Classical and Quantum Superintegrability of St\"ackel Systems

In this paper we discuss maximal superintegrability of both classical and quantum St\"ackel systems. We prove a sufficient condition for a flat or constant curvature St\"ackel system to be maximally superintegrable. Further, we prove a sufficient condition for a St\"ackel transform to preserve maximal superintegrability and we apply this condition to our class of St\"ackel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.


Introduction
A real-valued function h 1 on a 2n-dimensional manifold (phase space) M = T * Q is called a classical maximally superintegrable Hamiltonian if it belongs to a set of n Poisson-commuting functions h 1 , . . . , h n (constants of motion, so that {h i , h j } = 0 for all i, j = 1, . . . , n) and for which there exist n − 1 additional functions h n+1 , . . . , h 2n−1 on M that Poisson-commute with the Hamiltonian h 1 and such that all the functions h 1 , . . . , h 2n−1 constitute a functionally independent set of functions. Analogously, a quantum maximally superintegrable Hamiltonian is a self-adjoint differential operatorĥ 1 acting in an appropriate Hilbert space of functions on the configuration space Q (square integrable with respect to some metric) belonging to a set of n commuting self-adjoint differential operatorsĥ 1 , . . . ,ĥ n acting in the same Hilbert space (so that [ĥ i ,ĥ j ] = 0 for all i, j = 1, . . . , n) and such that it also commutes with an additional set of n − 1 differential operatorsĥ n+1 , . . . ,ĥ 2n−1 of finite order. Besides, in analogy with the classical case, it is required that all the operatorsĥ 1 , . . . ,ĥ 2n−1 are algebraically independent [19]. Throughout the paper it is tacitly assumed that n > 1 as the case n = 1 is not interesting from the point of view of our theory. This paper is devoted to n-dimensional maximally superintegrable classical and quantum Stäckel systems with all constants of motion quadratic in momenta. Although superintegrable systems of second order, both classical and quantum, have been intensively studied (see for example [1,2,11,14,16,17] and the review paper [19]), nevertheless all the results about superintegrable Stäckel systems (including the important classification results) were mainly restricted to two or three dimensions or focused on the situation when the Hamiltonian is a sum of one degree of freedom terms and therefore itself separates in the original coordinate arXiv:1608.04546v2 [nlin.SI] 30 Jan 2017 system (see for example [3,12] or [15]). Here we present some general results concerning ndimensional classical separable superintegrable systems in flat spaces, constant curvature spaces and conformally flat spaces. We also present how to separately quantize all considered classical systems. We stress, however, that we do not develop spectral theory of the obtained quantum systems, as it requires a separate investigation.
The paper is organized as follows. In Section 2 we briefly describe -following previous references, for example [18] and [5] -flat and constant curvature Stäckel systems that we consider in this paper. In Section 3 we prove (Theorem 3.3) a sufficient condition for this class of Stäckel system to be maximally superintegrable by finding a linear in momenta function P = n s=1 y s p s on M such that {h 1 , P } = c (it also means that the vector field Y = n s=1 y s ∂ ∂qs in P is a Killing vector for the metric generated by h 1 ) which yields additional n − 1 functions h n+i = {h i+1 , P } commuting with h 1 and thus turning h 1 into a maximally superintegrable Hamiltonian. In Section 4 we briefly remind the notion of Stäckel transform (a functional transform that preserves integrability) and prove (Theorem 4.2) conditions that guarantee that a Stäckel transform transforms maximally superintegrable system into another maximally superintegrable system (i.e., preserves maximal superintegrability). In Section 5 we apply this result to our class of maximally superintegrable Stäckel systems, obtaining Theorem 5.2 stating when the Stäckel transform applied to the considered class of systems yields a Stäckel system that is flat, of constant curvature or conformally flat. We also demonstrate (Theorem 5.4) that the additional integrals h n+i of systems after Stäckel transform can be obtained in two equivalent ways. Section 6 is devoted to the procedure of minimal quantization of considered Stäckel systems. As the procedure of minimal quantization depends on the choice of the metric on the configurational space, we remind first the result obtained in [5] explaining how to choose the metric in which a minimal quantization is performed so that the integrability of the quantized system is preserved (Theorem 6.1) and then apply Lemma 6.3 to obtain Corollary 6.4 stating under which conditions the procedure of minimal quantization of a classical Stäckel system, considered in previous sections, yields a quantum superintegrable and quantum separable system. The paper is furnished with several examples that continue throughout sections. The examples are all 3-dimensional in order to make the formulas readable but our theory works in arbitrary dimension.

A class of f lat and constant curvature Stäckel systems
Let us first introduce the class of Hamiltonian systems that we will consider in this paper. Consider a 2n-dimensional manifold M = T * Q (we remind the reader that n > 1) equipped with a set of (smooth) coordinates (λ, µ) = (λ 1 , . . . , λ n , µ 1 , . . . , µ n ) defined on an open dense set of M and such that λ are the coordinates on the base manifold Q while µ are fibre coordinates. Define the bivector Then the bivector Π satisfies the Jacobi identity so it becomes a Poisson operator (Poisson tensor), our manifold M becomes Poisson manifold and the coordinates (λ, µ) become Darboux (canonical) coordinates for the Poisson tensor (2.1). Consider also a set of n algebraic equations on M where we normalize γ n = 0 and where σ and f are arbitrary functions of one variable. The relations (2.2) constitute a system of n equations linear in the unknowns h j . Solving these equations with respect to h j we obtain n functions h j = h j (λ, µ) on M of the form where we denote λ = (λ 1 , . . . , λ n ) T and µ = (µ 1 , . . . , µ n ) T . The functions h j can be interpreted as n quadratic in momenta µ Hamiltonians on the manifold M = T * Q while the n × n symmetric matrices A j (λ) can be interpreted as n twice contravariant symmetric tensors on Q. The Hamiltonians h j commute with respect to Π  [23] of this system. This is the class we will consider throughout our paper. Note that by the very construction of h i the variables (λ, µ) are separation variables for all the Hamiltonians in (2.3) in the sense that the Hamilton-Jacobi equations associated with h j admit a common additively separable solution.
Let us now treat the matrix A 1 as a contravariant form of a metric tensor on Q: A 1 = G, which turns Q into a Riemannian space. The covariant form of G will be denoted by g (so that g = G −1 ). It turns out that the (1, 1)-tensors K j defined by (so that A j = K j G and K 1 = I) are Killing tensors of the metric g.
In this article we will focus on a particular subclass of systems (2.2) that is given by the separation relations (systems of the above class are known in literature as Benenti systems) where moreover where I ⊂ Z is some finite index set (i.e., σ is a Laurent polynomial). Note that taking k ∈ {0, . . . , n − 1} will only yield trivial terms in solutions (2.3) of (2.5), see the end of this section. Also, the parameters α k will play a crucial roll in the sequel, when we discuss the Stäckel transform of the above systems. The metric tensor G attains in this case, due to (2.6), the form where L = diag(λ 1 , . . . , λ n ) is a (1, 1)-tensor (the so called special conformal Killing tensor, see for example [10]) on Q, while Remark 2.1. The metric (2.8) is flat for m ≤ n and of constant curvature for m = n + 1 (see for example [9, p. 788]). For higher m it would have a non-constant curvature.
Further, the Killing tensors K i in (2.4) are in this case given by (2.10) Here and below q i = q i (λ) are Viète polynomials in the variables λ 1 , . . . , λ n : that can also be considered as new coordinates on our Riemannian manifold Q (we will call them Viète coordinates on Q). Notice that q i are coefficients of the characteristic polynomial of the tensor L. Notice also that the first form of K i in (2.10) is of course valid in any coordinate system while the second form of K i is valid in separation coordinates λ only. Further, due to (2.7), the potentials U j (λ) in (2.3) are for the subclass (2.5) given by where the "basic" potentials V k i (k ∈ Z) satisfy the linear system and can be computed by the recursive formula [4,8] T , V (0) = (0, 0, . . . , 0, −1) T and where F is an n × n matrix given with q i (λ) given by (2.11). Note that the formulas (2.13), (2.14) are non tensorial in that they are the same in an arbitrary coordinate system, not only in the separation variables λ i . As we mentioned above, the first potentials, i.e., V (1) = (0, 0, . . . , 0, −1, 0) T up to V (n−1) = (−1, 0, . . . , 0) T are constant, V (n) = (q 1 , . . . , q n ) is the first nonconstant positive potential while V (−1) = (1/q n , q 1 /q n , . . . , q n−1 /q n ) T . The potentials V (k) are for k < 0 rational functions of q that quickly become complicated with decreasing k.
To summarize, the Hamiltonians h i generated by (2.5)-(2.7) can be explicitly written as 3 Maximally superintegrable f lat and constant curvature Stäckel systems Suppose that we have an integrable system, i.e., n functionally independent Hamiltonians on a 2n-dimensional phase space M that pairwise commute: {h i , h j } = 0 for all i, j = 1, . . . , n. If there exists an additional function P commuting to a constant with one of the Hamiltonians, say with h 1 (so that {h 1 , P } = c) and if the n − 1 functions together with all h i are functionally independent, then the system becomes maximally superintegrable (with respect to this particular Hamiltonian h 1 ) since then by the Jacobi identity If moreover the first n integrals of motion h i are quadratic in momenta and if P is linear in momenta, then the resulting n − 1 extra integrals of motion h n+i are also quadratic in momenta. Thus, in order to distinguish those constant curvature Stäckel systems that are maximally superintegrable and have quadratic in momenta extra integrals of motion we have to find P that commutes with h 1 up to a constant and that is linear in momenta. To do it in a systematic way, we need the following well-known result.
where Y is the vector field on Q given by Y = One can thus say that h and P commute if the corresponding vector field Y is the Killing vector for the metric defined by the (2, 0)-tensor Consider now the Stäckel system given by (2.5). The coordinates (λ, µ) are Darboux (so that the above lemma applies to this situation) but the components of the metric (2.8) expressed in λ-coordinates are rational functions making computations very complicated. We will therefore perform the search for the function P in the coordinates (q, p) on M such that q i are Viète coordinates (2.11) and such that are the conjugated momenta. Since the transformation from (λ, µ) to (q, p) is a point transformation the coordinates (q, p) are also Darboux coordinates four our Poisson tensor. It can be shown [7] that in the (q, p)-coordinates and moreover where we set q 0 ≡ 1 and q r = 0 for r > n. An advantage of these new coordinates is that the geodesic parts of h i are polynomial in q.
Example 3.2. For n = 3 and in Viète coordinates (2.11) we have and hence the metric tensors G j have the form In accordance with Remark 2.1, the metric tensors G 0 , . . . , G 3 are flat, while the metric G 4 is of constant curvature.
We are now in position to perform our search for P . We do this in the case when σ is the Laurent polynomial (2.7) and allow f to be polynomial as in (2.6); a particular case of f = λ m of the theorem below was formulated in [6].
is maximally superintegrable in the following cases: (iii) case m = n + 1 (case of constant curvature) and b The additional integrals h n+r commuting with h 1 are given in (q, p)-coordinates by where Y is a vector field on Q given by and where L Y denotes the Lie derivative along Y .
Proof . We will search for a function P that commutes with h 1 and we will perform this search in the (q, p)-coordinates (2.11), (3.1). The Hamiltonian h 1 has in these coordinates the form with G given by (2.8) and further by (3.2), (3.3) and with the potential V = c. In consequence, due to Lemma 3.1, the function for k = n, . . . , 2n − m − 2 and  (3.12). Moreover, if b i = 0 for i = 2, . . . , r ≤ n − 1 then (3.12) is satisfied also for k = −1, . . . , −r + 1.
(iii) For m = n + 1 there is no constant solution of L Y G = 0. This equation has a simple linear in q solution (3.11) provided that b 0 = b 1 = 0 but Y is not a symmetry for any nontrivial potential V (k) 1 . In consequence, the function Poisson commutes only with the geodesic part Finally, the form of additional integrals h n+r in (3.8) is obtained through h n+r = {h r+1 , P } by using Lemma 3.1. Due to their form, the functions h 1 , . . . , h 2n−1 are functionally independent. It is possible to calculate explicitly the structure of the geodesic parts E n+r of the extra integrals h n+r in the separation coordinates (λ, µ).
where q r = q r (λ) are given by (2.11), f (λ) are given by (2.6) while ∆ i by (2.9), (ii) for m = n, n + 1 Let us illustrate the above considerations by some examples.
Then, according to (2.10), (3.4), (3.6) and to (2.13), (2.14) the corresponding Stäckel Hamiltonians attain in the (q, p) coordinates (2.11), (3.1) the form Hence, the system is maximally superintegrable with additional constants of motion for h 1 given by: for k = −1 and b 0 = 0 and for k = 4 Example 3.7. Consider the case n = 3, m = 1, with the monomial f (λ) = λ, given by the separation relations so that I = {−1, 3, 4} and satisfies the condition in part (i) of Theorem 3.3. The system is thus maximally superintegrable and has a three-parameter family of potentials (cf. Remark 3.4). Consider now the point transformation from (q, p)-coordinates (2.11), (3.1) to non-orthogonal coordinates (r, s) such that r i are given by [7] q 1 = r 1 , while are new conjugated momenta. Then r i are flat coordinates for the metric G 1 = A 1 in h 1 . In these coordinates we get in this case while the first three commuting Hamiltonians in (r, s)-variables become 1 (r), 2 (r), Then again, according to (2.10), (3.4)-(3.6) and to (2.13), (2.14), the corresponding Stäckel Hamiltonians attain in the (q, p)-variables the form Now, according to part (iii) of Theorem 3.3, P = q 3 p 1 , Y = q 3 ∂ ∂q 1 and {h 1 , P } = 0 so the additional constants of motion are

Stäckel transforms preserving maximal superintegrability
In this chapter we apply a 1-parameter Stäckel transform to our systems (2.5)-(2.7) to produce new maximally superintegrable Stäckel systems. As the transformation parameter α we will always use one of the α i from (2.7). Stäckel transform is a functional transform that maps a Liouville integrable systems into a new integrable system. It was first introduced in [13] (where it was called the couplingconstant metamorphosis) and later developed in [9]. When applied to a Stäckel separable system, this transformation yields a new Stäckel separable system, which explains its name. In the original paper [13] the authors used only one parameter (one coupling constant). In [22] the authors introduced a multiparameter generalization of this transform. This idea has been further developed in [8] and later in [4].
In this section we prove a theorem (Theorem 4.2) that yields sufficient conditions for Stäckel transform to preserve maximal superintegrability of a Stäckel system.
Let us first, following [4], remind the definition of the multiparameter Stäckel transform. Unless we extend the manifold M this operation cannot be obtained by any coordinate change of variables. Moreover, if we perform again the Stäckel transform on the functionsh i with respect toh s i we will receive back the functions h i in (4.1). In this sense the Stäckel transform is a reciprocal transform. Note also that neither r nor k are related to the dimension of the manifold M .
In [4] we proved that if dim M = 2n, k = r = n and if all h i are functionally independent then also allh i will be functionally independent and if all h i are pairwise in involution with respect to Π then also allh i will pairwise Poisson-commute. That means that if the functions h i , i = 1, . . . , n constitute a Liouville integrable system then alsoh i will constitute a Liouville integrable system. In other words, Stäckel transform preserves Liouville integrability. But what about superintegrability? Theorem 4.2. Consider a maximally superintegrable system on a 2n-dimensional Poisson manifold, i.e., a set of 2n − 1 functionally independent Hamiltonians h 1 , . . . , h 2n−1 such that the first n Hamiltonians pairwise commute, and assume that all the Hamiltonians depend on k ≤ n parameters α i : Note that the Hamiltonian h 1 is now distinguished as the one that commutes with all the remaining h i and as it can only depend on one parameter. Note also that the first n functions h i pairwise commute with each other and therefore constitute a Liouville integrable system. The same is true about the first n functionsh i . h s 1 (x,α 1 , . . . ,α k ), . . . ,h s k (x,α 1 , . . . ,α k ) =α i , i = 1, . . . , k with respect to x we get

Proof . Differentiating the identity
while differentiation of (4.3) yields The transformation (4.5), (4.6) can be written in a matrix form as where we denote dh = (dh 1 , . . . , dh 2n−1 ) T and dh = (dh 1 , . . . , dh 2n−1 ) T and where the (2n − 1) ×(2n − 1) matrix A has the form is not zero and since h i are by assumption functionally independent on M we conclude that also the functionsh i are functionally independent on M . Further, since s k ≤ n (the Stäckel transform is taken with respect to the Hamiltonians belonging to the Liouville integrable system h 1 , . . . , h n ) the columns with derivatives of h i with respect to parameters α j all lie in the left hand side of the matrix A. Moreover, the fact that h 1 = h 1 (x, α 1 ) also means that the first row of A is zero except and due to the aforementioned structure of A we have D ij = 0 for i, j = 1, . . . , n (meaning thath 1 , . . . ,h n constitute a Liouville integrable system) and moreover that D 1i = D i1 = 0 for i = 1, . . . , 2n − 1, so that {h 1 ,h i } = 0 for all i. That concludes the proof.

Remark 4.3.
A similar statement with an analogous proof is valid for any superintegrable system of the form (4.4), not only the maximally superintegrable one.

Stäckel transform of maximally superintegrable Stäckel systems
In this section we perform those Stäckel transforms of our systems (2.5)-(2.7) that preserve maximal superintegrability. According to Theorem 4.2, the Hamiltonian h 1 of the considered system can only depend on one parameter h 1 = h 1 (x, α). It is then natural to choose one of the a k in (2.7) as this parameter. Consider thus a maximally superintegrable system (h 1 , . . . , h 2n−1 ) with the first n commuting Hamiltonians h 1 , . . . , h n defined by our separation relations where the index set I satisfies the assumptions of Theorem 3.3 and where the higher integrals h n+r are constructed as usual through h n+r = {h r+1 , P } with P constructed as in Theorem 3.3.
Let us now choose one of the parameters α s , with s ∈ I, say α k , (we will suppose that k ≥ n or k < 0 otherwise the corresponding potential is trivial, as explained earlier) and define the functions H r , r = 1, . . . , 2n − 1, through Then V (k) r for r = 1, . . . , n obviously coincide with V (k) r defined through (2.5)-(2.7) or equivalently through (2.13), (2.14).
We now perform the Stäckel transform on this system (h 1 , . . . , h 2n−1 ) with respect to the chosen parameter α k as described in Theorem 4.2. It means that we first solve the relation h 1 =α, i.e., H 1 + α k V (k) 1 =α with respect to α k which yields and then replace α k withh 1 in all the remaining Hamiltonians: We obtain in this way a new superintegrable system (h 1 , . . . ,h 2n−1 ) where the first n commuting Hamiltoniansh r are defined by (see [4]) the following separation relations  whereh r for r = 1, . . . , n are defined by (5.4) whileh r for r = n + 1, . . . , 2n − 1 are obtained as usual throughh n+r = h r+1 , P , we see from (5.3) that and from (5.2) it also follows that the geodesic partẼ 1 ofh 1 has the form It means that the metricG is a conformal deformation of either a flat or a constant curvature metric G. In the following theorem we list the cases when the metricG is actually flat or of constant curvature as well. The theorem is formulated only for f in (2.6) being a monomial, f = λ m (in this case there is a maximum number of flat metricsG). If f is a polynomial then the admissible values of k must satisfy the above type of bonds for all powers of λ in f , not only for the highest power m so we choose not to present this more general theorem, only to maintain the simplicity of the picture. In order to prove Theorem 5.2 we need one more lemma. Lemma 5.3 ([20]). The Ricci scalars R andR of the conformally related (covariant) metric tensors g andg = σg are related through where G = g −1 and where where x i are any coordinates on the manifold.
Proof of Theorem 5.2. The values of k for which (5.1) is maximally superintegrable follows from the specification of Theorem 3.3 to the case f = λ m . For (i) and (ii) the metric G in (5.8) is flat so that its Ricci scalar R = 0. Therefore, according to (5.9),R = 0 if and only if s ij G ij = 0. This condition can be effectively calculated in flat coordinates r i of the metric G given by [7] q i = r i + 1 4 In these coordinates (G m ) kl = δ k+l n−m+1 + δ k+l 2n−m+1 and the condition s ij G ij = 0 yields both statements. The case in (i) whenG is of constant curvature (m = 1, k = −1) can be however more effectively proven using Lemma 5.1 since in this case the system (5.4) attains after the transformation (5.5) the form Due to Remark 2.1 the metricG of this system has constant curvature. Finally, in the case (iii) (m = n + 1) we have only negative potentials so by using Lemma 5.1 we transform this system toh where k < 0, and this is the system from case (i) with m = 1 and thereforeG is flat for k ≥ −[(n + 1)/2]. For other values of k the metricG is conformally flat.
Let us now pass to the issue of quantum superintegrability of considered Stäckel systems. We formulate now a quantum analogue of Lemma 3.1. A ij ∇ j ∇ k y k ∇ i − Y (U ).
One proves this lemma by a direct computation. Thus, a sufficient condition for [ h, Y ] = c is satisfied when Y is a Killing vector for both A and g and if moreover U is constant along Y , that is when ∇ k y k = 0). Corollary 6.4. Suppose we have a quantum integrable system on the configuration space Q, that is a set of n commuting and algebraically independent operators h 1 , . . . , h n of the form (6.1) acting in the Hilbert space L 2 (Q, |g| 1/2 dx) where g is some metric on Q. Suppose also that a vector field Y satisfies (6.8) with A 1 and U 1 instead of A and U (so that [ h 1 , Y ] = c). Then, analogously to the classical case, the operators h n+r = h r+1 , Y = 1 2 2 n i,j=1 ∇ i (L Y A r+1 ) ij ∇ j − Y (U r+1 ), r = 1, . . . , n − 1 (6.9) satisfy [ h n+r , h 1 ] = 0 and the system h 1 , . . . , h 2n−1 is algebraically independent; that is we obtain a quantum separable and quantum superintegrable system.
We can now apply this corollary to construct quantum superintegrable counterparts of classical systems considered in previous sections. According to Remark 6.2, for the systems generated by the separation relations (2.5) the most natural choice of the metric g is to take G = A 1 as in (6.5). Then, by construction, [ h i , h j ] = 0 for i, j = 1, . . . , n while the remaining operators h n+r are constructed by the formula (6.9) and are -up to a sign -identical with minimal quantization (in the metric G) of the extra integrals h n+r obtained in (3.8).