12(ε)V
(ε)
1 +B12(ε)V
(ε)
2 + C12(ε)V (ε),
will hold for each ε and in the limit. This means that the 4-dimensional solution space for the
potentials V will deform continuously into the 4-dimensional solution space for the potentials V ′.
Thus the target space of solutions V ′ (and of the functions W ′) is uniquely determined by the
free quadratic algebra contraction.
There is an apparent lack of uniqueness in this procedure, since for a nondegenerate super-
integrable system one typically chooses a basis V (j), j = 1, . . . , 4 for the potential space and
expresses a general potential as V =
4∑
j=1
ajV
(j). Of course the choice of basis for the source
system is arbitrary, as is the choice for the target system. Thus the structure equations for
the quadratic algebras and the dependence aj(ε) of the contraction constants on ε will vary
depending on these choices. However, all such possibilities are related by a basis change matrix.
3.4 Relation to separation of variables and Bôcher’s limit procedures
Bôcher’s analysis [1, 26] involves symbols of the form [n1, n2, . . . , np], where n1 + · · · + np = 4.
These symbols are used to define coordinate surfaces as follows. Consider the quadratic forms
Ω = x2
1 + x2
2 + x2
3 + x2
4 = 0, Φ =
x2
1
λ− e1
+
x2
2
λ− e2
+
x2
3
λ− e3
+
x2
4
λ− e4
= 0. (3.10)
If e1, e2, e3, e4 are pairwise distinct, the elementary divisors of these two forms are denoted by
the symbol [1, 1, 1, 1], see [2]. Given a point in 2D flat space with Cartesian coordinates (x0, y0),
Bôcher Contractions of Conformally Superintegrable Laplace Equations 15
there corresponds a set of tetraspherical coordinate (x0
1, x
0
2, x
0
3, x
0
4), unique up to multiplication
by a nonzero constant. If we substitute these coordinates into expressions (3.10) we can verify
that there are exactly 2 roots λ = ρ, µ such that Φ = 0. These are elliptic coordinates. It can
be verified that they are orthogonal with respect to the metric ds2 = dx2 + dy2 and that they
are R-separable for the Laplace equations (∂2
x + ∂2
y)Θ = 0 or
( 4∑
j=1
∂2
xj
)
Θ = 0. Now consider the
potential V[1,1,1,1] = a1
x21
+ a2
x22
+ a3
x23
+ a4
x24
. It can be verified that this is the only possible potential V
such that the Laplace equation
( 4∑
j=1
∂2
xj +V
)
Θ = 0 is R-separable in elliptic coordinates for all
choices of the parameters ej . The separation is characterized by 2nd order conformal symmetry
operators that are linear in the parameters ej . In particular the symmetries span a 3-dimensional
subspace of symmetries as the ej are varied, so the system
( 4∑
j=1
∂2
xj + V[1,1,1,1]
)
Θ = 0 must be
conformally superintegrable. We can write this as
H = (x3 + ix4)2
(
∂2
x1 + ∂2
x2 + ∂2
x3 + ∂2
x4 +
a1
x2
1
+
a2
x2
2
+
a3
x2
3
+
a4
x2
4
)
,
or in terms of flat space coordinates x, y as
H = ∂2
x + ∂2
y +
a1
x2
+
a2
y2
+
4a3
(x2 + y2 − 1)2
− 4a4
(x2 + y2 + 1)2
.
For the coordinates si, i = 1, 2, 3 we obtain
H = (1 + s3)2
(
∂2
s1 + ∂2
s2 + ∂2
s3 −
a1
s2
1
− a2
s2
2
− a3
s2
3
− a4
)
.
The coordinate curves are described by [1, 1, 1,
∞
1 ] (because we can always transform to equivalent
coordinates for which e4 = ∞) and the corresponding HΘ = 0 system is proportional to S9,
the eigenvalue equation for the generic potential on the 2-sphere, which separates variables in
elliptic coordinates s2
i = (ρ−ei)(µ−ei)
(ei−ej)(ei−ek) , where (ei − ej)(ei − ek) 6= 0 and i, j, k = 1, 2, 3. The
quantum Hamiltonian when written using these coordinates is equivalent to
H =
1
ρ− µ
[
P 2
ρ − P 2
µ −
3∑
i=1
ai
(ei − ej)(ei − ek)
(ρ− ei)(µ− ei)
]
,
where Pλ =
√
3∏
i=1
(λ− ei)∂λ.
3.5 [1, 1, 1, 1] to [2, 1, 1] contraction
Bôcher provides a recipe to derive separable coordinates in the cases, where some of the ei
become equal. In particular, Bôcher shows that the process of making e1 → e2 together with
suitable transformations of the a′is produces a conformally equivalent H. This corresponds to
the choice of coordinate curves obtained by the Bôcher limiting process [1, 1, 1, 1]→ [2, 1, 1], i.e.,
e1 = e2 + ε2, x1 → iy1
ε , x2 → y1
ε + εy2, xj → yj , j = 3, 4, which results in the pair of quadratic
forms
Ω = 2y1y2 + y2
3 + y2
4 = 0, Φ =
y2
1
(λ− e2)2
+
2y1y2
(λ− e2)
+
y2
3
(λ− e3)
+
y2
4
(λ− e4)
= 0.
16 E.G. Kalnins, W. Miller Jr. and E. Subag
The coordinate curves with e4 = ∞ correspond to cyclides with elementary divisors [2, 1,
∞
1 ],
see [2], i.e.,
Φ =
y2
1
(λ− e2)2
+
2y1y2
(λ− e2)
+
y2
3
(λ− e3)
= 0.
The λ roots of Φ yield planar elliptic coordinates. In order to identify “Cartesian” coordinates
on the cone we can choose y1 = 1√
2
(x′1 + ix′2), y2 = 1√
2
(x′1 − ix′2), y3 = x′3, y4 = x′4. Note that
the composite linear coordinate mapping
x1 + ix2 =
i
√
2
ε
(x′1 + ix′2) +
iε√
2
(x′1 − ix′2), x1 − ix2 = − iε√
2
(x′1 − ix′2),
x3 = x′3, x4 = x′4, (3.11)
satisfies lim
ε→0
4∑
j=1
x2
j =
4∑
j=1
x′2j = 0, preserving the null cone, and it induces a contraction of the
Lie algebra so(4,C) to itself. An explicit computation yields the Bôcher contraction [1, 1, 1, 1]→
[2, 1, 1]:
L′12 = L12, L′13 = − i√
2ε
(L13 − iL23)− iε√
2
L13,
L′23 = − i√
2ε
(L13 − iL23)− ε√
2
L13, L′34 = L34,
L′14 = − i√
2ε
(L14 − iL24)− iε√
2
L14, L′24 = − i√
2ε
(L14 − iL24)− ε√
2
L14.
Now under the contraction [1, 1, 1, 1]→ [2, 1, 1] we have V[1,1,1,1]
ε→0
=⇒ V[2,1,1], where
V[2,1,1] =
b1
(x′1 + ix′2)2
+
b2(x′1 − ix′2)
(x′1 + ix′2)3
+
b3
x′3
2 +
b4
x′4
2 ,
a1 = −1
2
(
b1
ε2
+
b2
2ε4
)
, a2 = − b2
4ε4
, a3 = b3, a4 = b4. (3.12)
Basis of conformal symmetries for original system: Let H0 =
4∑
j=1
∂2
xj . A basis is
{H0 + V[1,1,1,1], Q12, Q13}, where Qjk = L2
jk + aj
x2k
x2j
+ ak
x2j
x2k
, 1 ≤ j < k ≤ 4.
Contraction of basis: Using the notation of (3.12), we have
H0 + V[1,1,1,1] → H ′0 + V[2,1,1],
Q′12 = Q12 −
b1
2ε2
− b2
2ε4
= (L′12)2 + b1(
x′1 − ix′2
x′1 + ix′2
) + b2
(
x′1 − ix′2
x′1 + ix′2
)2
,
Q′13 = 2ε2Q13 = (L′23 − iL′13)2 +
b2x
′
3
2
(x′1 + ix′2)2
− b3(x′1 + ix′2)2
x′3
2 .
If we apply the same [1, 1, 1, 1]→ [2, 1, 1] contraction to the [2, 1, 1] system, the system contracts
to itself, but with parameters c1, . . . , c4, where b1 = −2c1
ε2
, b2 = c1
ε2
+ 4c2
ε4
, b3 = c3, b4 = c4.
If we apply the same contraction to the [2, 2] system, the system contracts to itself, but with
altered parameters. If we apply the same contraction to the [3, 1] system, the system contracts
Bôcher Contractions of Conformally Superintegrable Laplace Equations 17
to V (1). If we apply the same contraction to the [4] system the system contracts to a system
with potential
V [0] =
c1
(x′1 + ix′2)2
+
c2x
′
3 + c3x
′
4
(x′1 + ix′2)3
+ c4
x′23 + x′24
(x′1 + ix′2)4
.
If we apply this same contraction to the [0], (1) and (2) systems they contract to themselves,
but with altered parameters.
The remaining contractions are derived from the Bôcher recipe [1, 26].
3.6 [1, 1, 1, 1] to [2, 2] contraction
L′12 = L12, L′34 = L34, L′24 + L′13 = L24 + L13,
L′24 − L′13 =
(
ε2 +
1
ε2
)
L13 −
1
ε2
(iL14 − L24 − iL23),
L′23 − L′14 = 2L23 + iL13 − iL24,
L′23 + L′14 = i
((
ε2 − 1
ε2
)
L13 +
1
ε2
(iL14 + L24 + iL23)
)
.
Coordinate implementation:
x1 =
i√
2ε
(x′1 + ix′2), x2 =
1√
2
(
x′1 + ix′2
ε
+ ε(x′1 − ix′2)
)
,
x3 =
i√
2ε
(x′3 + ix′4), x4 =
1√
2
(
x′3 + ix′4
ε
+ ε(x′3 − ix′4)
)
.
Limit of 2D potential: V[1,1,1,1]
ε→0
=⇒ V[2,2], where
V[2,2] =
b1
(x′1 + ix′2)2
+
b2(x′1 − ix′2)
(x′1 + ix′2)3
+
b3
(x′3 + ix′4)2
+
b4(x′3 − ix′4)
(x′3 + ix′4)3
,
and a1 = −1
2
b1
ε2
− b2
4ε4
, a2 = − b2
4ε4
, a3 = −1
2
b3
ε2
− b4
4ε4
, a4 = − b4
4ε4
.
Contracted basis:
Q12 −
b2
2ε4
− b1
2ε2
→ Q′1 = L′
2
12 + b1
x′1 − ix′2
x′1 + ix′2
+ b2
(x′1 − ix′2)2
(x′1 + ix′2)2
,
4ε4Q13 → Q′2 = (L′13 + iL′14 + iL′23 − L′24)2 − b2
(x′3 + ix′4)2
(x′1 + ix′2)2
− b4
(x′1 + ix′2)2
(x′3 + ix′4)2
.
3.7 [2, 1, 1] to [3, 1] contraction
L′24 =
√
2i
2ε
(L14 + iL24)− L34, L
′
14 + iL′34 = −iε(L14 + iL24),
L′14 − iL′34 =
1
ε
(
iL14
(
1 +
1
2ε2
)
+ L24(1− 1
2ε2
)−
√
2
ε
L34
)
,
L′13 = −L12 − 2
√
2L13
(
ε+ 2ε3
)
, L′23 + iL′12 = 4ε3L13,
L′23 − iL′12 =
(
2
√
2−
√
2
ε2
)
L12 +
(
8ε3 + 4ε− 2
ε
+
1
2ε3
)
L13 +
i
2ε3
L23.
Coordinate implementation:
x1 + ix2 = − i
√
2ε
2
x′2 +
(ix′1 − x′3)
ε
, x1 − ix2 = −ε(x′3 + ix′1) +
3i
√
2x′2
4ε
+
1
2
(ix′1 − x′3)
ε3
,
18 E.G. Kalnins, W. Miller Jr. and E. Subag
x3 = −1
2
x′2 −
√
2
2
(x′1 + ix′3)
ε2
, x4 = x′4.
Limit of 2D potential: V[2,1,1]
ε→0
=⇒ V[3,1], where
V[3,1] =
c1
(x′1 + ix′3)2
+
c2x
′
2
(x′1 + ix′3)3
+
c3(4x′2
2 + x′4
2)
(x′1 + ix′3)4
+
c4
x′4
2 , (3.13)
b1 =
c3
ε6
+
√
2c2
4ε4
− c1
ε2
, b2 = −c3
ε4
−
√
2c2
2ε2
, b3 =
c3
4ε8
, b4 = c4.
Basis of conformal symmetries for original system H0 + V[2,1,1]:
Q12 = (L12)2 + b1
(
x1 − ix2
x1 + ix2
)
+ b2
(
x1 − ix2
x1 + ix2
)2
,
Q13 = (L23 − iL13)2 +
b2x3
2
(x1 + ix2)2
− b3(x1 + ix2)2
x3
2
.
Contraction of basis:
H0 + V[2,1,1] → H ′0 + V[3,1],
Q′12 = −2ε4Q12 +
c3
2ε4
− c1 = (L′12 − iL′23)2 +
c2x
′
2
x′1 + ix′3
+
4c3x
′
2
2
(x′1 + ix′3)2
,
Q′13 = −
√
2
4
(
Q13 + 2ε2Q12 −
3c3
2ε6
−
√
2c2
4ε4
+ c1
)
=
1
2
{L′13, L
′
23 + iL′12}+
c1x
′
2
x′1 + ix′3
+
c2(x′4
2 + 4x′2
2)
4(x′1 + ix′3)2
+
2c3x
′
2(x′4
2 + 2x′2
2)
(x′1 + ix′3)3
.
3.8 [1, 1, 1, 1] to [4] contraction
In this case there is a 2-parameter family of contractions, but all lead to the same result. Let A, B
be constants such that AB(1−A)(1−B)(A−B) 6= 0.
Coordinate implementation:
x1 =
i√
2ABε3
(x′1 + ix′2),
x2 =
(x′1 + ix′2) + ε2(x′3 + ix′4) + ε4(x′3 − ix′4) + ε6(x′1 − ix′2)√
2(A− 1)(B − 1)ε3
,
x3 =
(x′1 + ix′2) +Aε2(x′3 + ix′4) +A2ε4(x′3 − ix′4) +A3ε6(x′1 − ix′2)√
2A(A− 1)(A−B)ε3
,
x4 =
(x′1 + ix′2) +Bε2(x′3 + ix′4) +B2ε4(x′3 − ix′4) +B3ε6(x′1 − ix′2)√
2B(B − 1)(B −A)ε3
,
iL′14 + iL′23 + L′13 − L′24 = −2iε4
√
AB(A− 1)(B − 1)L12,
iL′14 − iL′23 − L′13 − L′24 = 2iε2
(√
B(A− 1)(A−B)L13 −
√
AB(A− 1)(B − 1)L12
)
,
L′12 =
√
AB√
(A− 1)(B − 1)
L12 +
√
B√
(A− 1)(A−B)
L13 −
i
√
A√
(B − 1)(A−B)
L14,
L′34 =
√
B(B − 1)√
A(A− 1)
L12 −
√
B(A−B)√
(A− 1)
L13 + i
√
(B − 1)(A−B)√
A
L23,
Bôcher Contractions of Conformally Superintegrable Laplace Equations 19
−iL′14 + iL′23 − L′13 − L′24 =
2
ε2
(
i(A+B − 1)√
AB(A− 1)(B − 1)
L12 +
i
√
B√
(A− 1)(A−B)
L13
−
√
A√
B(B − 1)(A−B)
L14 +
√
(B − 1)√
A(A−B)
L23 −
i
√
(A− 1)√
B(A−B)
L24
)
,
iL′14 + iL′23 − L′13 + L′24 =
2i
ε4
(
− 1√
AB(A− 1)(B − 1)
(L12 + L34)
+
i√
A(B − 1)(A−B)
(L14 + L23)− 1√
B(A− 1)(A−B)
(L13 − L24)
)
.
Limit of 2D potential: V[1,1,1,1]
ε→0
=⇒ V[4], where
V[4] =
d1
(x′1 + ix′2)2
+
d2(x′3 + ix′4)
(x′1 + ix′2)3
+ d3
(
3(x′3 + ix′4)2
(x′1 + ix′2)4
− 2
(x′1 + ix′2)(x′3 − ix′4)
(x′1 + ix′2)4
)
+ d4
4(x′1 + ix′2)(x′1
2 + x′2
2) + 2(x′3 + ix′4)3
(x′1 + ix′2)5
,
a1 = − d4
4A2B2ε12
− d3
2AB2ε10
− d2
4ABε8
− d1
2ABε6
,
a2 = − d4
4(1−A)2(1−B)2ε12
+
d3
2(1−A)(1−B)2ε10
− d2
4(1−A)(1−B)ε8
,
a3 = − d4
4A2(1−A)2(A−B)2ε12
,
a4 = − d4
4B2(1−B)2(A−B)2ε12
− d3
2B2(1−A)2(A−B)ε10
.
In these coordinates a basis for the conformal symmetry algebra is H, Q1, Q2, where
Q1 =
1
4
(L14 + L23 − iL13 + iL24)2 + 4a3
(
x1 + ix2
x3 + ix4
)
+ 4a4
(
x1 + ix2
x3 + ix4
)2
,
Q2 =
1
2
{L23 + L14 − iL13 + iL24, L12 + L34}+
1
4
(L14 − L23 + iL13 + iL24)2
+ 2a1
(
x1 + ix2
x3 + ix4
)
+ a2
(
2
x1 − ix2
x3 + ix4
−
(
x1 + ix2
x3 + ix4
)2
)
+ 2a3
(
6
(
x2
1 + x2
2
(x3 + ix4)2
)
−
(
x1 + ix2
x3 + ix4
)3
)
− 4a4
((
x1 − ix2
x3 + ix4
)2
− 3
(
(x1 + ix2)2(x1 − ix2)
(x3 + ix4)3
+
1
4
(
x1 + ix2
x3 + ix4
)4
))
.
Basis of conformal symmetries for original system: {H0 + V[1,1,1,1], Q12, Q13}, where
Qjk = (xj∂xk − xk∂xj )2 + aj
x2k
x2j
+ ak
x2j
x2k
, 1 ≤ j < k ≤ 4.
Contraction of basis: H0 + V[1,1,1,1] → H ′0 + V[4],
ε8Q12 ∼
−1
4(A− 1)(B − 1)AB
(L′13 − L′24 + iL′23 + iL′14)2 +
4d3(x′3 + ix′4)
AB(A− 1)(B − 1)(x′1 + ix′2)
+
d4
4AB(A− 1)(B − 1)
[
(x′3 + ix′4)2
(x′1 + ix′2)2
+ 2
x′3 − ix′4
x′1 + ix′2
]
,
20 E.G. Kalnins, W. Miller Jr. and E. Subag
ε5
(
Q12 −
B −A
(1−B)A
Q13
)
∼ −i
4AB(B − 1)
× {L′13 − L′24 + iL′23 + iL′14, L
′
14 + iL′13 − L′23 + iL′14}
+
(A+ 1)d1
2(B − 1)A2
+
d2
2(B − 1)AB
x′3 + ix′4
x′1 + ix′2
+
d3
2(B − 1)AB
[
3
(x′3 + ix′4)2
(x′1 + ix′2)2
− 2
x′3 − ix′4
x′1 + ix′2
]
+
d4
(B − 1)(A− 1)B
[
(x′3 + ix′4)3
(x′1 + ix′2)3
− 2
x′3
2 + x′4
2
(x′1 + ix′2)2
]
.
The second limit is equivalent to the contracted Hamiltonian, not an independent basis element.
3.9 [2, 2] to [4] contraction
L′12 = i
(
1 +
2
ε
− 1
2ε2
)
L12 +
1
ε
(
1− 3
4ε
+
1
4ε2
)
L13 +
i
4ε2
(
3− 1
ε
)
L14
+
i
4ε2
(
3− 1
ε
)
L23 +
(
3− ε+
3
4ε2
− 1
4ε3
)
L24 + i
(
3ε
2
− 2 +
1
ε
− 1
2ε2
)
L34,
L′12 + iL′24 = ε(L13 − iL14), L′13 + iL′34 = ε(L23 − iL24),
L′14 = (−1 + ε)L12 + i(1− ε)L13 + (1 + ε)L14, L′23 − L′14 = −L14 + L23,
L′13 + L′24 =
(
1
2
− 1
ε
)
L12 +
i
ε
L13 +
1
2
L14 +
1
2
L23 +
(
2 +
i
ε
)
L24 +
(
ε− 1
2
+
1
ε
)
L34.
Coordinate implementation:
x1 =
1
2
(
1
ε
+
1
ε2
)
(x′1 − ix′4) +
ε
2
(x′1 + ix′4)−
(
1 +
1
2ε
)
(x′2 − ix′3) +
1
2
(ε− 1)(x′2 + ix′3),
x2 =
i
2
(
1
ε
− 1
ε2
)
(x′1 − ix′4)− iε
2
(x′1 + ix′4)− i
(
1− 1
2ε
)
(x′2 − ix′3) +
i
2
(ε+ 1)(x′2 + ix′3),
x3 =
1
2
(
1
ε
− 1
ε2
)
(x′1 − ix′4) +
(
−1
2
+
1
ε
)
(x′2 − ix′3),
x4 =
i
2
(
1
ε
+
1
ε2
)
(x′1 − ix′4)− i
(
1
2
+
1
ε
)
(x′2 − ix′3).
Limit of 2D potential: V[2,2]
ε→0
=⇒ V ′[4],
V ′[4] =
e1
(x′1 − ix′4)2
+
e2(x′2 − ix′3)
(x′1 − ix′4)3
+ e3
(
3(x′2 − ix′3)2
(x′1 − ix′4)4
+ 2
(x′1 − ix′4)(x′2 + ix′3)
(x′1 − ix′4)4
)
+ e4
(
4(x′1 − ix′4)(x′2
2 + x′3
2) + 2(x′2 − ix′3)3
(x′1 − ix′4)5
)
(conformally equivalent to V [4]),
b1 =
e1
ε4
+ 2
e4
ε7
, b2 = − e2
4ε6
− e3
2ε7
− e4
ε8
, b3 = 2
e3
ε6
− 2
e4
ε7
, b4 = − e2
4ε6
+
3e3
2ε7
− e4
ε8
.
Basis of conformal symmetries for original system: {H0 + V[2,2], Q1, Q3}.
Contraction of basis: H0 + V[2,2] → H ′0 + V ′[4],
−4ε4
(
Q1 +
k4
ε6
− k3
2ε5
)
→ (iL′13 − L′12 − iL′24 − L′34)2
Bôcher Contractions of Conformally Superintegrable Laplace Equations 21
+ k2 + 4k3
x′2 − ix′3
x′1 − ix′4
− 4k4
(x′2 − ix′3)2
(x′1 − ix′4)2
,
ε3
(
Q3 −
2k4
ε7
+
k3
ε6
+
k1
2ε4
)
→ i
2
{L′23 − L′14, (L
′
12 − iL′13 + L′24 + L′34}
+ k1
(x′2 − ix′3)
(x′1 − ix′4)
+ k2
(x′2 − ix′3)2
(x′1 − ix′4)2
+ k3
3(x′2 − ix′3)3 + 2(x′2
2 + x′3
2)(x′1 − ix′4)
(x′1 − ix′4)3
− 2k4(x′2 − ix′3)
(x′2 − ix′3)3 + 2(x′2
2 + x′3
2)(x′1 − ix′4)
(x′1 − ix′4)4
.
The second limit is equivalent to the contracted Hamiltonian, not an independent basis element.
3.10 [3, 1] to [4] contraction
This contraction is not needed because the [1, 1, 1, 1]→ [4] contraction takes the V [3, 1] to V [4].
3.11 [2, 1, 1] to [4] contraction
This contraction is not needed because the [1, 1, 1, 1]→ [4] contraction takes V [2, 1, 1] to V [4].
3.12 [1, 1, 1, 1] to [3, 1] contraction
−L′12 + iL′24 = −a
√
2a2 − 2εL12, L′13 = − i√
a2 − 1
(L13 + aL12),
L′14 + iL′34 =
√
2aεL14, −L′12 + iL′23 = i
√
2aεL23, L′24 = i(
√
a2 − 1L24 − iaL14),
−L′14 + iL′34 =
√
2
εa
√
a2 − 1
(
L34 −
√
a2 − 1L14 − iaL24
)
.
Coordinate implementation:
x1 =
1√
2aε
(x′1 + ix′3) +
x′2
a
+
aε√
2
(x′1 − ix′3), x2 =
i(x′1 + ix′3)√
2a2 − 2ε
,
x3 = − (x′1 + ix′3)√
2a2 − 2aε
+
√
a2 − 1
a
x′2, x4 = x′4, a(a− 1) 6= 0.
Limit of 2D potential: V[1,1,1,1]
ε→0
=⇒ V[31], where V [31] is given by (3.13) and
a1 =
c1
2ε2
+
c3
4a4ε4
, a2 =
c2
4
√
2(a2 − 1)2ε3
+
c3
4(a2 − 1)2ε4
,
a3 =
c2
4
√
2(a2 − 1)2a2ε3
+
c3
4(a2 − 1)2a4ε4
, a4 = c4.
Basis of conformal symmetries for original system: H0 + V[1,1,1,1], Q12, Q13, where
Qjk = (xj∂xk − xk∂xj )
2 + aj
x2
k
x2
j
+ ak
x2
j
x2
k
, 1 ≤ j < k ≤ 4.
Contracted basis: H0 + V[1,1,1,1] → H ′0 + V[3,1],
ε2
(
Q12 +
c3
2a2(a2 − 1)ε4
+
√
2c2
a2(a2 − 1)ε3
)
→ − c1
2(a2 − 1)
22 E.G. Kalnins, W. Miller Jr. and E. Subag
− 2c3x
′
2
2
a2(a2 − 1)(x′1 + ix′3)2
− c2
2a2(a2 − 1)(x′1 + ix′3)
− 1
2a2(a2 − 1)
(L′12 − iL′23)2,
ε
(
Q13 + a2Q12 +
(a2 − 1)c3
2a4ε4
+
√
2c2
8a2ε3
+
c1(a2 − 1)
2ε2
)
→
√
2c1x
′
2
x′1 + ix′3
+
√
2c2(4x′2
2 + x′4
2)
4(x′1 + ix′3)2
+
2
√
2c3x
′
2(2x′2
2 + x′4
2)
(x′1 + ix′3)3
+
i
√
2
2
{L′13, L
′
12 − iL′23}.
3.13 [2, 2] to [4] contraction
L′12 = i
(
1 +
2
ε
− 1
2ε2
)
L12 +
1
ε
(
1− 3
4ε
+
1
4ε2
)
L13 +
i
4ε2
(
3− 1
ε
)
L14
+
i
4ε2
(
3− 1
ε
)
L23 +
(
3− ε+
3
4ε2
− 1
4ε3
)
L24 + i
(
3ε
2
− 2 +
1
ε
− 1
2ε2
)
L34,
L′12 + iL′24 = ε(L13 − iL14), L′13 + iL′34 = ε(L23 − iL24),
L′14 = (−1 + ε)L12 + i(1− ε)L13 + (1 + ε)L14, L′23 − L′14 = −L14 + L23,
L′13 + L′24 =
(
1
2
− 1
ε
)
L12 +
i
ε
L13 +
1
2
L14 +
1
2
L23 +
(
2 +
i
ε
)
L24 +
(
ε− 1
2
+
1
ε
)
L34.
Coordinate implementation:
x1 =
1
2
(
1
ε
+
1
ε2
)
(x′1 − ix′4) +
ε
2
(x′1 + ix′4)−
(
1 +
1
2ε
)
(x′2 − ix′3) +
1
2
(ε− 1)(x′2 + ix′3),
x2 =
i
2
(
1
ε
− 1
ε2
)
(x′1 − ix′4)− iε
2
(x′1 + ix′4)− i
(
1− 1
2ε
)
(x′2 − ix′3) +
i
2
(ε+ 1)(x′2 + ix′3),
x3 =
1
2
(
1
ε
− 1
ε2
)
(x′1 − ix′4) +
(
−1
2
+
1
ε
)
(x′2 − ix′3),
x4 =
i
2
(
1
ε
+
1
ε2
)
(x′1 − ix′4)− i
(
1
2
+
1
ε
)
(x′2 − ix′3).
Limit of 2D potential: V[2,2]
ε→0
=⇒ V ′[4]. Conformally equivalent to V [4],
V ′[4] =
e1
(x′1 − ix′4)2
+
e2(x′2 − ix′3)
(x′1 − ix′4)3
+ e3
(
3(x′2 − ix′3)2
(x′1 − ix′4)4
+ 2
(x′1 − ix′4)(x′2 + ix′3)
(x′1 − ix′4)4
)
+ e4
(
4(x′1 − ix′4)(x′2
2 + x′3
2) + 2(x′2 − ix′3)3
(x′1 − ix′4)5
)
,
b1 =
e1
ε4
+ 2
e4
ε7
, b2 = − e2
4ε6
− e3
2ε7
− e4
ε8
, b3 = 2
e3
ε6
− 2
e4
ε7
, b4 = − e2
4ε6
+
3e3
2ε7
− e4
ε8
.
Basis of conformal symmetries for original system: {H0 + V[2,2], Q1, Q3}.
Contraction of basis: H0 + V[2,2] → H ′0 + V ′[4],
−4ε4
(
Q1 +
k4
ε6
− k3
2ε5
)
→ (iL′13 − L′12 − iL′24 − L′34)2
+ k2 + 4k3
x′2 − ix′3
x′1 − ix′4
− 4k4
(x′2 − ix′3)2
(x′1 − ix′4)2
,
ε3
(
Q3 −
2k4
ε7
+
k3
ε6
+
k1
2ε4
)
→ i
2
{L′23 − L′14, (L
′
12 − iL′13 + L′24 + L′34}
+ k1
(x′2 − ix′3)
(x′1 − ix′4)
+ k2
(x′2 − ix′3)2
(x′1 − ix′4)2
+ k3
3(x′2 − ix′3)3 + 2(x′2
2 + x′3
2)(x′1 − ix′4)
(x′1 − ix′4)3
Bôcher Contractions of Conformally Superintegrable Laplace Equations 23
− 2k4(x′2 − ix′3)
(x′2 − ix′3)3 + 2(x′2
2 + x′3
2)(x′1 − ix′4)
(x′1 − ix′4)4
.
The second limit is equivalent to the contracted Hamiltonian, not an independent basis element.
3.14 Summary of Bôcher contractions of Laplace systems
This is a summary of the results of applying each of the Bôcher contractions to each of the Laplace
conformally superintegrable systems. In many cases a single contraction gives rise to rise to more
than one result, due to the fact that the indices of the image potential can be permuted and
image potential may not be permutation invariant. The details can be found in [27].
1. [1, 1, 1, 1]→ [2, 1, 1] contraction:
V[1,1,1,1] ↓ V[2,1,1]; V[2,1,1] ↓ V[2,1,1], V[2,2], V[3,1]; V[2,2] ↓ V[2,2], V[0];
V[3,1] ↓ V(1), V[3,1]; V[4] ↓ V[0], V(2); V[0] ↓ V[0]; V(1) ↓ V(1), V(2); V(2) ↓ V(2).
2. [1, 1, 1, 1]→ [2, 2] contraction:
V[1,1,1,1] ↓ V[2,2]; V[2,1,1] ↓ V[2,2] (special case of E15); V[2,2] ↓ V[2,2], V[0];
V[3,1] ↓ V(1) (special case of E15); V[4] ↓ V(2); V[0] ↓ V[0];
V(1) ↓ V(1) (special case of E15); V(2) ↓ V(2).
3. [2, 1, 1]→ [3, 1] contraction:
V[1,1,1,1] ↓ V[3,1]; V[2,1,1] ↓ V[3,1], V[0]; V[2,2] ↓ V[0]; V[3,1] ↓ V[3,1], V[0]; V[4] ↓ V[0];
V[0] ↓ V[0]; V(1) ↓ V(2); V(2) ↓ V(2).
4. [1, 1, 1, 1]→ [4] contraction:
V[1,1,1,1] ↓ V[4]; V[2,1,1] ↓ V[4]; V[2,2] ↓ V[0]; V[3,1] ↓ V[4]; V[4] ↓ V[0], V[4];
V[0] ↓ V[0]; V(1) ↓ V(2); V(2) ↓ V(2).
5. [2, 2]→ [4] contraction:
V[1,1,1,1] ↓ V[4]; V[2,1,1] ↓ V[4], V(2); V[2,2] ↓ V[4], V[0]; V[3,1] ↓ V(2); V[4] ↓ V(2);
V[0] ↓ V[0], V(2); V(1) ↓ V(2); V(2) ↓ V(2).
6. [1, 1, 1, 1]→ [3, 1] contraction:
V[1,1,1,1] ↓ V[3,1]; V[2,1,1] ↓ V[3,1], V[0]; V[2,2] ↓ V[0]; V[3,1] ↓ V[3,1], V[0];
V[4] ↓ V[0]; V[0] ↓ V[0]; V(1) ↓ V(2); V(2) ↓ V(2).
3.15 Conformal Stäckel transforms of the Laplace systems
We give the details of the description of the Helmholtz systems that follow from the Laplace
system [1, 1, 1, 1] by conformal Stäckel transform
V[1,1,1,1] =
a1
x2
1
+
a2
x2
2
+
a3
x2
3
+
a4
x2
4
.
We write the parameters aj defining the potential V[1,1,1,1] as a vector: (a1, a2, a3, a4). A Stäckel
transform is generated by the potential
U =
b1
x2
1
+
b2
x2
2
+
b3
x2
3
+
b4
x2
4
corresponding to the vector (b1, b2, b3, b4).
24 E.G. Kalnins, W. Miller Jr. and E. Subag
1. The potentials (1, 0, 0, 0), and any permutation of the indices bj generate conformal Stäckel
transforms to S9.
2. The potentials (1, 1, 0, 0) and (0, 0, 1, 1) generate conformal Stäckel transforms to S7.
3. The potentials (1, 1, 1, 1), (0, 1, 0, 1), (1, 0, 1, 0), (0, 1, 1, 0) and (1, 0, 0, 1) generate conformal
Stäckel transforms to S8.
4. The potentials (b1, b2, 0, 0), b1b2 6= 0, b1 6= b2, and any permutation of the indices bj
generate conformal Stäckel transforms to D4B.
5. The potentials (1, 1, a, a), a 6= 0, 1, and any permutation of the indices bj . generate
conformal Stäckel transforms to D4C.
6. Each potential not proportional to one of these must generate a conformal Stäckel trans-
form to a superintegrable system on a Koenigs space in the family K[1, 1, 1, 1].
Similar details for all of the other Laplace systems are given in [27]. Here, we simply list the
Helmholtz systems in each equivalence class.
3.16 Summary of Stäckel equivalence classes of Helmholtz systems
1. [1, 1, 1, 1]: S9, S8, S7, D4B, D4C, K[1, 1, 1, 1].
2. [2, 1, 1]: S4, S2, E1, E16, D4A, D3B, D2B, D2C, K[2, 1, 1].
3. [2, 2]: E8, E17, E7, E19, D3C, D3D, K[2, 2].
4. [3, 1]: S1, E2, D1B, D2A, K[3, 1].
5. [4]: E10, E9, D1A, K[4].
6. [0]: E20, E11, E3′, D1C, D3A, K[0].
7. (1): special cases of E15.
8. (2): special cases of E15.
4 Helmholtz contractions from Bôcher contractions
We describe how Bôcher contractions of conformal superintegrable systems induce contractions
of Helmholtz superintegrable systems. The basic idea here is that the procedure of taking a con-
formal Stäckel transform of a conformal superintegrable system, followed by a Helmholtz con-
traction yields the same result as taking a Bôcher contraction followed by an ordinary Stäckel
transform: The diagrams commute [28]. To describe this process we recall that each of the
Bôcher systems classified above can be considered as an equivalence class of Helmholtz super-
integrable systems under the Stäckel transform. We now determine the Helmholtz systems in
each equivalence class and how they are related.
Consider the conformal Stäckel transforms of the conformal system [1, 1, 1, 1] with poten-
tial V[1,1,1,1]. The various possibilities are listed in Section 3.15. Let H be the initial Hamil-
tonian. In terms of tetraspherical coordinates the conformal Stäckel transformed potential will
take the form
V =
a1
x21
+ a2
x22
+ a3
x23
+ a4
x24
A1
x21
+ A2
x22
+ A3
x23
+ A4
x24
=
V[1,1,1,1]
F (x,A)
, F (x,A) =
A1
x2
1
+
A2
x2
2
+
A3
x2
3
+
A4
x2
4
,
and the transformed Hamiltonian will be Ĥ = 1
F (x,A)H, where the transform is determined by
the fixed vector (A1, A2, A3, A4). Now we apply the Bôcher contraction [1, 1, 1, 1] → [2, 1, 1] to
Bôcher Contractions of Conformally Superintegrable Laplace Equations 25
this system. In the limit as ε → 0 the potential V[1,1,1,1] → V[2,1,1], (3.12), and H → H ′ of the
[2, 1, 1] system. Now consider F (x(ε),A) = V ′(x′, A)εα + O(εα+1), where the integer exponent
α depends upon our choice of A. We will provide the theory to show that the system defined
by Hamiltonian Ĥ ′ = lim
ε→0
εαĤ(ε) = 1
V ′(x′,A)H
′ is a superintegrable system that arises from the
system [2, 1, 1] by a conformal Stäckel transform induced by the potential V ′(x′, A). Thus the
Helmholtz superintegrable system with potential V = V1,1,1,1/F contracts to the Helmholtz
superintegrable system with potential V[2,1,1]/V
′. The contraction is induced by a generalized
Inönü–Wigner Lie algebra contraction of the conformal algebra so(4,C). In this case the pos-
sibilities for V ′ can be computed easily from the limit expressions (3.11). Then the V ′ can be
identified with a [2, 1, 1] potential from the list in Section 3.2. The results follow. For each A
corresponding to a constant curvature or Darboux superintegrable system O we list the con-
tracted system O′ and α. For Koenigs spaces we will not go into detail but merely give the
contraction for a “generic” Koenigs system: One for which there are no rational numbers rj , not
all 0, such that
4∑
j=1
rjAj = 0. This ensures that the contraction is also “generic”. The schematic
to keep in mind that relates conformal and regular Stäckel transforms, Bôcher contractions,
Helmholtz and Laplace superintegrable systems is Fig. 1.
Figure 1. The bigger picture.
26 E.G. Kalnins, W. Miller Jr. and E. Subag
Example 4.1. In Section 3.15, consider Stäckel transform (1, 0, 0, 0), i.e., U = 1/x2
1. The
transformed system is
H =
1
1
x21
(
4∑
i=1
∂2
xi
)
+
1
1
x21
(
a1
x2
1
+
a2
x2
2
+
a3
x2
3
+
a4
x2
4
)
,
which is S9. Now take the [1, 1, 1, 1]→ [2, 1, 1] Bôcher contraction, equation (3.12). The sum of
the derivatives in H goes to
4∑
i=1
∂2
x′i
and the numerator of the potential goes to equation (3.12).
However, the denominator 1/x2
1 goes as 1/x2
1 = −2ε2/((x′1 + ix′2)2 + O(ε6), so α = 2. Thus,
if we set H ′ = ε2H and go to the limit as ε → 0, we get a contracted system with potential
b1 + b2(x2 + y2) + b3/x
2 + b4/y
2 in Cartesian coordinates, up to a scalar factor −2. This is E1.
The complicated details of the possible Helmholtz contractions induced by Bôcher contrac-
tions of Laplace systems are presented in [27]. Here, we summarize the results. In many cases
a single contraction gives rise to more than one result, due to the fact that the indices of the
image potential can be permuted and image potential may not be permutation invariant.
4.1 Summary of Helmholtz contractions
The superscript for each targeted Helmholtz system is the value of α. In each table, correspon-
ding to a single Laplace equation equivalence class, the top line is a list of the Helmholtz systems
in the class, and the lower lines are the target systems under the Bôcher contraction.
Table 1. [1, 1, 1, 1] equivalence class contractions.
contraction S9 S7 S8 D4B D4C K[1111]
[1111] ↓ [211] E2
1 S0
4 S0
4 E2
1 S0
4 D4A
0
S0
2 S0
2 E0
16 D4A
0 D4A
0
S0
2
[1111] ↓ [22] E2
7 E4
19 E4
17 E2
7 E1
19 E2
7
E2
7 E4
19
E2
17
[1111] ↓ [31] E2
2 S0
1 S0
1 S0
1 S0
1 S0
1
S0
1 E2
2 E2
2 E2
2
[1111] ↓ [4] E6
10 E6
10 E6
10 E6
10 E6
10 E6
10
[22] ↓ [4] E4
10 E6
9 E5
10 E4
10 E5
10 E4
10
E4
10
E5
9
[211] ↓ [31] E6
2 S0
1 S0
1 S0
1 S0
1 S0
1
E4
2 E4
2 E8
2 E6
2
S0
1 E4
2
5 Conclusions and discussion
The use of Lie algebra contractions based on the symmetry groups of constant curvature spaces
to construct quadratic algebra contractions of 2nd order 2D Helmholtz superintegrable systems
is esthetically pleasing but incomplete, because it doesn’t satisfactorily account for Darboux
and Koenigs spaces. Also the hierarchy of contractions is confusing. The situation is clarified
Bôcher Contractions of Conformally Superintegrable Laplace Equations 27
Table 2. [2, 1, 1] equivalence class contractions.
contraction S4 S2 E1 E16 D4A D3B D2B D2C K[211]
[1111] ↓ [211] S0
4 S0
2 E2
1 E4
16 D4A
0 E2
1 S0
2 S0
4 S0
4
E4
17 E2
8 E0
8 E0
17 E2
8 D3C
0 E0
8 E0
17 D3C
0
S0
1 S0
1 E2
2 E2
2 S0
1 E2
2 S0
1 S0
1 S0
1
E2
2 D1B
3 E2
2
[1111] ↓ [22] E4
17 E2
8 E2
8 E4
17 E2
7 E2
8 E2
7 E4
19 E2
7
E2
8 E2
17 E2
8 E4
17
[1111] ↓ [31] S0
1 S0
1 E2
2 E2
2 S0
1 E2
2 E2
1 S0
1 S0
1
D1B
3
E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2
D1C
3 D1C
3 D1C
3
[1111] ↓ [4] E6
10 E6
10 E6
10 E6
10 E6
10 E6
10 E6
10 E6
10 E6
10
E8
9 E8
9 E8
9 E8
9
[22] ↓ [4] E5
10 E4
10 E4
10 E5
10 E4
10 E4
10 E4
10 E4
10 E4
10
E5
10 E5
10
Stäckel transforms of V (2)
[211] ↓ [31] S0
1 S0
1 E6
2 E8
2 S0
1 E6
2 S0
1 S0
1 S0
1
E5
2 E5
2
E′3
8 E′3
6 E′3
4 E′3
4 E′3
6 E′3
6 E′3
4 E′3
4 E′3
4
Table 3. [2, 2] equivalence class contractions.
contraction E8 E17 E7 E19 D3C D3D K[22]
[1111] ↓ [211] E0
8 E0
17 E0
7 E0
19 D3C
0 E2
7 D3C
0
E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2
[1111] ↓ [22] E2
8 E4
17 E2
7 E4
19 E2
8 E2
8 E2
7
E′3
2 E2
11 E′3
2 E2
11 E2
11 E2
11 E2
11
[1111] ↓ [31] E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2
E4
11 D1C
3 D1C
3
E4
20
[1111] ↓ [4] E′63 E′63 E′63 E′63 E′63 E′63 E′63
E8
11 E8
11 E8
11 E8
11
[22] ↓ [4] E4
10 E5
10 E4
10 E5
10 E4
10 E4
10 E4
10
E5
9 E6
9
E′3
2 E1
11 E′3
2 E1
11 E1
11 E1
11 E1
11
E3
11 E4
20
[211] ↓ [31] E′3
4 E′43 E′3
2 E′3
2 E′3
4 D1C
2 D1C
2
E′3
6 E′3
6 E′3
6 E20
4 E′3
6 E′3
6 E′3
6
D1C
9
when one extends these systems to 2nd order Laplace conformally superintegrable systems with
conformal symmetry algebra. Classes of Stäckel equivalent Helmholtz superintegrable systems
are now recognized as corresponding to a single Laplace superintegrable system on flat space
with underlying conformal symmetry algebra so(4,C). The conformal Lie algebra contractions
are induced by Bôcher limits of so(4,C) to itself associated with invariants of quadratic forms.
28 E.G. Kalnins, W. Miller Jr. and E. Subag
Table 4. [3, 1] equivalence class contractions.
contraction S1 E2 D1B D2A K[31]
[1111] ↓ [211] Stäckel transforms of V (1)
S0
1 E2
2 E2
2 E2
2 S0
1
D1B
3 D2A
4
[1111] ↓ [22] Stäckel transforms of V (1)
[1111] ↓ [31] S0
1 E2
2 E2
2 E2
2 S0
1
D1B
3
E′3
2 E′3
2 E′3
2 E′3
2 E′3
2
D1C
3
[1111] ↓ [4] E6
10 E6
10 E6
10 E6
10 E6
10
E8
9
[22] ↓ [4] Stäckel transforms of V (2)
[211] ↓ [31] S0
1 E6
2 E6
2 E6
2 S0
1
E2
2 S1
1 S0
1
E′3
4 E′3
6 E′3
6 E′3
6 E′3
4
Table 5. [4] equivalence class contractions.
contraction E10 E9 D1A K[4]
[1111] ↓ [211] E′3
2 E2
11 E4
20 E′3
2
E′3
2 E′3
2
Stäckel transforms of V (2)
[1111] ↓ [22] Stäckel transforms of V (2)
E′3
2 E′3
2 D1C
2 D3A
2
[1111] ↓ [31] E′3
2 E′3
2 E′3
2 E′3
2
E2
11
[1111] ↓ [4] E′3
6 E′3
6 E′3
6 E′3
6
E8
11
E6
10 E6
10 E6
10 E6
10
E8
9
[22] ↓ [4] Stäckel transforms of V (2)
[211] ↓ [31] E′3
1 E′3
1 E′3
−1 E′3
−1
E′3
4 E′3
5 E′3
4 E′3
3
E′3
6 E′3
6 E′3
6 E′3
6
Except for one special class they generalize all of the Helmholtz contractions derived earlier.
In particular, contractions of Darboux and Koenigs systems can be described easily. All of the
concepts introduced in this paper are clearly also applicable for dimensions n ≥ 3, see [4]. The
conceptual picture is Fig. 1.
The special class that is missing in the present paper is the class of contractions to systems
with degenerate Hamiltonians, i.e., systems for which the determinant of the metric tensor is
zero. In a paper under preparation we will show that these limits correspond to contractions of
so(4,C) to e(3,C) and lead to time-dependent conformally superintegrable systems (Schrödinger
equations) with potential. We will examine the relations between the contractions in classified
in [8, 21] and show that they are properly contained in those induced by so(4,C). From Theo-
rem 1.1 we know that the potentials of all Helmholtz superintegrable systems are completely
Bôcher Contractions of Conformally Superintegrable Laplace Equations 29
Table 6. [0] equivalence class contractions.
contraction E20 E11 E′3 D1C D3A K[0]
[1111] ↓ [211] E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2
E3
11 E3
11 D1C
3 D1C
3
[1111] ↓ [22] E2
11 E2
11 E′3
2 E2
11 E2
11 E2
11
E′3
2 E′3
2
[1111] ↓ [31] E′3
2 E′3
2 E′3
2 E′3
2 E′3
2 E′3
2
D1C
3 D1C
3
[1111] ↓ [4] E′3
6 E′3
6 E′3
6 E′3
6 E′3
6 E′3
6
E8
11 E8
11 E8
11 E8
11
[22] ↓ [4] E′3
4 E′3
4 E′3
4 E′3
4 E′3
4 E′3
4
E5
11 E5
11 E5
11
[211] ↓ [31] E′3
6 E′3
6 E′3
6 E′3
6 E′3
6 E′3
6
D1C
9
determined by their free quadratic algebras, i.e., the symmetry algebra that remains when the
parameters in the potential are set equal to 0. Thus for classification purposes it is enough to
classify free abstract quadratic algebras. We will give a classification of abstract free nonde-
generate quadratic algebras and their abstract contractions and discuss which of these abstract
systems and contractions correspond to physical systems.
In papers under preparation we will 1) give a precise definition of Böcher contractions and
introduce other methods of constructing them, 2) apply the Bôcher construction to degenerate
(1-parameter) Helmholtz superintegrable systems (which admit a 1st order symmetry), 3) give
a complete classification of free abstract degenerate quadratic algebras and identify which of
those correspond to free 2nd order superintegrable systems, 4) classify abstract contractions of
degenerate quadratic algebras and identify which of those correspond to geometric contractions
of Helmholtz superintegrable systems.
We note that by taking contractions step-by-step from a model of the S9 quadratic algebra we
can recover the Askey scheme [25]. However, the contraction method is more general. It applies
to all special functions that arise from the quantum systems via separation of variables, not just
polynomials of hypergeometric type, and it extends to higher dimensions. The functions in the
Askey scheme are just those hypergeometric polynomials that arise as the expansion coefficients
relating two separable eigenbases that are both of hypergeometric type. Thus, there are some con-
tractions which do not fit in the Askey scheme since the physical system fails to have such a pair
of separable eigenbases. In another paper we will analyze the Laplace 2nd order conformally
superintegrable systems, determine which of them is exactly solvable or quasi-exactly solvable,
identify the spaces of polynomials that arise and examine their behavior under contraction.
Acknowledgements
This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard
Miller Jr).
References
[1] Bôcher M., Ueber die Reihenentwickelungen der Potentialtheorie, B.G. Teubner, Leipzig, 1894.
[2] Bromwich T.J.I., Quadratic forms and their classification by means of invariant-factors, Cambridge Univer-
sity Press, Cambridge, 1906.
30 E.G. Kalnins, W. Miller Jr. and E. Subag
[3] Capel J.J., Kress J.M., Invariant classification of second-order conformally flat superintegrable systems,
J. Phys. A: Math. Theor. 47 (2014), 495202, 33 pages, arXiv:1406.3136.
[4] Capel J.J., Kress J.M., Post S., Invariant classification and limits of maximally superintegrable systems in
3D, SIGMA 11 (2015), 038, 17 pages, arXiv:1501.06601.
[5] Daskaloyannis C., Tanoudis Y., Quantum superintegrable systems with quadratic integrals on a two dimen-
sional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058.
[6] Evans N.W., Super-integrability of the Winternitz system, Phys. Lett. A 147 (1990), 483–486.
[7] Fordy A.P., Quantum super-integrable systems as exactly solvable models, SIGMA 3 (2007), 025, 10 pages,
math-ph/0702048.
[8] Heinonen R., Kalnins E.G., Miller Jr. W., Subag E., Structure relations and Darboux contractions for 2D
2nd order superintegrable systems, SIGMA 11 (2015), 043, 33 pages, arXiv:1502.00128.
[9] Inönü E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA
39 (1953), 510–524.
[10] Izmest’ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation
of variables, J. Phys. A: Math. Gen. 29 (1996), 5949–5962.
[11] Izmest’ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and the separa-
tion of variables: interbase expansions, J. Phys. A: Math. Gen. 34 (2001), 521–554.
[12] Kalnins E.G., Kress J.M., Miller Jr. W., Second-order superintegrable systems in conformally flat spaces.
I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
[13] Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces.
II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
[14] Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces.
III. Three-dimensional classical structure theory, J. Math. Phys. 46 (2005), 103507, 28 pages.
[15] Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces.
IV. The classical 3D Stäckel transform and 3D classification theory, J. Math. Phys. 47 (2006), 043514,
26 pages.
[16] Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces.
V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
[17] Kalnins E.G., Kress J.M., Miller Jr. W., Nondegenerate 2D complex Euclidean superintegrable systems and
algebraic varieties, J. Phys. A: Math. Theor. 40 (2007), 3399–3411, arXiv:0708.3044.
[18] Kalnins E.G., Kress J.M., Miller Jr. W., Post S., Laplace-type equations as conformal superintegrable
systems, Adv. in Appl. Math. 46 (2011), 396–416, arXiv:0908.4316.
[19] Kalnins E.G., Kress J.M., Miller Jr. W., Winternitz P., Superintegrable systems in Darboux spaces, J. Math.
Phys. 44 (2003), 5811–5848, math-ph/0307039.
[20] Kalnins E.G., Kress J.M., Pogosyan G.S., Miller Jr. W., Completeness of superintegrability in two-
dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705–4720, math-ph/0102006.
[21] Kalnins E.G., Miller Jr. W., Quadratic algebra contractions and second-order superintegrable systems, Anal.
Appl. (Singap.) 12 (2014), 583–612, arXiv:1401.0830.
[22] Kalnins E.G., Miller Jr. W., Post S., Wilson polynomials and the generic superintegrable system on the
2-sphere, J. Phys. A: Math. Theor. 40 (2007), 11525–11538.
[23] Kalnins E.G., Miller Jr. W., Post S., Models for quadratic algebras associated with second order superinte-
grable systems in 2D, SIGMA 4 (2008), 008, 21 pages, arXiv:0801.2848.
[24] Kalnins E.G., Miller Jr. W., Post S., Coupling constant metamorphosis and Nth-order symmetries in classical
and quantum mechanics, J. Phys. A: Math. Theor. 43 (2010), 035202, 20 pages, arXiv:0908.4393.
[25] Kalnins E.G., Miller Jr. W., Post S., Contractions of 2D 2nd order quantum superintegrable systems and the
Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013), 057, 28 pages, arXiv:1212.4766.
[26] Kalnins E.G., Miller Jr. W., Reid G.J., Separation of variables for complex Riemannian spaces of constant
curvature. I. Orthogonal separable coordinates for SnC and EnC, Proc. Roy. Soc. London Ser. A 394 (1984),
183–206.
[27] Kalnins E.G., Miller Jr. W., Subag E., Bôcher contractions of conformally superintegrable Laplace equations:
detailed computations, arXiv:1601.02876.
http://dx.doi.org/10.1088/1751-8113/47/49/495202
http://arxiv.org/abs/1406.3136
http://dx.doi.org/10.3842/SIGMA.2015.038
http://arxiv.org/abs/1501.06601
http://dx.doi.org/10.1063/1.2746132
http://arxiv.org/abs/math-ph/0607058
http://dx.doi.org/10.1016/0375-9601(90)90611-Q
http://dx.doi.org/10.3842/SIGMA.2007.025
http://arxiv.org/abs/math-ph/0702048
http://dx.doi.org/10.3842/SIGMA.2015.043
http://arxiv.org/abs/1502.00128
http://dx.doi.org/10.1088/0305-4470/29/18/024
http://dx.doi.org/10.1088/0305-4470/34/3/314
http://dx.doi.org/10.1063/1.1897183
http://dx.doi.org/10.1063/1.1894985
http://dx.doi.org/10.1063/1.2037567
http://dx.doi.org/10.1063/1.2191789
http://dx.doi.org/10.1063/1.2337849
http://dx.doi.org/10.1088/1751-8113/40/13/008
http://arxiv.org/abs/0708.3044
http://dx.doi.org/10.1016/j.aam.2009.11.014
http://arxiv.org/abs/0908.4316
http://dx.doi.org/10.1063/1.1619580
http://dx.doi.org/10.1063/1.1619580
http://arxiv.org/abs/math-ph/0307039
http://dx.doi.org/10.1088/0305-4470/34/22/311
http://arxiv.org/abs/math-ph/0102006
http://dx.doi.org/10.1142/S0219530514500377
http://dx.doi.org/10.1142/S0219530514500377
http://arxiv.org/abs/1401.0830
http://dx.doi.org/10.1088/1751-8113/40/38/005
http://dx.doi.org/10.3842/SIGMA.2008.008
http://arxiv.org/abs/0801.2848
http://dx.doi.org/10.1088/1751-8113/43/3/035202
http://arxiv.org/abs/0908.4393
http://dx.doi.org/10.3842/SIGMA.2013.057
http://arxiv.org/abs/1212.4766
http://dx.doi.org/10.1098/rspa.1984.0075
http://arxiv.org/abs/1601.02876
Bôcher Contractions of Conformally Superintegrable Laplace Equations 31
[28] Kalnins E.G., Miller Jr. W., Subag E., Laplace equations, conformal superintegrability and Bôcher contrac-
tions, Acta Polytechnica, to appear, arXiv:1510.09067.
[29] Koenigs G., Sur les géodésiques a intégrales quadratiques, in Darboux G., Lecons sur la théorie générale des
surfaces et les applications geométriques du calcul infinitesimal, Vol. 4, Chelsea, New York, 1972, 368–404.
[30] Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007),
560–566.
[31] Miller Jr. W., Li Q., Wilson polynomials/functions and intertwining operators for the generic quantum
superintegrable system on the 2-sphere, J. Phys. Conf. Ser. 597 (2015), 012059, 11 pages, arXiv:1411.2112.
[32] Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A:
Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
[33] Nesterenko M., Popovych R., Contractions of low-dimensional Lie algebras, J. Math. Phys. 47 (2006),
123515, 45 pages, math-ph/0608018.
[34] NIST digital library of mathematical functions, available at http://dlmf.nist.gov/.
[35] Post S., Models of quadratic algebras generated by superintegrable systems in 2D, SIGMA 7 (2011), 036,
20 pages, arXiv:1104.0734.
[36] Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42
(2001), 4248–4257, hep-th/0011209.
[37] Tempesta P., Winternitz P., Harnad J., Miller W., Pogosyan G., Rodriguez M. (Editors), Superintegra-
bility in classical and quantum systems, CRM Proceedings and Lecture Notes, Vol. 37, Amer. Math. Soc.,
Providence, RI, 2004.
[38] Weimar-Woods E., The three-dimensional real Lie algebras and their contractions, J. Math. Phys. 32 (1991),
2028–2033.
http://arxiv.org/abs/1510.09067
http://dx.doi.org/10.1134/S1063778807030167
http://dx.doi.org/10.1088/1742-6596/597/1/012059
http://arxiv.org/abs/1411.2112
http://dx.doi.org/10.1088/1751-8113/46/42/423001
http://dx.doi.org/10.1088/1751-8113/46/42/423001
http://arxiv.org/abs/1309.2694
http://dx.doi.org/10.1063/1.2400834
http://arxiv.org/abs/math-ph/0608018
http://dlmf.nist.gov/
http://dx.doi.org/10.3842/SIGMA.2011.036
http://arxiv.org/abs/1104.0734
http://dx.doi.org/10.1063/1.1386927
http://arxiv.org/abs/hep-th/0011209
http://dx.doi.org/10.1063/1.529222
1 Introduction
1.1 The Helmholtz nondegenerate superintegrable systems
1.2 Lie algebras and quadratic algebras
2 2D conformal superintegrability of the 2nd order
2.1 The conformal Stäckel transform
3 Tetraspherical coordinates
3.1 Classification of nondegenerate conformally superintegrable systems
3.2 The 8 Laplace superintegrable systems with nondegenerate potentials
3.3 Contractions of conformal superintegrable systems with potential induced by generalized Inönü–Wigner contractions
3.4 Relation to separation of variables and Bôcher's limit procedures
3.5 [1,1,1,1] to [2,1,1] contraction
3.6 [1,1,1,1] to [2,2] contraction
3.7 [2,1,1] to [3,1] contraction
3.8 [1,1,1,1] to [4] contraction
3.9 [2,2] to [4] contraction
3.10 [3,1] to [4] contraction
3.11 [2,1,1] to [4] contraction
3.12 [1,1,1,1] to [3,1] contraction
3.13 [2,2] to [4] contraction
3.14 Summary of Bôcher contractions of Laplace systems
3.15 Conformal Stäckel transforms of the Laplace systems
3.16 Summary of Stäckel equivalence classes of Helmholtz systems
4 Helmholtz contractions from Bôcher contractions
4.1 Summary of Helmholtz contractions
5 Conclusions and discussion
References
|