Acta Polytechnica doi:10.14311/AP.2016.56.0214 Acta Polytechnica 56(3):214–223, 2016 © Czech Technical University in Prague, 2016 available online at http://ojs.cvut.cz/ojs/index.php/ap LAPLACE EQUATIONS, CONFORMAL SUPERINTEGRABILITY AND BÔCHER CONTRACTIONS Ernest Kalninsa, Willard Miller, Jr.b, ∗, Eyal Subagc a Department of Mathematics, University of Waikato, Hamilton, New Zealand b School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA c Department of Mathematics, Pennsylvania State University, State College 16802, Pennsylvania USA ∗ corresponding author: miller@ima.umn.edu Abstract. 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 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. The systems can be best understood by transforming them to Laplace conformally superintegrable systems and using ideas introduced in the 1894 thesis of Bôcher 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 submitted and under preparation. Keywords: conformal superintegrability; contractions; Laplace equations. 1. Introduction A quantum superintegrable system is an inte- grable Hamiltonian system on an n-dimensional Riemannian/pseudo-Riemannian manifold with poten- tial: H = ∆n + V that admits 2n−1 algebraically in- dependent partial differential operators Lj commuting with H, the maximum possible. [H,Lj] = 0, j = 1, 2, · · · , 2n− 1. Superintegrability captures the prop- erties of quantum Hamiltonian systems that allow the Schrödinger eigenvalue problem (or Helmholtz equa- tion) HΨ = EΨ to be solved exactly, analytically and algebraically [1–5]. A system is of order K if the maximum order of the symmetry operators, other than H, is K. For n = 2, K = 1, 2 all systems are known, see, e.g., [6, 7] We review quickly the facts for free 2nd order su- perintegrable 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 in 1896 [8]. They are: • The two constant curvature spaces (flat space and the complex sphere), six linearly independent 2nd order symmetries and three 1st order symmetries, • The four Darboux spaces (one with a parameter), four 2nd order symmetries and one 1st order sym- metry [9], ds2 = 4x(dx2 + dy2), ds2 = x2 + 1 x2 (dx2 + dy2), ds2 = ex + 1 e2x (dx2 + dy2), ds2 = 2 cos 2x + b sin2 2x (dx2 + dy2). • Eleven 4-parameter Koenigs spaces. No 1st order symmetries. An example is ds2 = ( c1 x2 + y2 + c2 x2 + c3 y2 + c4 ) (dx2 + dy2). For 2nd order systems with non-constant potential, K = 2, the following is true [6, 7, 10–12]. • The symmetry operators of each system close under commutation to generate a quadratic algebra, and the irreducible representations of this algebra deter- mine the eigenvalues of H and their multiplicity. • All the 2nd order superintegrable systems are limit- ing cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing [13], or are obtained from these limits by a Stäckel trans- form (an invertible structure preserving mapping of superintegrable systems [6]). Analogously all quadratic symmetry algebras of these systems are limits of that of S9. S9 : H = ∆2 + a1 s21 + a2 s22 + a3 s23 , s21 + s 2 2 + s 2 3 = 1, L1 = (s2∂s3 −s3∂s2 ) 2 + a3s 2 2 s23 + a2s 2 3 s22 , L2, L3, • 2nd order superintegrable systems are multisepara- ble. 214 http://dx.doi.org/10.14311/AP.2016.56.0214 http://ojs.cvut.cz/ojs/index.php/ap vol. 56 no. 3/2016 Laplace Equations, Conformal Superintegrability and Bôcher Contractions Here we consider only the nondegenerate superinte- grable systems: Those with 4-parameter potentials (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 in- dependent set. For these the symmetry algebra gener- ated by H,L1,L2 always closes under commutation and gives the following quadratic algebra structure: Define 3rd order commutator R by R = [L1,L2]. Then [Lj,R] = A (j) 1 L 2 1 + A (j) 2 L 2 2 + A (j) 3 H 2 + 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 = b1L31 +b2L 3 2 +b3H 3 +b4{L21,L2}+b5{L1,L 2 2} + b6L1L2L1 + b7L2L1L2 + b8H{L1,L2} + b9HL21 + b10HL22 + b11H 2L1 + b12H2L2 + b13L21 + b14L22 + b15{L1,L2} + b16HL1 + b17HL2 + b18H2 + b19L1 + b20L2 + b21H + b22, where {L1,L2} = L1L2 + L2L1 and A (j) i , bk are con- stants. All 2nd order 2D superintegrable systems with po- tential and their quadratic algebras are known. There are 44 nondegenerate systems, on a variety of mani- folds (just the manifolds classified by Koenigs), but under the Stäckel transform they divide into 6 equiv- alence classes with representatives on flat space and the 2-sphere [14]. 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 example is S9 where H = J21 + J 2 2 + J 2 3 + a1 s21 + a2 s22 + a3 s23 , where J3 = s1∂s2 −s2∂s1 and J2,J3 are obtained by cyclic permutations of indices. Basis symmetries are (J3 = s2∂s1 −s1∂s2, . . .) L1 = J21 + a3s 2 2 s23 + a2s 2 3 s22 , L2 = J22 + a1s 2 3 s21 + a3s 2 1 s23 , L3 = J23 + a2s 2 1 s22 + a1s 2 2 s21 . Theorem 1. There is a bijection between quadratic algebras generated by 2nd order elements in the en- veloping algebra of o(3,C), called free, and 2nd order nondegenerate superintegrable systems on the com- plex 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 nonde- generate superintegrable systems on the 2D complex flat space. Remark. This theorem is constructive [15]. Given a free quadratic algebra Q̃ one can compute the poten- tial 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 expan- sion coefficients relating distinct separable coordinate eigenbases [16, 17, 19, 20]. Most of the classical spe- cial functions in the Digital Library of Mathematical Functions, as well as Wilson polynomials, appear in these ways [21]. 1.1. The big picture: Contractions and special functions • Taking coordinate limits starting from quantum system S9 we can obtain other superintegrable sys- tems. • These coordinate limits induce limit relations be- tween the special functions associated as eigenfunc- tions of the superintegrable systems. • The limits induce contractions of the associated quadratic algebras, and via the models, limit rela- tions 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) [22–24] • The Askey scheme for orthogonal functions of hy- pergeometric 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 con- traction of A if for every � ∈ (0; 1] there exists a linear invertible map t� : B → A such that for every X,Y ∈ B, lim�→0 t−1� [t�X,t�Y ]A = [X,Y ]B. 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 J1 = s2∂s3 − s3∂s2, J2 = s3∂s1 − s1∂s3, J3 = s1∂s2 − s2∂s1 , commutation re- lations [J2,J1] = J3, [J3,J2] = J1, [J1,J3] = J2, and Hamiltonian H = J21 +J22 +J23 . Here s21 +s22 +s23 = 1. We introduce the basis change: {J′1,J ′ 2,J ′ 3} = {�J1, �J2, J3}, 0 < � ≤ 1, with coordinate implementation x = s1 � ,y = s2 � ,s3 ≈ 1. The structure relations become [J′2,J ′ 1] = � 2J′3, [J ′ 3,J ′ 2] = J ′ 1, [J ′ 1,J ′ 3] = J ′ 2, 215 E. Kalnins, W. Miller, E. Subag Acta Polytechnica As � → 0 these converge to [J′2,J ′ 1] = 0, [J ′ 3,J ′ 2] = J ′ 1, [J ′ 1,J ′ 3] = J ′ 2, the Lie algebra e(2,C). Contractions of quadratic algebras. Just as for Lie algebras we can define a contraction of a quadratic al- gebra in terms of 1-parameter families of basis changes in the algebra: As � → 0 the 1-parameter family of ba- sis transformations becomes singular but the structure constants go to a finite limit [15]. Motivating idea — Lie algebra contractions induce quadratic algebra contractions. For constant curva- ture spaces we have the following theorem. 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 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. 1.2. The problems and the proposed solutions The various limits of 2nd order superintegrable sys- tems on constant curvature spaces and their appli- cations, 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 case- by-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 is- sues 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 classifica- tion problems in the case n = 3. It is the aim of this paper to describe the Laplace equation mecha- nism 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 superinte- grable systems on a variety of manifolds correspond to a single conformally superintegrable Laplace equa- tion 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 ∆nΨ = 0 in n dimensions. (That is, he constructed separable solutions of the form Ψ = R(u)Πnj=1ψj(uj) where R is a fixed gauge function and ψj depends only on the variable uj and the separation constants.) We show that his limit pro- cedure 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 clas- sified 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 [32]. 2. The Laplace equation Systems of Laplace type are of the form HΨ ≡ ∆nΨ + V Ψ = 0. Here ∆n is the Laplace-Beltrami operator on a conformally flat nD Riemannian or pseudo-Riemannian manifold. A conformal symme- try 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 op- erator RS. The system is maximally conformally superintegrable (or Laplace superintegrable) for n ≥ 2 if there are 2n − 1 functionally independent confor- mal symmetries, S1, · · · ,S2n−1 with S1 = H [26]. It is second order conformally superintegrable if each symmetry Si can be chosen to be a differential opera- tor 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) ≡ (x1,x2) such that the Helmholtz eigenvalue equation takes the form H̃Ψ = ( 1 λ(x,y) (∂2x + ∂ 2 y) + Ṽ (x) ) Ψ = EΨ. (1) However, this equation is equivalent to the flat space Laplace equation HΨ ≡ ( ∂2x + ∂ 2 y + V (x) ) Ψ = 0, V (x) = λ(x)(Ṽ (x) −E). (2) In particular, the symmetries of (1) correspond to the conformal symmetries of (2). Indeed, if [S,H̃] = 0 then [S,H] = [S,λ(H̃−E)] = [S,λ](H̃−E) = [S,λ]λ−1H. Conversely, if S is an E-independent conformal sym- metry of H we find that [S,H̃] = 0. Further, the conformal symmetries of the system (H̃ −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 H = ∂xx + ∂yy + V (x,y) = 0, H = H0 + V, where V (x,y) = W(x,y) − E U(x,y) for arbitrary parameter E. 216 vol. 56 no. 3/2016 Laplace Equations, Conformal Superintegrability and Bôcher Contractions Theorem 3 [26]. The transformed (Helmholtz) sys- tem H̃Ψ = EΨ, H̃ = 1 U (∂xx + ∂yy) + Ṽ is superin- tegrable (not just conformally superintegrable), where Ṽ = W U . 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 super- integrable system admitting a nonconstant potential U can be Stäckel transformed to a Helmholtz super- integrable system. By choosing all possible special potentials U associated with the fixed Laplace system we generate the equivalence class of all Helmholtz su- perintegrable systems obtainable through this process. Theorem 4. There is a one-to-one relationship be- tween flat space conformally superintegrable Laplace systems with nondegenerate potential and Stäckel equivalence classes of superintegrable Helmholtz sys- tems with nondegenerate potential. Indeed, for a Stäckel transform induced by the func- tion U(1), we can take the original Helmholtz system to have Hamiltonian H = H0 + V = H0 + U(1)α1 + U(2)α2 + U(3)α3 + α4, (3) where {U(1),U(2),U(3), 1} is a basis for the 4- dimensional potential space. A 2nd order symmetry S would have the form S = S0 + W (1)α1 + W (2)α2 + W (3)α3, where S0 is a symmetry of the potential free Hamil- tonian, H0. The Stäckel transformed symmetry and Hamiltonian take the form S̃ = S −W (1)H̃ and H̃ = 1 U(1) H0 + U(1)α1 + U(2)α2 + U(3)α3 + α4 U(1) . Note that the parameter α1 cancels out of the expres- sion for S̃; it is replaced by a term −α4W (1)/U(1). Now suppose that Ψ is a formal eigenfunction of H (not required to be normalizable): HΨ = EΨ. With- out loss of generality we can absorb the energy eigen- value into α4 so that α4 = −E in (3) and, in terms of this redefined H, we have HΨ = 0. It follows im- mediately that S̃Ψ = SΨ. Thus, for the 3-parameter 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) , we have H′Ψ = EΨ and H̃′Ψ = −α1Ψ. The effect of the Stäckel transform is to replace α1 by −E and E by −α1. Further, S and S̃ 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 [R,Sj] = f(j)(S1,S2,α1,α2,α3,H′), j = 1, 2, R2 = f(3)(S1,S2,α1,α2,α3,H′), (4) where {S1,S2,H} is a basis for the 2nd order sym- metries and α1,α2,α3 are the parameters for the po- tential [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 [R̃, S̃j] = f(j)(S̃1, S̃2,−H̃′,α2,α3,−α1), j = 1, 2, R2 = f(3)(S̃1, S̃2,−H̃′,α2,α3,−α1), (5) Of course, the switch of α1 and H′ is only for illustra- tion; there is a Stäckel transform that replaces any αj by −H′ and H′ by −αj 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 be- longing to the equivalence class of a single nondegen- erate conformally superintegrable Hamiltonian Ĥ = ∂xx + ∂yy + 4∑ j=1 αjV (j)(x,y). (6) Let Ŝ1, Ŝ2,Ĥ be a basis of 2nd order conformal symme- tries of Ĥ. From the above discussion we can conclude the following. Theorem 5. The symmetries of the 2D nondegener- ate conformal superintegrable Hamiltonian 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), (7) where R̂ = [Ŝ1, Ŝ2] and all identities hold mod(Ĥ). A conformal Stäckel transform generated by the po- tential V (j)(x,y) yields a nondegenerate Helmholtz superintegrable Hamiltonian H̃ with quadratic alge- bra relations identical to (7), except that we make the replacements Ŝ` → S̃` for ` = 1, 2 and αj →−H̃. These modified relations (6) are now true identities, not mod(Ĥ). Every 2nd order conformal symmetry is of the form S = S0 + W where S0 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 11- dimensional subspace of symmetries congruent to 0 mod(H0) where H0 = P 21 + P 22 . Thus mod(H0) the space of 2nd order symmetries is 10-dimensional. 217 E. Kalnins, W. Miller, E. Subag Acta Polytechnica 3. The Bôcher Method In his 1894 thesis Bôcher [28], developed a geometrical method for finding and classifying the R-separable or- thogonal coordinate systems for the flat space Laplace equation ∆nΨ = 0 in n dimensions. It was based on the conformal symmetry of these equations. The con- formal 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 poten- tial HΨ ≡ (∂2x+∂2y +V )Ψ = 0. The so(4,C) conformal symmetry algebra in the case n = 2 has the basis P1 = ∂x, P2 = ∂y, J = x∂y −y∂x, D = x∂x + y∂y, K1 = (x2 − y2)∂x + 2xy∂y, K2 = (y2 − x2)∂y + 2xy∂x. Bôcher linearizes this action by introducing tetras- pherical coordinates. These are 4 projective complex coordinates (x1,x2,x3,x4) confined to the null cone x21 + x22 + x23 + x24 = 0. They are related to complex Cartesian coordinates (x,y) via x = − x1 x3 + ix4 , y = − x2 x3 + ix4 , H = ∂xx + ∂yy + Ṽ = (x3 + ix4)2 ( 4∑ k=1 ∂2xk + V ) , where Ṽ = (x3 + ix4)2V . We define Ljk = xj∂xk − xk∂xj , 1 ≤ j,k ≤ 4, j 6= k, where Ljk = −Lkj. These operators are clearly a basis for so(4,C). The generators for flat space conformal symmetries are related to these via P1 = ∂x = L13 + iL14, P2 = ∂y = L23 + iL24, D = iL34, J = L12, K1 = L13 − iL14, K2 = L23 − iL24. 3.1. Relation to separation of variables Bôcher uses symbols of the form [n1,n2, ..,np] where n1 + ... + np = 4, to define coordinate surfaces as follows. Consider the quadratic forms Ω = x21 + x 2 2 + x 2 3 + x 2 4 = 0, Φ = x21 λ−e1 + x22 λ−e2 + x23 λ−e3 + x24 λ−e4 . If the parameters ej are pairwise distinct, the el- ementary divisors of these two forms are denoted by [1, 1, 1, 1]. Given a point in 2D flat space with Cartesian coordinates (x0,y0), there corresponds a set of tetraspherical coordinates (x01,x02,x03,x04), 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 e4 → ∞ these correspond to elliptic coordinates on the 2- sphere.) They are orthogonal with respect to the metric ds2 = dx2 + dy2 and are R-separable for the Laplace equations (∂2x+∂2y)Θ = 0 or ( ∑4 j−1 ∂ 2 xj )Θ = 0. Example. Consider the potential V[1,1,1,1] = a1x21 + a2 x22 + a3 x23 + a4 x24 . It is the only potential V such that equation ( ∑4 j−1 ∂ 2 xj + V )Θ = 0 is R-separable in el- liptic coordinates for all choices of the parameters ej. The separation is characterized by 2nd order confor- mal symmetry operators that are linear in the pa- rameters ej. 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 confor- mally superintegrable. 3.2. Bôcher limits Suppose some of the ei 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 e1 → e2 is described by the mapping, which in the limit as � → 0 takes the null cone to the null cone: e1 = e2 + �2, x1 → i(x′1 + ix′2)√ 2� , x2 → (x′1 + ix′2)√ 2� + � (x′1 − ix′2)√ 2 , x3 → x′3, x4 → x ′ 4, In the limit we have Ω = x′1 2 + x′2 2 + x′3 2 + x′4 2 = 0, Φ = (x′1 + ix′2)2 2(λ−e2)2 + x′1 2 + x′2 2 λ−e2 + x′3 2 λ−e3 + x′4 2 λ−e4 , which has elementary divisors [2, 1, 1], see [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: L′12 = L12, L ′ 13 = − i √ 2 � (L13 − iL23) − i � √ 2 L13, L′23 = − i √ 2 � (L13 − iL23) − � √ 2 L13, L ′ 34 = L34, L′14 = − i √ 2 � (L14 − iL24) − i � √ 2 L14, L′24 = − i √ 2 � (L14 − iL24) − � √ 2 L14. We call this the Bôcher contraction [1, 1, 1, 1] → [2, 1, 1]. There are analogous Bôcher contractions of so(4,C) to itself corresponding to limits from [1, 1, 1, 1] to [2, 2], [3, 1], [4]. Similarly, there are Bôcher contrac- tions [2, 1, 1] → [2, 2], etc. If we apply the contraction [1, 1, 1, 1] → [2, 1, 1] to the potential V[1,1,1,1] we get a finite limit 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 , (8) provided the parameters transform as a1 = − 1 2 (b1 �2 + b2 2�4 ) , a2 = − b2 4�4 , a3 = b3, a4 = b4. 218 vol. 56 no. 3/2016 Laplace Equations, Conformal Superintegrability and Bôcher Contractions Note: We know from theory that the 4-dimensional vector space of potentials V[1,1,1,1] maps to the 4- dimensional vector space of potentials V[2,1,1] 1-1 under the contraction [15]. The reason for the �- dependence 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 aj = bj, 1 ≤ j ≤ 4. Bôcher contractions obey a composition law: Theorem 6. Let A: (∆x + VA(x))Ψ = 0, B : (∆y + VB(y))Ψ = 0 C : (∆z + VC(z))Ψ = 0, be Bôcher superintegrable systems such that A Bôcher- contracts to B and B Bôcher-contracts to C. Then there is a one-parameter contraction of A to C. A fundamental advantage in recognizing Bôcher’s limit procedure as contractions is that whereas the Bôcher limits had a fixed starting and ending point, say [1, 1, 1, 1] → [2, 1, 1], contractions can be applied to any nondegenerate conformally superintegrable sys- tem and are guaranteed to result in another nonde- generate conformally superintegrable system. This greatly increases the range of applicability of the lim- its. 4. The 8 classes of nondegenerate conformally superintegrable systems The possible Laplace equations (in tetraspherical co- ordinates) are ( ∑4 j=1 ∂ 2 xj + V )Ψ = 0 with potentials: V[1,1,1,1] = 4∑ j=1 aj x2j , (9) V[2,1,1] = a1 x21 + a2 x22 + a3(x3 − ix4) (x3 + ix4)3 + a4 (x3 + ix4)2 , V[2,2] = a1 (x1 + ix2)2 + a2(x1 − ix2) (x1 + ix2)3 + a3 (x3 + ix4)2 + a4(x3 − ix4) (x3 + ix4)3 , V[3,1] = a1 (x3 + ix4)2 + a2x1 (x3 + ix4)3 + a3(4x12 + x22) (x3 + ix4)4 + a4 x22 , 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 , V[0] = a1 (x3 + ix4)2 + a2x1 + a3x2 (x3 + ix4)3 +a4 x21 + x22 (x3 + ix4)4 , 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 , 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 . (The last 3 systems do not correspond to elementary divisors; they appear as Bôcher contractions of sys- tems that do correspond to elementary divisors.) Each of the 44 Helmholtz nondegenerate superintegrable (i.e., 3-parameter) eigenvalue systems is Stäckel equiv- alent 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 VS = (x3 +ix4)−2h(x1+ix2x3+ix4 ) where h is an arbitrary analytic function except that VS 6= V (1),V (2). This family is unrelated to the other systems. Expressed as flat space Laplace equations (∂2x +∂2y + Ṽ )Ψ = 0 in Cartesian coordinates, the potentials are Ṽ[1,1,1,1] = a1 x2 + a2 y2 + 4a3 (x2 + y2 − 1)2 − 4a4 (x2 + y2 + 1)2 , Ṽ[2,1,1] = a1 x2 + a2 y2 −a3(x2 + y2) + a4, Ṽ[2,2] = a1 (x + iy)2 + a2(x− iy) (x + iy)3 +a3 −a4(x2 + y2), Ṽ[3,1] = a1 −a2x + a3(4x2 + y2) + a4 y2 , Ṽ[4] = a1 −a2(x + iy) +a3 ( 3(x + iy)2 + 2(x− iy) ) −a4 ( 4(x2 + y2) + 2(x + iy)3 ) , Ṽ[0] = a1 − (a2x + a3y) + a4(x2 + y2), Ṽ (1) = a1 (x + iy)2 + a2 − a3 (x + iy)3 + a4 (x + iy)4 , Ṽ (2) = a1 + a2(x + iy) + a3(x + iy)2 + a4(x + iy)3. (10) 4.1. Summary of Bôcher contractions of Laplace superintegrable systems Table 1 contains a partial list of contractions. The full list is presented in [32]. We have omitted some contractions, such as [3, 1] → [4], because they are consequences of other contractions in the table. 5. Helmholtz contractions from Bôcher contractions We describe how Bôcher contractions of confor- mal 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]. 219 E. Kalnins, W. Miller, E. Subag Acta Polytechnica [1, 1, 1, 1] → [2, 1, 1] V[1,1,1,1] ↓ V[2,1,1] V[2,1,1] ↓ V[2,1,1] V[2,2] ↓ V[2,2] V[3,1] ↓ V(1) V[4] ↓ V[0] V[0] ↓ V[0] V (1) ↓ V (1) V (2) ↓ V (2) [1, 1, 1, 1] → [2, 2] V[1,1,1,1] ↓ V[2,2] V[2,1,1] ↓ V[2,2] V[2,2] ↓ V[2,2] V[3,1] ↓ V (1) V[4] ↓ V (2) V[0] ↓ V[0] V (1) ↓ V (1) V (2) ↓ V (2) [2, 1, 1] → [3, 1] V[1,1,1,1] ↓ V[3,1] V[2,1,1] ↓ V[3,1] V[2,2] ↓ V[0] V[3,1] ↓ V[3,1] V[4] ↓ V[0] V[0] ↓ V[0] V (1) ↓ V (2) V (2) ↓ V (2) [1, 1, 1, 1] → [4] 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[0] ↓ V[0] V (1) ↓ V (2) V (2) ↓ V (2) [2, 2] → [4] V[1,1,1,1] ↓ V[4] V[2,1,1] ↓ V[4] V[2,2] ↓ V[4] V[3,1] ↓ V (2) V[4] ↓ V (2) V[0] ↓ V[0] V (1) ↓ V (2) V (2) ↓ V (2). [1, 1, 1, 1] → [3, 1] V[1,1,1,1] ↓ V[3,1] V[2,1,1] ↓ V[3,1] V[2,2] ↓ V[0] V[3,1] ↓ V[3,1] V[4] ↓ V[0] V[0] ↓ V[0] V (1) ↓ V (2) V (2) ↓ V (2) Table 1. Bôcher contractions of Laplace superintegrable systems. As we show explicitly in [32], the various possibilities are S9 above and 2 more Helmholtz systems on the sphere, S7 and S8, 2 Darboux systems D4B and D4C, and a family of Koenigs systems. Example 1. Using Cartesian coordinates x,y, we consider the [1, 1, 1, 1] Hamiltonian H = ∂2x + ∂ 2 y + a1 x2 + a2 y2 + 4a3 (x2 + y2 − 1)2 + 4a4 (x2 + y2 + 1)2 . Dividing on the left by 1/x2 we obtain Ĥ = x2(∂2x + ∂ 2 y) + a1 + a2 x2 y2 + 4a3 x2 (x2 + y2 − 1)2 − 4a4 x2 (x2 + y2 + 1)2 , the Stäckel transform corresponding to the case (a1,a2,a3,a4) = (1, 0, 0, 0). This becomes more trans- parent if we introduce variables x = e−a,y = r. The Hamiltonian Ĥ can be written Ĥ = ∂2a + e −2a∂2r + a1 + a2 e−2a r2 + a3 4 (e−a + ea(r2 − 1))2 −a4 4 (e−a + ea(r2 + 1))2 . Recalling horospherical coordinates on the complex two sphere, viz. s1 = i 2 (e−a + (r2 + 1)ea), s2 = rea, s3 = 1 2 (e−a + (r2 − 1)ea) we see that the Hamiltonian Ĥ can be written as Ĥ = ∂2s1 + ∂ 2 s2 + ∂2s3 + a1 + a2 s22 + a3 s23 + a4 s21 , and this is explicitly the superintegrable system S9. More generally, let H be the initial Hamiltonian. In terms of tetraspherical coordinates a general 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) , where F(x, A) = A1 x21 + A2 x22 + A3 x23 + A4 x24 , 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 contrac- tion [1, 1, 1, 1] → [2, 1, 1] to this system. In the limit as � → 0 the potential V[1,1,1,1] → V[2,1,1], (8), and H → H′ the [2, 1, 1] system. Now consider F(x(�), A) = V ′(x′,A)�α + O(�α+1), 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 Ĥ′ = lim �→0 �αĤ(�) = 1 V ′(x′,A) H′ is a superintegrable system that arises from the sys- tem [2, 1, 1] by a conformal Stäckel transform induced by the potential V ′(x′,A). Thus the Helmholtz su- perintegrable system with potential V = V[1,1,1,1]/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 contrac- tion of the conformal algebra so(4,C). Always the V ′ can be identified with a specialization of the [2, 1, 1] potential . Thus a conformal Stäckel transform of [1, 1, 1, 1] has been contracted to a conformal Stäckel 220 vol. 56 no. 3/2016 Laplace Equations, Conformal Superintegrability and Bôcher Contractions Figure 1. Relationship between conformal Stäckel transforms and Bôcher contractions. transform of [2, 1, 1]. The results follow and generalize to all Laplace systems. The basic idea is that the procedure of taking a conformal Stäckel transform of a conformal superin- tegrable system, followed by a Helmholtz contraction yields the same result as taking a Bôcher contraction followed by an ordinary Stäckel transform: The dia- grams commute. The possible Helmholtz contractions obtainable from these Bôcher contractions number well over 100; they will be classified in another pa- per. All quadratic algebra contractions are induced by Lie algebra contractions of so(4,C), even those for Darboux and Koenigs spaces. 6. Conclusions and discussion We have pointed out that the use of Lie algebra con- tractions based on the symmetry groups of constant curvature spaces to construct quadratic algebra con- tractions 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 ob- tained 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 su- perintegrable system on flat space with underlying conformal symmetry algebra so(4,C). The confor- mal Lie algebra contractions are induced by Bôcher limits associated with invariants of quadratic forms. They generalize all of the Helmholtz contractions de- rived earlier. In particular, contractions of Darboux Figure 2. The bigger picture. and Koenigs systems can be described easily. All of the concepts introduced in this paper are clearly also applicable for dimensions n ≥ 3 [33]. In papers submitted [32], and under preparation we will: (1.) give a complete detailed classification of 2D non- degenerate 2nd order conformally superintegrable systems and their relation to Bôcher contractions; (2.) present a detailed classification of all Bôcher con- tractions of 2D nondegenerate 2nd order confor- mally superintegrable systems; (3.) present tables describing the contractions of non- degenerate 2nd order Helmholtz superintegrable systems and how they are induced by Bôcher con- tractions; (4.) introduce so(4,C) → e(3,C) contractions of Laplace systems and show how they produce confor- mally 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 classifi- cation purposes it is enough to classify free abstract quadratic algebras. In a second paper under prepara- tion we will: (1.) apply the Bôcher construction to degenerate 221 E. Kalnins, W. Miller, E. Subag Acta Polytechnica (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 cor- respond free 2nd order superintegrable systems; (3.) classify abstract contractions of degenerate quadratic algebras and identify which of those cor- respond 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 the abstract contractions of nondegener- ate quadratic algebras. 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 contrac- tion method is more general. It applies to all special functions that arise from the quantum systems via separation of variables, not just polynomials of hyper- geometric type, and it extends to higher dimensions. The functions in the Askey Scheme are just those hy- pergeometric polynomials that arise as the expansion 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 prepara- tion 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. 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Subag, Bôcher contractions of conformally superintegrable Laplace equations: Detailed computations, arXiv:1601.02876, 2016 [33] Capel J.J., Kress J.M. and Post S., Invariant Classification and Limits of Maximally Superintegrable Systems in 3D, SIGMA, 11 (2015), 038, 17 pages arXiv:1501.06601, doi:10.3842/SIGMA.2015.038 223 http://arxiv.org/abs/0801.2848 http://dx.doi.org/10.3842/SIGMA.2008.008 http://dx.doi.org/10.3842/SIGMA.2011.051 http://arxiv.org/abs/1104.0734 http://dx.doi.org/10.3842/SIGMA.2011.036 http://dx.doi.org/10.1073/pnas.39.6.510 http://dx.doi.org/10.1063/1.529222 http://dx.doi.org/10.3842/SIGMA.2013.057 http://dx.doi.org/10.1088/1751-8113/43/3/035202 http://dx.doi.org/10.1098/rspa.1984.0075 http://arxiv.org/abs/1512.09315 http://arxiv.org/abs/1601.02876 http://arxiv.org/abs/1501.06601 http://dx.doi.org/10.3842/SIGMA.2015.038 Acta Polytechnica 56(3):214–223, 2016 1 Introduction 1.1 The big picture: Contractions and special functions 1.2 The problems and the proposed solutions 2 The Laplace equation 3 The Bôcher Method 3.1 Relation to separation of variables 3.2 Bôcher limits 4 The 8 classes of nondegenerate conformally superintegrable systems 4.1 Summary of Bôcher contractions of Laplace superintegrable systems 5 Helmholtz contractions from Bôcher contractions 6 Conclusions and discussion Acknowledgements References