Laplace equations, conformal superintegrability and B\^ocher contractions

Quantum superintegrable systems are solvable eigenvalue problems. Their solvability 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 and their algebras are related by geometric limits, induced by generalized In\"on\"u-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. The systems can be best understood by transforming them to Laplace conformally superintegrable systems and using ideas introduced in the 1894 thesis of B\^ocher to study separable solutions of the wave equation. The contractions can be subsumed into contractions of the conformal algebra $so(4,C)$ to itself. Here we announce main findings, with detailed classifications in papers under preparation.

• The symmetry operators of each system close under commutation to generate a quadratic algebra, and the irreducible representations of this algebra determine the eigenvalues of H and their multiplicity • All the 2nd order superintegrable systems are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S 9 in our listing, [13], or are obtained from these limits by a Stäckel transform (an invertible structure preserving mapping of superintegrable systems, [6]). Analogously all quadratic symmetry algebras of these systems are limits of that of S 9 . , L 2 , L 3 , • 2nd order superintegrable systems are multiseparable.
Here we consider only the nondegenerate superintegrable systems: Those with 4-parameter potentials (the maximum possible):

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where {V (1) (x), V (2) (x), V (3) (x), 1} is a linearly independent set. For these the symmetry algebra generated by H, L 1 , L 2 always closes under commutation and gives the following quadratic algebra structure: Define 3rd order commutator R by R = [L 1 , L 2 ]. Then All 2nd order 2D superintegrable systems with potential and their quadratic algebras are known. There are 44 nondegenerate systems, on a variety of manifolds (just the manifolds classified by Koenigs), but under the Stäckel transform they divide into 6 equivalence classes with representatives on flat space and the 2-sphere, [14]. Every 2nd order symmetry operator on a constant curvature space takes the form where K is a 2nd order element in the enveloping algebra of o (3, C) or e(2, C). An example is S 9 where where J 3 = s 1 ∂ s2 − s 2 ∂ s1 and J 2 , J 3 are obtained by cyclic permutations of indices. Basis symmetries are 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.
Remark : This theorem is constructive, [15]. Given a free quadratic algebraQ one can compute the potential V and the symmetries of the quadratic algebra Q of the nondegenerate superintegrable system. Special functions arise from these systems in two distinct ways: 1) As separable eigenfunctions of the quantum Hamiltonian. Second order superintegrable systems are multiseparable, [6]. 2) As interbasis expansion coefficients relating distinct separable coordinate eigenbases, [16,17,19,20]. Most of the classical special functions in the Digital Library of Mathematical Functions, as well as Wilson polynomials, appear in these ways, [21].

The big picture: Contractions and special functions
• Taking coordinate limits starting from quantum system S 9 we can obtain other superintegrable systems.
• These coordinate limits induce limit relations between the special functions associated as eigenfunctions of the superintegrable systems.
• The limits induce contractions of the associated quadratic algebras, and via the models, limit relations between the associated special functions.
• 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), [22][23][24] • The Askey scheme for orthogonal functions of hypergeometric type fits nicely into this picture. [25] 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 ∈ (0; 1] there exists a linear invertible map t : B → A such that for every Thus, as → 0 the 1-parameter family of basis transformations can become nonsingular but the structure constants go to a finite limit. Contractions of e(2, C) and o(3, C): These are the symmetry Lie algebras of free (zero potential) systems on constant curvature spaces. Their contractions have long since been classified, [15]. There are 6 nontrivial contractions of e(2, C) and 4 of o (3, C). They are each induced by coordinate limits.
Example: An Inönü-Wigner-contraction of o(3, C). We use the classical realization for o(3, C) acting on the 2-sphere, with basis Here s 2 1 + s 2 2 + s 2 3 = 1. We introduce the basis change: As → 0 these converge to the Lie algebra e(2, C).
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 PREPRINT vol.
no. / Laplace equations in the algebra: As → 0 the 1-parameter family of basis transformations becomes singular but the structure constants go to a finite limit, [15]. Motivating idea: Lie algebra contractions induce quadratic algebra contractions. For constant curvature spaces we have Theorem 2. [15], Every Lie algebra contraction of A = e(2, C) or A = o(3, C) induces a contraction of a free (zero potential) quadratic algebraQ 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.

The problems and the proposed solutions
The various limits of 2nd order superintegrable systems on constant curvature spaces and their applications, such as to the Askey-Wilson scheme, can be classified and understood in terms of generalized Inönü-Wigner contractions [15]. However, there are complications for spaces not of constant curvature. For Darboux spaces the Lie symmetry algebra is only 1-dimensional so limits must be determined on a caseby-case basis. 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 relations. 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 [26] and applied in [27] to solve important classification problems in the case n = 3. It is the aim of this paper to describe the Laplace equation mechanism and how it can be applied to systematize the classification of Helmholtz superintegrable systems and their relations via limits. 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 potential on flat space is the conformal algebra so(n + 2, C). In his 1894 thesis [28] 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. We show that his 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 all of the limits of the Helmholtz systems classified before for n = 2, [15], are induced by the larger class of Bôcher contractions. Here we present the main constructions and findings. Detailed proofs and the lengthy classifications are in papers under preparation.

The Laplace equation
Systems of Laplace type are of the form HΨ ≡ ∆ n Ψ + V Ψ = 0. Here ∆ n is the Laplace-Beltrami operator on a conformally flat nD Riemannian or pseudo-Riemannian manifold. A conformal symmetry of this equation is a partial differential operator S in the variables x = (x 1 , · · · , x n ) such that [S, H] ≡ SH − HS = R S H for some differential operator R S . The system is maximally conformally superintegrable (or Laplace superintegrable) for n ≥ 2 if there are 2n − 1 functionally independent conformal symmetries, S 1 , · · · , S 2n−1 with S 1 = H, [26]. It is second order conformally superintegrable if each symmetry S i can be chosen to be a differential operator of at most second order. Every 2D Riemannian manifold is conformally flat, so we can always find a Cartesian-like coordinate system with coordinates (x, y) ≡ (x 1 , x 2 ) such that the Helmholtz eigenvalue equation takes the form However, this equation is equivalent to the flat space Laplace equation (2) In particular, the symmetries of (1) correspond to the conformal symmetries of (2). Indeed, if [S,H] = 0 then Conversely, if S is an E-independent conformal symmetry of H we find that [S,H] = 0. Further, the conformal symmetries of the system (H − E)Ψ = 0 are identical with the conformal symmetries of (2). Thus without loss of generality we can assume the manifold is flat space with λ ≡ 1.
The conformal Stäckel transform: Suppose we have a second order conformal superintegrable system [26]. There is a similar definition of ordinary Stäckel transforms of Helmholtz superintegrable systems HΨ = EΨ which takes superintegrable systems to superintegrable systems, essentially preserving the quadratic algebra structure, [29].
Thus any second order conformal Laplace superintegrable system admitting a nonconstant potential U can be Stäckel transformed to a Helmholtz superintegrable system. By choosing all possible special E. Kalnins, W. Miller, E. Subag

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PREPRINT potentials U associated with the fixed Laplace system we generate the equivalence class of all Helmholtz superintegrable systems obtainable through this process.

Theorem 4.
There is a one-to-one relationship between flat space conformally superintegrable Laplace systems with nondegenerate potential and Stäckel equivalence classes of superintegrable Helmholtz systems with nondegenerate potential.
Indeed, for a Stäckel transform induced by the function U (1) , we can take the original Helmholtz system to have Hamiltonian 1} is a basis for the 4dimensional potential space. A 2nd order symmetry S would have the form where S 0 is a symmetry of the potential free Hamiltonian, H 0 . The Stäckel transformed symmetry and Hamiltonian take the formS = S − W (1)H and Note that the parameter α 1 cancels out of the expression forS; it is replaced by a term −α 4 W (1) /U (1) . Now suppose that Ψ is a formal eigenfunction of H (not required to be normalizable): HΨ = EΨ. Without loss of generality we can absorb the energy eigenvalue into α 4 so that α 4 = −E in (3) and, in terms of this redefined H, we have HΨ = 0. It follows immediately thatSΨ = SΨ. Thus, for the 3-parameter system H and the Stäckel transformH , we have H Ψ = EΨ andH Ψ = −α 1 Ψ. The effect of the Stäckel transform is to replace α 1 by −E and E by −α 1 . Further, S andS must 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 where {S 1 , S 2 , H} is a basis for the 2nd order symmetries and α 1 , α 2 , α 3 are the parameters for the potential, [6]. It follows from the above considerations that the effect of a Stäckel transform generated by the potential function U (1) is to determine a new superintegrable system with structure Of course, the switch of α 1 and H is only for illustration; there is a Stäckel transform that replaces any α j by −H and H by −α j and similar transforms that apply to any basis that we choose for the potential space.
Formulas (4) and (5) are just instances of the quadratic algebras of the superintegrable systems belonging to the equivalence class of a single nondegenerate conformally superintegrable Hamiltonian LetŜ 1 ,Ŝ 2 ,Ĥ be a basis of 2nd order conformal symmetries ofĤ. From the above discussion we can conclude the following.
Theorem 5. The symmetries of the 2D nondegenerate conformal superintegrable HamiltonianĤ generate a quadratic algebra [R, , and all identities hold mod(Ĥ).
A conformal Stäckel transform generated by the potential V (j) (x, y) yields a nondegenerate Helmholtz superintegrable HamiltonianH with quadratic algebra relations identical to (7), except that we make the replacementsŜ →S for = 1, 2 and α j → −H. These modified relations (6) are now true identities, not mod(Ĥ).
Every 2nd order conformal symmetry is of the form S = S 0 + W where S 0 is a 2nd order element of the enveloping algebra of so (4, C). The dimension of this space of 2nd order elements is 21 but there is an 11dimensional subspace of symmetries congruent to 0 mod(H 0 ) where H 0 = P 2 1 + P 2 2 . Thus mod(H 0 ) the space of 2nd order symmetries is 10-dimensional.

The Bôcher Method
In his 1894 thesis Bôcher, [28], developed a geometrical method for finding and classifying the R-separable orthogonal coordinate systems for the flat space Laplace equation ∆ n Ψ = 0 in n dimensions. It was based on the conformal symmetry of these equations. The conformal Lie symmetry algebra of the flat space complex Laplacian is so(n + 2, C). We will use his ideas for n = 2 , but applied to the Laplace equation with potential HΨ ≡ (∂ 2 x +∂ 2 y +V )Ψ = 0. The so(4, C) conformal symmetry algebra in the case n = 2 has the basis P 1 = ∂ x , P 2 = ∂ y , J = x∂ y − y∂ x , D = x∂ x + y∂ y , K 1 = (x 2 − y 2 )∂ x + 2xy∂ y , K 2 = (y 2 − x 2 )∂ y + 2xy∂ x . Bôcher linearizes this action by introducing tetraspherical coordinates. These are 4 projective complex PREPRINT vol.
no. / Laplace equations coordinates (x 1 , x 2 , x 3 , x 4 ) confined to the null cone x 2 1 + x 2 2 + x 2 3 + x 2 4 = 0. They are related to complex Cartesian coordinates (x, y) via 4 , These operators are clearly a basis for so(4, C). The generators for flat space conformal symmetries are related to these via

Relation to separation of variables
Bôcher uses symbols of the form [n 1 , n 2 , .., n p ] where n 1 + ... + n p = 4, to define coordinate surfaces as follows. Consider the quadratic forms If the parameters e j are pairwise distinct, the elementary divisors of these two forms are denoted by [1, 1, 1, 1]. Given a point in 2D flat space with Cartesian coordinates (x 0 , y 0 ), there corresponds a set of tetraspherical coordinates (x 0 1 , x 0 2 , x 0 3 , x 0 4 ), unique up to multiplication by a nonzero constant. If we substitute into Φ we see that there are exactly 2 roots λ = ρ, µ such that Φ = 0. (If e 4 → ∞ these correspond to elliptic coordinates on the 2sphere.) They are orthogonal with respect to the metric ds 2 = dx 2 + dy 2 and are R-separable for the Laplace equations (∂ 2 x +∂ 2 y )Θ = 0 or ( 4 j−1 ∂ 2 xj )Θ = 0.

Example 1.
Consider the potential V [1,1,1,1] = a1 . It is the only potential V such that equation ( 4 j−1 ∂ 2 xj + V )Θ = 0 is R-separable in elliptic coordinates for all choices of the parameters e j . The separation is characterized by 2nd order conformal symmetry operators that are linear in the parameters e j . In particular the symmetries span a 3-dimensional subspace of symmetries, so the system HΘ = ( 4 j=1 ∂ 2 xj + V [1,1,1,1] )Θ = 0 must be conformally superintegrable.

Bôcher limits
Suppose some of the e i become equal. To obtain separable coordinates we cannot just set them equal in Ω, Φ but must take limits, Bôcher develops a calculus to describe this. Thus the process of making e 1 → e 2 is described by the mapping, which in the limit as → 0 takes the null cone to the null cone.
In the limit we have which has elementary divisors [2, 1, 1], [30,31]. In the same way as for [1, 1, 1, 1], these forms define a new set of orthogonal coordinates R-separable for the Laplace equations. We can show that the coordinate limit induces a contraction of so(4, C) to itself: We provided the parameters transform as We know from theory that the 4-dimensional vector space of potentials V [1,1,1,1] maps to the 4dimensional vector space of potentials V [2,1,1] 1-1 under the contraction, [15]. The reason for thedependence of the parameters is the arbitrariness of choosing a basis. If we had chosen a basis for V [1,1,1,1] specially adapted to this contraction, we could have achieved a j = b j , 1 ≤ j ≤ 4.
Bôcher contractions obey a composition law:

The 8 classes of nondegenerate conformally superintegrable systems
The possible Laplace equations (in tetraspherical coordinates) are ( 4 j=1 ∂ 2 xj + V )Ψ = 0 with potentials: (The last 3 systems do not correspond to elementary divisors; they appear as Bôcher contractions of systems that do correspond to elementary divisors.) Each of the 44 Helmholtz nondegenerate superintegrable (i.e. 3-parameter) eigenvalue systems is Stäckel equivalent to exactly one of these systems. Thus, with one caveat, there are exactly 8 equivalence classes of Helmholtz systems. The caveat is the singular family of systems with potentials V S = (x 3 + ix 4 ) −2 h( x1+ix2 x3+ix4 ) where h is an arbitrary analytic function except that . This family is unrelated to the other systems.

Helmholtz contractions from Bôcher contractions
We describe how Bôcher contractions of conformal superintegrable systems induce contractions of Helmholtz superintegrable systems. We consider the conformal Stäckel transforms of the conformal system [1, 1, 1, 1] with potential V [1,1,1,1] . As we will show explicitly in another paper, the various possibilities are S 9 above and 2 more Helmholtz systems on the sphere, S 7 and S 8 , 2 Darboux systems D4B and D4C, and a family of Koenigs systems.
Recalling horospherical coordinates on the complex two sphere, viz.
we see that the HamiltonianĤ can be written aŝ , and this is explicitly the superintegrable system S 9 .
More generally, let H be the initial Hamiltonian. In terms of tetraspherical coordinates a general conformal Stäckel transformed potential will take the form , and the transformed Hamiltonian will bê where the transform is determined by the fixed vector where the the integer exponent α depends upon our choice of A. We will provide the theory to show that the system defined by Hamiltonian The basic idea is that the procedure of taking a conformal Stäckel transform of a conformal superintegrable system, followed by a Helmholtz contraction yields the same result as taking a Bôcher contraction followed by an ordinary Stäckel transform: The diagrams commute. The possible Helmholtz contractions obtainable from these Bôcher contractions number well over 100; they will be classified in another paper.

Figure 1. Relationship between conformal Stäckel transforms and Bôcher contractions
All quadratic algebra contractions are induced by Lie algebra contractions of so(4, C), even those for Darboux and Koenigs spaces.

Conclusions and discussion
We have pointed out that 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 incomplete, because it doesn't satisfactorily account for Darboux and Koenigs spaces, and because even for constant curvature spaces there are abstract quadratic algebra contractions that cannot be obtained from the Lie symmetry algebras. However, this gap is filled in 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 associated with invariants of quadratic forms. 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, [32].
In a paper under preparation we will 1) give a complete detailed classification of 2D nondegenerate 2nd order conformally superintegrable systems and their relation to Bôcher contractions, 2) present a detailed classification of all Bôcher contractions of 2D nondegenerate 2nd order conformally superintegrable systems., 3) present tables describing the contractions of nondegenerate 2nd order Helmholtz superintegrable systems and how they are induced by Bôcher contractions, 4) introduce so(4, C) → e(3, C) contractions of Laplace systems and show how they produce conformally 2nd order superintegrable 2D time-dependent Schrödinger equations.
From Theorem 1 we know that the potentials of all Helmholtz superintegrable systems are completely 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. In a second paper under preparation we will 1) apply the Bôcher construction to degenerate (1-parameter) Helmholtz superintegrable systems (which admit a 1st order symmetry), 2) give a classification of free abstract degenerate quadratic algebras and identify which of those correspond free 2nd order superintegrable systems. 3) classify abstract contractions of degenerate quadratic algebras and identify which of those correspond to geometric contractions of Helmholtz superintegrable systems, 4) classify free abstract nondegenerate quadratic algebras and identify those corresponding to free nondegenerate Helmholtz 2nd order superintegrable systems, 5) classify of abstract contractions of nondegenerate quadratic algebras.
We note that by taking contractions step-by-step from a model of the S 9 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 ex-PREPRINT vol.
no. / Laplace equations pansion coefficients relating two separable eigenbases that are both of hypergeometric type. Thus, there are some contractions which do not fit in the Askey scheme since the physical system fails to have such a pair of separable eigenbases. In a third paper under preparation we will analyze the Laplace 2nd order conformally superintegrable systems, determine which of them is exactly solvable or quasi-exactly solvable and identify the spaces of polynomials that arise. Again, multiple Helmholtz superintegrable systems will correspond to a single Laplace system. This will enable us to apply our results to characterize polynomial eigenfunctions not of Askey type and their limits.