Classical and Quantum Superintegrability of Stäckel Systems

In this paper we discuss maximal superintegrability of both classical and quantum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal sup...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2017
Hauptverfasser:	Błaszak, M., Marciniak, K.
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Sprache:	Englisch
Veröffentlicht:	Інститут математики НАН України 2017
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Zitieren:	Classical and Quantum Superintegrability of Stäckel Systems / M. Błaszak, // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ.

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author	Błaszak, M. Marciniak, K.
author_facet	Błaszak, M. Marciniak, K.
citation_txt	Classical and Quantum Superintegrability of Stäckel Systems / M. Błaszak, // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ.
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	In this paper we discuss maximal superintegrability of both classical and quantum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal superintegrability and we apply this condition to our class of Stäckel 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.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 008, 23 pages Classical and Quantum Superintegrability of Stäckel Systems Maciej B LASZAK † and Krzysztof MARCINIAK ‡ † Faculty of Physics, Division of Mathematical Physics, A. Mickiewicz University, Poznań, Poland E-mail: blaszakm@amu.edu.pl ‡ Department of Science and Technology, Campus Norrköping, Linköping University, Sweden E-mail: krzma@itn.liu.se Received September 18, 2016, in final form January 19, 2017; Published online January 28, 2017 https://doi.org/10.3842/SIGMA.2017.008 Abstract. In this paper we discuss maximal superintegrability of both classical and quan- tum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal superintegrability and we apply this condition to our class of Stäckel 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. Key words: Hamiltonian systems; classical and quantum superintegrable systems; Stäckel systems; Hamilton–Jacobi theory; Stäckel transform 2010 Mathematics Subject Classification: 70H06; 70H20; 81S05; 53B20 1 Introduction A real-valued function h1 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 h1, . . . , hn (constants of motion, so that {hi, hj} = 0 for all i, j = 1, . . . , n) and for which there exist n − 1 additional functions hn+1, . . . , h2n−1 on M that Poisson-commute with the Hamiltonian h1 and such that all the functions h1, . . . , h2n−1 constitute a functionally independent set of functions. Analogously, a quantum maximally superintegrable Hamiltonian is a self-adjoint differential operator ĥ1 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 operators ĥ1, . . . , ĥn acting in the same Hilbert space (so that [ĥi, ĥj ] = 0 for all i, j = 1, . . . , n) and such that it also commutes with an additional set of n− 1 differential operators ĥn+1, . . . , ĥ2n−1 of finite order. Besides, in analogy with the classical case, it is required that all the operators ĥ1, . . . , ĥ2n−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 blaszakm@amu.edu.pl krzma@itn.liu.se https://doi.org/10.3842/SIGMA.2017.008 2 M. B laszak and K. Marciniak system (see for example [3, 12] or [15]). Here we present some general results concerning n- dimensional 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 refe- rences, 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 ysps on M such that {h1, P} = c (it also means that the vector field Y = n∑ s=1 ys ∂ ∂qs in P is a Killing vector for the metric generated by h1) which yields additional n− 1 functions hn+i = {hi+1, P} commuting with h1 and thus turning h1 into a maximally superintegrable Hamiltonian. In Sec- tion 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 trans- forms 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 ap- plied 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 hn+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 re- mind 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 (Theo- rem 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. 2 A class of flat 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) = (λ1, . . . , λn, µ1, . . . , µn) 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 Π = n∑ i=1 ∂ ∂λi ∧ ∂ ∂µi . (2.1) 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 σ(λi) + n∑ j=1 hjλ γj i = 1 2 f(λi)µ 2 i , i = 1, . . . , n, γi ∈ N, (2.2) Classical and Quantum Superintegrability of Stäckel Systems 3 where we normalize γn = 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 hj . Solving these equations with respect to hj we obtain n functions hj = hj(λ, µ) on M of the form hj = 1 2 µTAj(λ)µ+ Uj(λ), j = 1, . . . , n, (2.3) where we denote λ = (λ1, . . . , λn)T and µ = (µ1, . . . , µn)T . The functions hj can be interpreted as n quadratic in momenta µ Hamiltonians on the manifold M = T ∗Q while the n × n sym- metric matrices Aj(λ) can be interpreted as n twice contravariant symmetric tensors on Q. The Hamiltonians hj commute with respect to Π {hi, hj} ≡ Π(dhi, dhj) = 0 for all i, j = 1, . . . , n, since the right-hand sides of relations (2.2) commute. Thus, the Hamiltonians in (2.3) constitute a Liouville integrable Hamiltonian system (as they are moreover functionally independent). The Hamiltonians (2.3) constitute a wide class of the so called Stäckel systems [24] on M while the relations (2.2) are called separation relations [23] of this system. This is the class we will consider throughout our paper. Note that by the very construction of hi the variables (λ, µ) are separation variables for all the Hamiltonians in (2.3) in the sense that the Hamilton–Jacobi equations associated with hj admit a common additively separable solution. Let us now treat the matrix A1 as a contravariant form of a metric tensor on Q: A1 = 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 Kj defined by Kj = Ajg, j = 1, . . . , n (2.4) (so that Aj = KjG and K1 = 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 σ(λi) + n∑ j=1 hjλ n−j i = 1 2 f(λi)µ 2 i , i = 1, . . . , n, (2.5) (systems of the above class are known in literature as Benenti systems) where moreover f(λ) = m∑ j=0 bjλ j , bj ∈ R, m ∈ {0, . . . , n+ 1}, (2.6) σ(λ) = ∑ k∈I αkλ k, αk ∈ R, (2.7) 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 αk 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 G = m∑ j=0 bjGj = m∑ j=0 bjL jG0, (2.8) where L = diag(λ1, . . . , λn) is a (1, 1)-tensor (the so called special conformal Killing tensor, see for example [10]) on Q, while Gj = diag ( λj1 ∆1 , . . . , λjn ∆n ) , j ∈ Z, ∆i = ∏ j 6=i (λi − λj). (2.9) 4 M. B laszak and K. Marciniak 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 Ki in (2.4) are in this case given by Ki = i−1∑ r=0 qrL i−1−r = −diag ( ∂qi ∂λ1 , . . . , ∂qi ∂λn ) , i = 1, . . . , n. (2.10) Here and below qi = qi(λ) are Viète polynomials in the variables λ1, . . . , λn: qi(λ) = (−1)i ∑ 1≤s1<s2<···<si≤n λs1 · · ·λsi , i = 1, . . . , n, (2.11) that can also be considered as new coordinates on our Riemannian manifold Q (we will call them Viète coordinates on Q). Notice that qi are coefficients of the characteristic polynomial of the tensor L. Notice also that the first form of Ki in (2.10) is of course valid in any coordinate system while the second form of Ki is valid in separation coordinates λ only. Further, due to (2.7), the potentials Uj(λ) in (2.3) are for the subclass (2.5) given by Uj = ∑ k∈I αkV (k) j , j = 1, . . . , n, (2.12) where the “basic” potentials V k i (k ∈ Z) satisfy the linear system λki + n∑ j=1 V (k) j λn−ji = 0, i = 1, . . . , n, k ∈ Z, and can be computed by the recursive formula [4, 8] V (k) = F kV (0), k ∈ Z, (2.13) where V (k) = ( V (k) 1 , . . . , V (k) n )T , V (0) = (0, 0, . . . , 0,−1)T and where F is an n× n matrix given by F =  −q1(λ) 1 −q2(λ) . . . ... 1 −qn(λ) 0 · · · 0  (2.14) with qi(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 λi. 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) = (q1, . . . , qn) is the first nonconstant positive potential while V (−1) = (1/qn, q1/qn, . . . , qn−1/qn)T . The potentials V (k) are for k < 0 rational functions of q that quickly become complicated with decreasing k. To summarize, the Hamiltonians hi generated by (2.5)–(2.7) can be explicitly written as hr(λ) = −1 2 n∑ i=0 ∂qr ∂λi f(λi)µ 2 i − σ(λi) ∆i = −1 2 n∑ i=0 ∂qr ∂λi f(λi)µ 2 i ∆i + Ur(λ), r = 1, . . . , n. Classical and Quantum Superintegrability of Stäckel Systems 5 3 Maximally superintegrable flat 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: {hi, hj} = 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 h1 (so that {h1, P} = c) and if the n− 1 functions hn+i = {hi+1, P}, i = 1, . . . , n− 1 together with all hi are functionally independent, then the system becomes maximally superin- tegrable (with respect to this particular Hamiltonian h1) since then by the Jacobi identity {hn+i, h1} = −{{P, h1}, hi+1} − {{h1, hi+1}, P} = 0, i = 1, . . . , n− 1. If moreover the first n integrals of motion hi are quadratic in momenta and if P is linear in momenta, then the resulting n−1 extra integrals of motion hn+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 h1 up to a constant and that is linear in momenta. To do it in a systematic way, we need the following well-known result. Lemma 3.1. Suppose that (q, p) = (q1, . . . , qn, p1, . . . , pn) are Darboux (canonical) coordinates on a 2n-dimensional phase space M = T ∗Q. Consider two functions on M : h = 1 2 n∑ i,j=1 piA ij(q)pj + U(q) with A = AT and P = n∑ i=1 yi(q)pi. Then {h, P} = 1 2 n∑ i,j=1 pi(LYA)ijpj + Y (U), where Y is the vector field on Q given by Y = n∑ i=1 yi(q) ∂ ∂qi and where LY is the Lie derivative (on Q) along 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 A (i.e., if LYA = 0) and if moreover Y is symmetry of U (i.e., if Y (U) = 0). 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 qi are Viète coordinates (2.11) and such that pi = − n∑ k=1 (λk) n−iµk ∆k (3.1) are the conjugated momenta. Since the transformation from (λ, µ) to (q, p) is a point transfor- mation the coordinates (q, p) are also Darboux coordinates four our Poisson tensor. It can be shown [7] that in the (q, p)-coordinates (L)ij = −δ1 j qi + δi+1 j , (G0)ij = n−1∑ k=0 qkδ i+j n+k+1, (3.2) 6 M. B laszak and K. Marciniak and moreover (Gr) ij =  n−r−1∑ k=0 qkδ i+j n−r+k+1, i, j = 1, . . . , n− r, − n∑ k=n−r+1 qkδ i+j n−r+k+1, i, j = n− r + 1, . . . , n, 0 otherwise, r = 1, . . . , n, (3.3) (Gn+1)ij = qiqj − qi+j , i, j = 1, . . . , n, where we set q0 ≡ 1 and qr = 0 for r > n. An advantage of these new coordinates is that the geodesic parts of hi are polynomial in q. Example 3.2. For n = 3 and in Viète coordinates (2.11) we have L = −q1 1 0 −q2 0 1 −q3 0 0  , G0 = 0 0 1 0 1 q1 1 q1 q2  , (3.4) and hence the metric tensors Gj have the form G1 = 0 1 0 1 q1 0 0 0 −q3  , G2 = 1 0 0 0 −q2 −q3 0 −q3 0  , (3.5) G3 = −q1 −q2 −q3 −q2 −q3 0 −q3 0 0  , G4 =  q2 1 − q2 q1q2 − q3 q1q3 q1q2 − q3 q2 2 q2q3 q1q3 q2q3 q2 3  . (3.6) In accordance with Remark 2.1, the metric tensors G0, . . . , G3 are flat, while the metric G4 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 = λm of the theorem below was formulated in [6]. Theorem 3.3. The Stäckel system ∑ k∈I αkλ k i + n∑ j=1 hjλ n−j i = 1 2 f(λi)µ 2 i , i = 1, . . . , n, (where I ⊂ Z is a finite index set) with f(λi) given by f(λ) = m∑ j=0 bjλ j , bj ∈ R, m ∈ {0, . . . , n+ 1} (3.7) is maximally superintegrable in the following cases: (i) case m ∈ {0, . . . , n − 1}: if I ⊂ {n, . . . , 2n −m − 1} ∪ {−1, . . . ,−r − 1}, where r is such that bi = 0 for i = 0, . . . , r ≤ m − 1 (if all bi 6= 0, then there is no such r and no second component in I); (ii) case m = n and b0 = b1 = 0: if I ⊂ {n,−1, . . . ,−r + 1}, where r is such that bi = 0 for i = 2, . . . , r ≤ n− 1; (iii) case m = n + 1 (case of constant curvature) and b0 = b1 = 0: if I ⊂ {−1, . . . ,−r + 1}, where r is such that bi = 0 for i = 2, . . . , r ≤ n. Classical and Quantum Superintegrability of Stäckel Systems 7 The additional integrals hn+r commuting with h1 are given in (q, p)-coordinates by hn+r = 1 2 n∑ i,j=1 pi(LYAr+1)ijpj + Y (Ur+1), r = 1, . . . , n− 1, (3.8) where Y is a vector field on Q given by (i) for m ∈ {0, . . . , n− 1} Y = m∑ i=0 bm−i ∂ ∂qn−m+i , (3.9) (ii) for m = n Y = qn n∑ i=2 bn−i+2 ∂ ∂qi , (3.10) (iii) for m = n+ 1 Y = qn n∑ i=1 bn−i+2 ∂ ∂qi , (3.11) and where LY denotes the Lie derivative along Y . Proof. We will search for a function P that commutes with h1 and we will perform this search in the (q, p)-coordinates (2.11), (3.1). The Hamiltonian h1 has in these coordinates the form h1 = 1 2 n∑ i,j=1 Gij(q)pipj + V (k) 1 (q) with G given by (2.8) and further by (3.2), (3.3) and with the potential V (k) 1 (q) defined by (2.13) and (2.14). (i) For m = 0, . . . , n− 1, the Killing equation LYG = 0 has a unique (up to a multiplicative constant) constant solution (3.9) which also satisfies Y ( V (k) 1 ) = 0 for k = n, . . . , 2n−m− 2 and Y ( V (2n−m−1) 1 ) = c. In consequence, due to Lemma 3.1, the function P = bmpn−m + bm−1pn−m+1 + · · ·+ b0pn satisfies {h1, P} = 0 (3.12) for k = n, . . . , 2n−m− 2 and {h1, P} = c for k = 2n−m− 1. Moreover, if bi = 0 for i = 0, . . . , r ≤ m− 1, then (3.12) is satisfied also for k = −1, . . . ,−r − 1. (ii) For m = n, there is no constant solution of LYG = 0. This equation has a simple linear in q solution (3.10) provided that b0 = b1 = 0; Y is then also a symmetry for the single nontrivial potential V (n) 1 , i.e., Y ( V (n) 1 ) = 0. In consequence, the function P = qn(bnp2 + bn−1p3 + · · ·+ b2pn) 8 M. B laszak and K. Marciniak satisfies (3.12). Moreover, if bi = 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 LYG = 0. This equation has a simple linear in q solution (3.11) provided that b0 = b1 = 0 but Y is not a symmetry for any nontrivial potential V (k) 1 . In consequence, the function P = qn(bn+1p1 + bnp2 + · · ·+ b2pn). Poisson commutes only with the geodesic part E1 of h1: {E1, P} = 0. However, if bi = 0 for i = 2, . . . , r ≤ n then (3.12) is satisfied for k = −1, . . . ,−r + 1. Finally, the form of additional integrals hn+r in (3.8) is obtained through hn+r = {hr+1, P} by using Lemma 3.1. Due to their form, the functions h1, . . . , h2n−1 are functionally indepen- dent. � Remark 3.4. The above theorem provides us with a sufficient condition for maximal super- integrability of Stäckel systems of constant curvature (flat in particular) in case when f(λ) is a polynomial of maximal order n + 1. In consequence, the case (i) of Theorem 3.3 yields an (n+ 1)-parameter family of maximally superintegrable systems, parametrized by {br, . . . , bm, α−r−1, . . . , α−1, αn, . . . , α2n−m−1}, r = 0, . . . ,m, where bj parametrize superintegrable metrics (2.6), (2.8) and αj parametrize families of non- trivial superintegrable potentials U (2.12) (in case there is no r, i.e., all bi 6= 0) then there is no α−j in the above set. Similarly, in the cases (ii) and (iii) Theorem 3.3 yields appropriate n-parameter families of superintegrable systems. A particular case of that classification (for the monomial case f(λ) = λm) was presented in [21]. It is possible to calculate explicitly the structure of the geodesic parts En+r of the extra integrals hn+r in the separation coordinates (λ, µ). Proposition 3.5. The geodesic parts En+r = 1 2 n∑ i,j=1 µiA ij n+r(λ)µj , r = 1, . . . , n− 1 of additional integrals of motion hn+r = {hr+1, P} with r = 1, . . . , n− 1 are given by (i) for 0 ≤ m ≤ n− 1 Aijn+r = − ∂2qr ∂λi∂λj f(λi)f(λj) ∆i∆j , i 6= j, Aiin+r = f(λi) ∆i n∑ j=1 ∂2qr ∂λi∂λj f(λj) ∆j , where qr = qr(λ) are given by (2.11), f(λ) are given by (2.6) while ∆i by (2.9), (ii) for m = n, n+ 1 Aijn+r = − ∂2qr ∂λi∂λj ∂qn ∂λi 1 λj f(λi)f(λj) ∆i∆j , i 6= j, Aiin+r = f(λi) ∆i n∑ j=1 ∂2qr ∂λi∂λj f(λj) ∆j ∂qn ∂λj 1 λj . Classical and Quantum Superintegrability of Stäckel Systems 9 Let us illustrate the above considerations by some examples. Example 3.6. Consider the flat case n = 3, m = 1, b1 = 1, (so that f(λ) = b0 + λ) with σ(λ) = αλk and where k = −1, 3 or 4. The commuting Hamiltonians hi are given by separation relations (2.5) αλki + h1λ 2 i + h2λi + h3 = 1 2 (b0 + λi)µ 2 i , i = 1, 2, 3. Then, according to (2.10), (3.4), (3.6) and to (2.13), (2.14) the corresponding Stäckel Hamilto- nians attain in the (q, p) coordinates (2.11), (3.1) the form h1 = p1p2 + b0p1p3 + b0q1p2p3 + 1 2 (q1 + b0)p2 2 + 1 2 (b0q2 − q3)p2 3 + αV (k) 1 (q), h2 = 1 2 p2 1 + 1 2 ( q2 1 + 2b0q1 − q2 ) p2 2 + 1 2 (b0q1q2 − q1q3 − b0q3)p2 3 + (q1 + b0)p1p2 + b0q1p1p3 + ( b0q 2 1 − q3 ) p2p3 + αV (k) 2 (q), h3 = 1 2 b0p 2 1 + 1 2 (b0q 2 1 − q3)p2 2 + 1 2 ( −b0q1q3 + b0q 2 2 − q2q3 ) p2 3 + b0q1p1p2 + (b0q2 − q3)p1p3 + (b0q1q2 − q1q3 − b0q3)p2p3 + αV (k) 3 (q), where V (−1) 1 = 1 q3 , V (−1) 2 = q1 q3 , V (−1) 3 = q2 q3 , V (3) 1 = q1, V (3) 2 = q2, V (3) 3 = q3, V (4) 1 = −q2 1 + q2, V (4) 2 = −q1q2 + q3, V (4) 3 = −q1q3. According to Theorem 3.3 Y = ∂ ∂q2 + b0 ∂ ∂q3 so that P = p2 + b0p3 and thus {h1, P} =  0 for k = −1 and b0 = 0, 0 for k = 3, α for k = 4 ( then Y ( V (4) 1 ) = 1 ) . Hence, the system is maximally superintegrable with additional constants of motion for h1 given by: for k = −1 and b0 = 0 h4 = {h2, P} = −1 2 p2 2, h5 = {h3, P} = −1 2 q3p 2 3 + α q3 q2 3 , for k = 3 h4 = {h2, P} = −1 2 p2 2 − 1 2 b20p 2 3 − b0p2p3 + α, h5 = {h3, P} = −1 2 b0p 2 2 + ( 1 2 b0q2 − 1 2 q3 − 1 2 b20q1 ) p2 3 − b20p2p3 + αb0, and for k = 4 h4 = {h2, P} = −1 2 p2 2 − 1 2 b20p 2 3 − b0p2p3 + α(b0 − q1), h5 = {h3, P} = −1 2 b0p 2 2 + 1 2 (b0q2 − q3 − b20q1)p2 3 − b20p2p3 − αb0q1. 10 M. B laszak and K. Marciniak Example 3.7. Consider the case n = 3, m = 1, with the monomial f(λ) = λ, given by the separation relations α4λ 4 i + α3λ 3 i + h1λ 2 i + h2λi + h3 + α−1λ −1 i = 1 2 λiµ 2 i , i = 1, 2, 3, (3.13) 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 ri are given by [7] q1 = r1, q2 = r2 + 1 4 r2 1, q3 = −1 4 r2 3, (3.14) while sj = 3∑ i=1 ∂qi ∂rj pi, j = 1, 2, 3 (3.15) are new conjugated momenta. Then ri are flat coordinates for the metric G1 = A1 in h1. In these coordinates we get in this case G = G1 = 0 1 0 1 0 0 0 0 1  , L = −1 2r1 1 0 −r2 −1 2r1 −1 2r3 −1 2r3 0 0  , (3.16) while the first three commuting Hamiltonians in (r, s)-variables become h1 = s1s2 + 1 2 s2 3 + α−1V (−1) 1 (r) + α3V (3) 1 (r) + α4V (4) 1 (r), h2 = 1 2 s2 1 − 1 2 r2s 2 2 + 1 2 r1s 2 3 + 1 2 r1s1s2 − 1 2 r3s2s3 + α−1V (−1) 2 (r) + α3V (3) 2 (r) + α4V (4) 2 (r), h3 = 1 8 r2 3s 2 2 + ( 1 2 r2 + 1 8 r2 1 ) s2 3 − 1 2 r3s1s3 − 1 4 r1r3s2s3 + α−1V (−1) 3 (r) + α3V (3) 3 (r) + α4V (4) 3 (r), (3.17) with V (−1) 1 = 4 r2 3 , V (−1) 2 = 4r1 r2 3 , V (−1) 3 = r2 1 + 4r2 r2 3 , (3.18) V (3) 1 = r1, V (3) 2 = ( r2 + 1 4 r2 1 ) , V (3) 3 = −1 4 r2 3, (3.19) V (4) 1 = r2 − 3 4 r2 1, V (4) 2 = − ( r1r2 + 1 4 r3 1 + 1 4 r2 3 ) , V (4) 3 = 1 4 r1r 2 3. (3.20) In accordance with Theorem 3.3 and after the transformation to (r, s)-coordinates we have P = s2, and Y = ∂ ∂r2 so the additional constants of motion hn+i of h1 are h4 = {h2, P} = −1 2 s2 2 + α3 − α4r1, h5 = {h3, P} = 1 2 s2 3 + 4α−1 r2 3 . (3.21) Example 3.8. Consider the constant curvature case n = 3, m = 4 and I = {−2,−1}. In order to apply part (iii) of Theorem 3.3 we have to put b0 = b1 = 0. Assume further that also Classical and Quantum Superintegrability of Stäckel Systems 11 b2 = b3 = 0 and b4 = 1 (so that f(λ) = λ4 is again a monomial). The commuting Hamiltonians are then given by the separation relations α−2λ −2 i + α−1λ −1 i + h1λ 2 i + h2λi + h3 = 1 2 λ4 iµ 2 i , i = 1, 2, 3. 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 h1 = 1 2 ( q2 1 − q2 ) p2 1 + 1 2 q2 2p 2 2 + 1 2 q2 3p 2 3 + (q1q2 − q3)p1p2 + q1q3p1p3 + q2q3p2p3 + α−2V (−2) 1 + α−1V (−1) 1 , h2 = 1 2 (q1q2 − q3)p2 1 + q2q3p 2 2 + q2 2p1p2 + q2q3p1p3 + q2 3p2p3 + α−2V (−2) 2 + α−1V (−1) 2 , h3 = 1 2 q1q3p 2 1 + 1 2 q2 3p 2 2 + q2q3p1p2 + q2 3p1p3 + α−2V (−2) 3 + α−1V (−1) 3 with V (−1) 1 = 1 q3 , V (−1) 2 = q1 q3 , V (−1) 3 = q2 q3 , V (−2) 1 = −q2 q2 3 , V (−2) 2 = 1 q3 − q1q2 q2 3 , V (−2) 3 = q1 q3 − q2 2 q2 3 . Now, according to part (iii) of Theorem 3.3, P = q3p1, Y = q3 ∂ ∂q1 and {h1, P} = 0 so the additional constants of motion are h4 = {h2, P} = −1 2 q2q3p 2 1 − q2 3p1p2 + α−1 − α−2 q2 q3 , h5 = {h3, P} = −1 2 q2 3p 2 1 + α−2. 4 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 αi 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 coupling- constant 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. Consider again a manifold M equipped with a Poisson tensor Π and the corresponding Poisson bracket {·, ·}. Suppose we have r smooth functions hi : M → R on M, each depending on k ≤ r parameters α1, . . . , αk so that hi = hi(x, α1, . . . , αk), i = 1, . . . , r, (4.1) where x ∈ M . Let us now from r functions in (4.1) choose k functions hsi , i = 1, . . . , k, where {s1, . . . , sk} ⊂ {1, . . . , r}. Assume also that the system of equations hsi(x, α1, . . . , αk) = α̃i, i = 1, . . . , k, 12 M. B laszak and K. Marciniak (where α̃i is another set of k free parameters, or values of Hamiltonians hsi) involving the functions hsi can be solved for the parameters αi yielding αi = h̃si(x, α̃1, . . . , α̃k), i = 1, . . . , k, (4.2) where the right hand sides of these solutions define k new functions h̃si on M , each depending on k parameters α̃i. Finally, let us define r− k functions h̃i for i = 1, . . . , r, i /∈ {s1, . . . , sk}, by substituting h̃si from (4.2) instead of αi in hi: h̃i = hi\|α1→h̃s1 ,...,αk→h̃sk , i = 1, . . . , r, i /∈ {s1, . . . , sk}. (4.3) Definition 4.1. The functions h̃i = h̃i(x, α̃1, . . . , α̃k), i = 1, . . . , r, defined through (4.2) and (4.3) are called the (generalized) Stäckel transform of the functions (4.1) with respect to the indices {s1, . . . , sk} (or with respect to the functions hs1 , . . . , hsk). 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 functions h̃i with respect to h̃si we will receive back the functions hi 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 dimM = 2n, k = r = n and if all hi are functionally independent then also all h̃i will be functionally independent and if all hi are pairwise in involution with respect to Π then also all h̃i will pairwise Poisson-commute. That means that if the functions hi, i = 1, . . . , n constitute a Liouville integrable system then also h̃i 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 h1, . . . , h2n−1 such that the first n Hamiltonians pairwise commute, and assume that all the Hamiltonians depend on k ≤ n parameters αi: hi = hi(x, α1, . . . , αk), i = 1, . . . , 2n− 1, {hi, hj} = 0, i, j = 1, . . . , n, for all αi, (4.4) {h1, hn+j} = 0, j = 1, . . . , n− 1, for all αi. Suppose that {s1, . . . , sk} ⊂ {1, . . . , 2n − 1} are chosen so that s1 = 1 and that {s2, . . . , sk} ⊂ {2, . . . , n} and moreover that h1 = h1(x, α1). Then the Stäckel transform h̃i, i = 1, . . . , 2n−1 given by (4.2), (4.3) also satisfy (4.4) and therefore constitute a maximally superintegrable system. Note that the Hamiltonian h1 is now distinguished as the one that commutes with all the remaining hi and as it can only depend on one parameter. Note also that the first n functions hi pairwise commute with each other and therefore constitute a Liouville integrable system. The same is true about the first n functions h̃i. Proof. Differentiating the identity hsi ( x, h̃s1(x, α̃1, . . . , α̃k), . . . , h̃sk(x, α̃1, . . . , α̃k) ) = α̃i, i = 1, . . . , k with respect to x we get dhsi = − k∑ j=1 ∂hsi ∂αj dh̃sj , i = 1, . . . , k, (4.5) Classical and Quantum Superintegrability of Stäckel Systems 13 while differentiation of (4.3) yields dhi = dh̃i − k∑ j=1 ∂hi ∂αj dh̃sj , i = 1, . . . , 2n− 1, i /∈ {s1, . . . , sk}. (4.6) The transformation (4.5), (4.6) can be written in a matrix form as dh = Adh̃, where we denote dh = (dh1, . . . , dh2n−1)T and dh̃ = (dh̃1, . . . , dh̃2n−1)T and where the (2n− 1) ×(2n− 1) matrix A has the form Aij = δij for j /∈ {s1, . . . , sk}, Aisj = − ∂hi ∂αj for j = 1, . . . , k. Since detA = ±det ( ∂hsi ∂αj ) is not zero and since hi are by assumption functionally independent on M we conclude that also the functions h̃i are functionally independent on M . Further, since sk ≤ n (the Stäckel transform is taken with respect to the Hamiltonians belonging to the Liouville integrable system h1, . . . , hn) the columns with derivatives of hi with respect to parameters αj all lie in the left hand side of the matrix A. Moreover, the fact that h1 = h1(x, α1) also means that the first row of A is zero except A11 = − ∂hi ∂α1 . Let us now introduce the (2n− 1)× (2n− 1) matrices C and D through Cij = {hi, hj} and Dij = {h̃i, h̃j}. A direct calculation yields { h̃i, h̃j } = 2n−1∑ l1,l2=1 ( A−1 ) il1 ( A−1 ) jl2 {hl1 , hl2}Π or in matrix form D = A−1C ( A−1 )T , and due to the aforementioned structure of A we have Dij = 0 for i, j = 1, . . . , n (meaning that h̃1, . . . , h̃n constitute a Liouville integrable system) and moreover that D1i = Di1 = 0 for i = 1, . . . , 2n− 1, so that {h̃1, h̃i} = 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. 5 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 h1 of the considered system can only depend on one parameter h1 = h1(x, α). It is then natural to choose one of the ak in (2.7) as this parameter. Consider thus a maximally superintegrable system (h1, . . . , h2n−1) with the first n commuting Hamiltonians h1, . . . , hn defined by our separation relations∑ s∈I αsλ s i + h1λ n−1 i + h2λ n−2 i + · · ·+ hn = 1 2 f(λi)µ 2 i , i = 1, . . . , n, 14 M. B laszak and K. Marciniak where the index set I satisfies the assumptions of Theorem 3.3 and where the higher integrals hn+r are constructed as usual through hn+r = {hr+1, P} with P constructed as in Theorem 3.3. Let us now choose one of the parameters αs, with s ∈ I, say αk, (we will suppose that k ≥ n or k < 0 otherwise the corresponding potential is trivial, as explained earlier) and define the functions Hr, r = 1, . . . , 2n− 1, through hr = Hr + αkV (k) r , r = 1, . . . , 2n− 1. (5.1) Then V (k) r for r = 1, . . . , n obviously coincide with V (k) r defined through (2.5)–(2.7) or equiva- lently through (2.13), (2.14). We now perform the Stäckel transform on this system (h1, . . . , h2n−1) with respect to the chosen parameter αk as described in Theorem 4.2. It means that we first solve the relation h1 = α̃, i.e., H1 + αkV (k) 1 = α̃ with respect to αk which yields h̃1 = αk = − 1 V (k) 1 H1 + α̃ 1 V (k) 1 , (5.2) and then replace αk with h̃1 in all the remaining Hamiltonians: h̃r = Hr − V (k) r V (k) 1 H1 + α̃ V (k) r V (k) 1 , r = 2, . . . , 2n− 1. (5.3) We obtain in this way a new superintegrable system (h̃1, . . . , h̃2n−1) where the first n commuting Hamiltonians h̃r are defined by (see [4]) the following separation relations h̃1λ k i + ∑ s∈I, s 6=k αsλ s i + α̃λn−1 i + h̃2λ n−2 i + · · ·+ h̃n = 1 2 f(λi)µ 2 i , i = 1, . . . , n, (5.4) as it is easy to see, since on the level of the separation relations our Stäckel transform replaces αk with h̃1 and h1 with α̃. For k ≥ n or k < −1 the system (5.4) is no longer in the class (2.5), while for k = −1 it can be easily transformed by a simple point transformation to the form (2.5). Lemma 5.1. The separable system αkλ k i + ∑ s∈I, s 6=k αsλ s i + h1λ n−1 i + h2λ n−2 i + · · ·+ hn = 1 2 λmi µ 2 i , i = 1, . . . , n attains after the Stäckel transform (5.2), (5.3) and after the consecutive point transformation on M given by λi → 1/λi, µi → −λ2 iµi, i = 1, . . . , n (5.5) the form α̃λ−1 i + ∑ s∈I, s 6=k αsλ n−2−s i + h̃1λ n−k−2 i + h̃nλ n−2 i + · · ·+ h̃2 = 1 2 λn−m+2 i µ2 i , i = 1, . . . , n. (5.6) Note that the transformation (5.5) on M does not change the separation web of the system on Q. Denoting, as before h̃r = H̃r + α̃Ṽr, r = 1, . . . , 2n− 1, (5.7) Classical and Quantum Superintegrability of Stäckel Systems 15 where h̃r for r = 1, . . . , n are defined by (5.4) while h̃r for r = n+ 1, . . . , 2n− 1 are obtained as usual through h̃n+r = { h̃r+1, P } , we see from (5.3) that Ṽr = Vr − V (k) r V (k) 1 V1, r = 2, . . . , 2n− 1, and from (5.2) it also follows that the geodesic part Ẽ1 of h̃1 has the form Ẽ1 = n∑ i,j=1 G̃ijpipj , G̃ = − 1 V (k) 1 G. (5.8) It means that the metric G̃ is a conformal deformation of either a flat or a constant curvature metric G. In the following theorem we list the cases when the metric G̃ is actually flat or of constant curvature as well. The theorem is formulated only for f in (2.6) being a monomial, f = λm (in this case there is a maximum number of flat metrics G̃). Theorem 5.2. Consider the system (5.4) with f = λm where m ∈ {0, . . . , n+ 1}. (i) For 0 ≤ m ≤ n− 1 the system (5.4) is maximally superintegrable for k ∈ {−m, . . . ,−1, n, . . . , 2n −m − 1}. The metric G̃ in (5.8) is flat for k ∈ {−[m/2], . . . ,−1, n, . . . , n − 1 + [(n−m)/2]}, where [·] denotes the integer part. Moreover, for m = 1 and k = −1 G̃ is of constant curvature. Otherwise G̃ is conformally flat. (ii) For m = n the system (5.4) is maximally superintegrable for k ∈ {−(n − 2), . . . ,−1, n}. The metric G̃ in (5.8) is flat for k ∈ {−[n/2], . . . ,−1}. Otherwise G̃ is conformally flat. (iii) For m = n+ 1 the system (5.4) is maximally superintegrable for k ∈ {−(n− 1), . . . ,−1}. The metric G̃ in (5.8) is flat for k ∈ {−[(n+ 1)/2], . . . ,−1}. Otherwise G̃ is conformally flat. 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 and R̃ of the conformally related (covariant) metric tensors g and g̃ = σg are related through R̃ = σ−1R− 1 2 (n− 1)σ−1sijG ij , (5.9) where G = g−1 and where sij = sji = 2∇isj − sisj + 1 2 gijsks k with si = σ−1 ∂σ ∂xi , where xi 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 = λm. For (i) and (ii) the metric G in (5.8) is flat so that its Ricci scalar R = 0. Therefore, according to (5.9), R̃ = 0 if and only if sijG ij = 0. This condition can be effectively calculated in flat coordinates ri of the metric G given by [7] qi = ri + 1 4 i−1∑ j=1 rjri−j , i = 1, . . . , n−m, qi = −1 4 n∑ j=i rjrn−j+i, i = n−m+ 1, . . . ,m. 16 M. B laszak and K. Marciniak In these coordinates (Gm)kl = δk+l n−m+1 + δk+l 2n−m+1 and the condition sijG ij = 0 yields both statements. The case in (i) when G̃ 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 h̃1λ n−1 i + ∑ s∈I, s 6=k αsλ n−2−s i + h̃nλ n−2 i + · · ·+ h̃2 + α̃λ−1 i = 1 2 λn+1 i µ2 i , i = 1, . . . , n. Due to Remark 2.1 the metric G̃ 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 to h̃1λ n−k−2 i + ∑ s∈I, s 6=k αsλ n−2−s i + h̃nλ n−2 i + · · ·+ h̃2 + α̃λ−1 i = 1 2 λiµ 2 i , i = 1, . . . , n, where k < 0, and this is the system from case (i) with m = 1 and therefore G̃ is flat for k ≥ −[(n+ 1)/2]. For other values of k the metric G̃ is conformally flat. � If Y (V (k) 1 ) = 0 then Y (1/V (k) 1 ) = 0 and due to (5.8) also LY G̃ = 0 so that {h̃1, P} = 0 as well and the same P as in the “non-tilde”-case (i.e., before the Stäckel transform) can be used as an alternative definition of extra Hamiltonians through h̄n+r = {h̃r+1, P}, r = 1, . . . , n − 1. This is however no longer true if Y ( V (k) 1 ) = c 6= 0 (according to Theorem 3.3, it happens only in the case when m < n and k = 2n−m− 1). It turns out that it leads to the same extra integrals of motion, as the following theorem states Theorem 5.4. If Y (V (k) 1 ) = 0 then both sets of extra integrals of motion: h̄n+r = {h̃r+1, P}, r = 1, . . . , n− 1 and h̃n+r = hn+r\|α=h̃1(α̃) , r = 1, . . . , n− 1 coincide. Proof. On one hand, according to (5.3) and due to the fact that {h̃1, P} = 0 we have h̄n+r = { h̃r+1, P } = { Hr+1 − V (k) r+1 V (k) 1 H1 + α̃ V (k) r+1 V (k) 1 , P } = {Hr+1, P} − H1 V (k) 1 { V (k) r+1, P } + α̃ V (k) 1 { V (k) r+1, P } = {Hr+1, P}+ h̃1 { V (k) r+1, P } . On the other hand, due to h̃n+r = hn+r\|α=h̃1(α̃) = {hr+1, P}\|α=h̃1(α̃) = {Hr+1, P}+ α { V (k) r+1, P }∣∣ α=h̃1(α̃) , which yields the same result. � Classical and Quantum Superintegrability of Stäckel Systems 17 Thus, if Y (V (k) 1 ) = 0, the diagram below commutes (h1, . . . , hn) P−→ (h1, . . . , h2n−1) with hn+r = {hr+1, P} \| \| Stäckel transform Stäckel transform ↓ ↓( h̃1, . . . , h̃n ) P−→ ( h̃1, . . . , h̃2n−1 ) with h̃n+r = { h̃r+1, P } . Example 5.5. Let us apply the relations (5.2), (5.3) to perform the Stäckel transform on the system from Example 3.7. To keep the formulas simple, we assume that all the αs in (2.7) are zero except the transformation parameter αk. Thus, we consider again the system given by the separation relations αkλ k i + h1λ 2 i + h2λi + h3 = 1 2 λiµ 2 i , i = 1, 2, 3 with k = −1, 3 or 4, respectively. Applying Stäckel transform to the resulting Hamiltonians (3.17)–(3.21) we obtain a maximally superintegrable system with the separation relations of the form: h̃1λ k i + α̃λ2 i + h̃2λi + h̃3 = 1 2 λiµ 2 i , i = 1, 2, 3. (5.10) Again we perform our calculations in the (r, s)-variables (3.14), (3.15). Explicitly, we obtain for k = −1 h̃1 = 1 8 r2 3s 2 3 + 1 4 r2 3s1s2 − 1 4 α̃r2 3, h̃2 = 1 2 s2 1 − 1 2 r2s 2 2 − 1 2 r1s1s2 − 1 2 r3s2s3 + α̃r1, h̃3 = 1 8 r2 3s 2 2 − ( 1 4 r2 1 + r2 ) s1s2 − 1 2 r3s1s3 − 1 4 r1r3s2s3 + 1 4 α̃ ( r2 1 + 4r2 ) , h̃4 = −1 2 s2 2, h̃5 = −s1s2 + α̃, (5.11) for k = 3 h̃1 = − 1 r1 s1s2 − 1 2 1 r1 s2 3 + α̃ 1 r1 , h̃2 = 1 2 s2 1 + 1 4 r2 1 − 4r2 r1 s1s2 − 1 2 r2s 2 2 − 1 2 r3s2s3 + 1 8 3r2 1 − 4r2 r1 s2 3 + 1 4 α̃ r2 1 + 4r2 r1 , h̃3 = 1 4 r2 3 r1 s1s2 − 1 2 r3s1s3 + 1 8 r2 3s 2 2 − 1 4 r1r3s2s3 + 1 8 r3 1 + 4r1r2 + r2 3 r1 s2 3 − 1 4 α̃ r2 3 r1 , h̃4 = − 1 r1 s1s2 − 1 2 s2 2 − 1 2 1 r1 s2 3 + α̃ 1 r1 , h̃5 = 1 2 s2 3, and for k = 4 h̃1 = − 1 r2 − 3 4r 2 1 s1s2 − 1 2 1 r2 − 3 4r 2 1 s2 3 + α̃ 1 r2 − 3 4r 2 1 , h̃2 = 1 2 s2 1 − 1 2 r2s 2 2 − 1 8 2r3 1 − 8r1r2 − r2 3 r2 − 3 4r 2 1 s2 3 − 1 8 r3 1 − 12r1r2 − 2r2 3 r2 − 3 4r 2 1 s1s2 − 1 2 r3s2s3 − α̃ r1r2 + 1 4r 3 1 + 1 4r 2 3 r2 − 3 4r 2 1 , 18 M. B laszak and K. Marciniak h̃3 = 1 8 r2 3s 2 2 − 1 32 3r4 1 + 8r2 1r2 + 4r1r 2 3 + 16r2 2 r2 − 3 4r 2 1 s2 3 + 1 4 r1r 2 3 r2 − 3 4r 2 1 s1s2 − 1 2 r3s1s3 − 1 4 r1r3s2s3 + 1 4 α̃ r1r 2 3 r2 − 3 4r 2 1 , h̃4 = −1 2 s2 2 + 1 2 r1 r2 − 3 4r 2 1 s2 3 + r1 r2 − 3 4r 2 1 s1s2 − α̃ r1 r2 − 3 4r 2 1 , h̃5 = 1 2 s2 3. (5.12) According to part (i) of Theorem 5.2 the metrics of h̃1 are of constant curvature, flat and conformally flat, respectively. 6 Quantization of maximally superintegrable Stäckel systems This section is devoted to separable quantizations of Stäckel systems that were considered in the classical setting in the previous sections. Let us consider, as in the classical case, an n- dimensional Riemannian space Q equipped with a matric tensor g and the quadratic in momenta Hamiltonian on the cotangent bundle T ∗Q: h = 1 2 n∑ i,j=1 piA ij(x)pj + U(x). By its minimal quantization [5] we mean the following self-adjoint operator ĥ = −1 2 }2 n∑ i,j=1 ∇iAij(x)∇j + U(x) = −1 2 }2 n∑ i,j=1 1√ \|g\| ∂i √ \|g\|Aij(x)∂j + U(x) (6.1) (both expressions on the right hand side of (6.1) are equivalent) acting in the Hilbert space H = L2(Q, dµ), dµ = \|g\|1/2dx, \|g\| = \|det g\|, where ∇ is the Levi-Civita connection of the metric g. Note that a priori there is no relation between the tensor A and the metric g. Let us now consider an arbitrary Stäckel system of the form (2.3) coming from the separation relations (2.2). Applying the procedure of minimal quantization to this system will in general yield a non-integrable and non-separable quantum system. In order to preserve integrability and separability we have to carefully choose the metric g. To do this, we will use the following theorem, proved in [5]. Theorem 6.1. Suppose that hj are Hamiltonian functions (2.3), defined by separation rela- tions (2.2). Suppose also that θ is an arbitrary function of one variable. Applying to hj the procedure of minimal quantization (6.1) with the metric tensor g = ϕ 2 n gθ, (6.2) where gθ = G−1 θ with Gθ given by Gθ = diag ( θ(λ1) ∆1 , . . . , θ(λn) ∆n ) , (6.3) and with ϕ being a particular function of λ1, . . . , λn, uniquely defined by (2.2) (see formula (27) in [5] for details), we obtain a quantum integrable and separable system. More precisely, we obtain n operators ĥi of the form (6.1) such that (i) [ĥi, ĥj ] = 0 for all i, j and (ii) eigenvalue problems for all ĥi ĥiΨ = εiΨ, i = 1, . . . , n Classical and Quantum Superintegrability of Stäckel Systems 19 have for each choice of eigenvalues εi of ĥi the common multiplicatively separable eigenfunction Ψ(λ1, . . . , λn) = n∏ i=1 ψ(λi) with ψ satisfying the following ODE (quantum separation relation)( ε1λ γ1 + ε2λ γ2 + · · ·+ εn ) ψ(λ) = −1 2 ~2f(λ) [ d2ψ(λ) dλ2 + ( f ′(λ) f(λ) − 1 2 θ′(λ) θ(λ) ) dψ(λ) dλ ] + σ(λ)ψ(λ). (6.4) Remark 6.2. For Stäckel systems defined by (2.5), when (γ1, . . . , γn) = (n− 1, . . . , 0) in (2.2), we have ϕ = 1 and the most natural choice in (6.3) is to put θ = f which yields the metric for quantization G = Gf = A1. (6.5) On the other hand, for Stäckel systems defined by (5.4) we have ϕ = −V (k) 1 (as it follows from (5.2) and the formula (27) in [5]) and again the simplest choice in (6.3) is to put θ = f which yields according to (6.2) and (5.8) the metric for quantization G = ϕ− 2 nGf = ϕ1− 2 n Ã1. (6.6) For the choice (6.5) and (6.6) the quantum separation equation (6.4) reduce to( ε1λ γ1 + ε2λ γ2 + · · ·+ εn ) ψ(λ) = −1 2 ~2f(λ) [ d2ψ(λ) dλ2 + 1 2 f ′(λ) f(λ) dψ(λ) dλ ] + σ(λ)ψ(λ), (6.7) where (γ1, . . . , γn) = (n−1, n−2, . . . , 0) in the first case (6.5) and (γ1, . . . , γn) = (k, n−2, n−3, . . . , 0) in the second case (6.6). Let us now pass to the issue of quantum superintegrability of considered Stäckel systems. We formulate now a quantum analogue of Lemma 3.1. Lemma 6.3. Suppose that ĥ is given by (6.1) and that Y = n∑ i=1 yi(x)∇i is a vector field on the Riemannian manifold Q with a metric g. Then[ ĥ, Y ] = 1 2 }2 n∑ i,j=1 ∇i(LYA)ij∇j + 1 2 }2 n∑ i,j,k=1 Aij ( ∇j∇kyk ) ∇i − Y (U). One proves this lemma by a direct computation. Thus, a sufficient condition for [ĥ, 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 LYA = 0, LY g = 0, Y (U) = c (6.8) (note that LY g = 0 implies n∑ i=1 ∇kyk = 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 ĥ1, . . . , ĥn of the form (6.1) acting in the Hilbert space L2(Q, \|g\|1/2dx) where g is some metric on Q. Suppose also that a vector field Y satisfies (6.8) with A1 and U1 instead of A and U (so that [ĥ1, Y ] = c). Then, analogously to the classical case, the operators ĥn+r = [ ĥr+1, Y ] = 1 2 }2 n∑ i,j=1 ∇i(LYAr+1)ij∇j − Y (Ur+1), r = 1, . . . , n− 1 (6.9) satisfy [ĥn+r, ĥ1] = 0 and the system ĥ1, . . . , ĥ2n−1 is algebraically independent; that is we obtain a quantum separable and quantum superintegrable system. 20 M. B laszak and K. Marciniak We can now apply this corollary to construct quantum superintegrable counterparts of classi- cal 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 = A1 as in (6.5). Then, by construction, [ĥi, ĥj ] = 0 for i, j = 1, . . . , n while the remaining operators ĥn+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 hn+r obtained in (3.8). Example 6.5. Consider again separation relations (3.13) from Example 3.7, so that f(λ) = λ and σ = α−1λ −1 +α3λ 3 +α4λ 4. Performing the minimal quantization of the Hamiltonians (3.17) in the metric G = A1, i.e., given by (3.16), we obtain, in the flat r-coordinates (3.14) ĥ1 = −1 2 }2 ( ∂1∂2 + 1 2 ∂2 3 ) + α−1V (−1) 1 (r) + α3V (3) 1 (r) + α4V (4) 1 (r), ĥ2 = −1 4 }2 ( ∂2 1 − ∂2r2∂2 + r1∂3∂3 + 1 2 ∂1r1∂2 + 1 2 r1∂2∂1 − r3 1 2 ∂2∂3 − 1 2 ∂3r3∂2 ) + α−1V (−1) 2 (r) + α3V (3) 2 (r) + α4V (4) 2 (r), ĥ3 = −1 8 }2 ( 1 2 r2 3∂ 2 2 + ( 2r2 + 1 2 r2 1 ) ∂2 3 − r3∂1∂3 − ∂3r3∂1 − 1 2 r1r3∂2∂3 − 1 2 r1∂3r3∂2 ) + α−1V (−1) 3 (r) + α3V (3) 3 (r) + α4V (4) 3 (r), where ∂i = ∂/∂ri and V (k) i are given by (3.18)–(3.20). The respective separation equation, according to (3.13) and (6.7), is of the form ( α−1λ −1 + α3λ 3 + α4λ 4 + ε1λ 2 + ε2λ+ ε3 ) ψ(λ) = −1 2 ~2 [ λ d2ψ(λ) dλ2 + 1 2 dψ(λ) dλ ] . Now Y = ∂2 satisfies the conditions (6.8) and the extra operators ĥ4, ĥ5 can be obtained either by using the formula (6.9) or directly by minimal quantization of functions h4, h5 in (3.21). The result is (up to a sign) ĥ4 = 1 4 }2∂2 2 − α3 + α4r1, ĥ5 = −1 4 }2∂2 3 + 4α−1 r2 3 . If we want to perform the separable quantization of superintegrable systems obtained by the Stäckel transform, as in Section 5, we have two cases: either the system – after the Stäckel transform – belongs again to the same class (2.5) or belongs to the other class, given by the separation relations (5.4) that are different from (2.5) as soon as k 6= −1. Again by Remark 6.2, in the first case the natural choice of the metric in which we perform the minimal quantization is to take G̃ = Ã1, i.e., G̃ as given by (5.8). In the second case we have to use the metric given by (6.2) which in our case is given by (6.6), i.e., by G = ϕ1− 2 n Ã1 with ϕ = −V (k) 1 . Example 6.6. Let us now minimally quantize the Stäckel Hamiltonians h̃1, h̃2, h̃3 given in (5.11), obtained through a Stäckel transform in Example 5.5, generated by the separation relations (5.10) with k = −1, that is by h̃1λ −1 i + α̃λ2 i + h̃2λi + h̃3 = 1 2 λiµ 2 i , i = 1, 2, 3. The metric associated with h̃1 G̃ = 1 4 r2 3 0 1 0 1 0 0 0 0 1  (6.10) Classical and Quantum Superintegrability of Stäckel Systems 21 is of constant curvature as – by Lemma 5.1 – after applying transformation (5.5), in the new separation coordinates the separation relations (5.10) turns to α̃λ−1 i + h̃1λ 2 i + h̃3λi + h̃2 = 1 2 λ4 iµ 2 i , i = 1, 2, 3 and belong again to the class (2.5). Thus, by Remark 6.2, we have to perform the minimal quan- tization of this system with respect to the original metric Ã1 of the system which is just (6.10). Observing that √ \|g̃\| = 8/r3 3, we obtain the following quantum superintegrable system (we use the second expression in (6.1)): ̂̃ h1 = −1 4 }2r2 3 ( 1 2 r3∂3 1 r3 ∂3 + ∂1∂2 ) − 1 4 α̃r2 3, ̂̃ h2 = 1 4 }2 ( −2∂2 1 + 2∂2r2∂2 + ∂1r1∂2 + r1∂2∂1 + r3∂2∂3 + r3 3∂3 1 r2 3 ∂2 ) + α̃r1, ̂̃ h3 = 1 8 }2 [ −r2 3∂ 2 2 + (r2 1 + 4r2)∂1∂2 + ∂1r 2 1∂2 + 4∂2r2∂1 + 2r3∂1∂3 + 2r3 3∂3 1 r2 3 ∂1 + r1r3∂2∂3 + r1r 3 3∂3 1 r2 3 ∂1 ] + 1 4 α̃ ( r2 1 + 4r2 ) , ̂̃ h4 = 1 2 }2∂2 2 , ̂̃ h5 = }2∂1∂2 + α̃. Example 6.7. Let us finally minimally quantize the Stäckel Hamiltonians h̃1, h̃2, h̃3 given in (5.12), obtained through a Stäckel transform in Example 5.5) and generated by separation relations (5.10) with k = 4 h̃1λ 4 i + α̃λ2 i + h̃2λi + h̃3 = 1 2 λiµ 2 i , i = 1, 2, 3. The metric associated with h̃1 G̃ = 1 3 4r 2 1 − r2 0 1 0 1 0 0 0 0 1  is conformally flat. By Remark 6.2, we have to perform minimal quantization of this system with respect to the metric (6.6) given by G = ( − V (4) 1 )1− 2 3 G̃ = ( r2 − 3 4 r2 1 )− 2 3 0 1 0 1 0 0 0 0 1  . Observing that √ \|g\| = V (4) 1 = r2 − 3 4r 2 1, we obtain the following quantum operators (we use again the second expression in (6.1)): ̂̃ h1 = 1 2 }2 ( r2 − 3 4 r2 1 )−1 ( 2∂1∂2 + ∂2 3 ) + α̃ r2 − 3 4r 2 1 , ̂̃ h2 = −1 2 }2 ( r2 − 3 4 r2 1 )−1∑ i,j ∂iB ij 2 ∂j − α̃ r1r2 + 1 4r 3 1 + 1 4r 2 3 r2 − 3 4r 2 1 , ̂̃ h3 = −1 2 }2 ( r2 − 3 4 r2 1 )−1∑ i,j ∂iB ij 3 ∂j + 1 4 α̃ r1r 2 3 r2 − 3 4r 2 1 , 22 M. B laszak and K. Marciniak ̂̃ h4 = −1 2 }2 ( r2 − 3 4 r2 1 )−1 [ ∂1r1∂2 + r1∂2∂1 − ∂2 ( r2 − 3 4 r2 1 ) ∂2 + r1∂ 2 3 ] − α̃ r1 r2 − 3 4r 2 1 , ̂̃ h5 = −1 2 }2∂2 3 , where B2 =  r2 − 3 4r 2 1 3 2r1r2 − 1 8r 3 1 + 1 4r 2 3 0 3 2r1r2 − 1 8r 3 1 + 1 4r 2 3 −r2 ( r2 − 3 4r 2 1 ) −1 2r3 ( r2 − 3 4r 2 1 ) 0 −1 2r3 ( r2 − 3 4r 2 1 ) 2r1r2 − 1 2r 3 1 + 1 4r 2 3  , B3 =  0 −1 4r1r 2 3 −1 2r3 ( r2 − 3 4r 2 1 ) −1 4r1r 2 3 1 4r 2 3 ( r2 − 3 4r 2 1 ) −1 4r1r3 ( r2 − 3 4r 2 1 ) −1 2r3 ( r2 − 3 4r 2 1 ) −1 4r1r3 ( r2 − 3 4r 2 1 ) −1 2r 2 1r2 + r2 2 − 1 4r1r 2 3 − 3 16r 4 1  with B = √ \|g\|A in (6.1). It can be checked that it is again a quantum superintegrable system. References [1] Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., Superintegrable anharmonic oscillators on N -dimen- sional curved spaces, J. Nonlinear Math. Phys. 15 (2008), suppl. 3, 43–52, arXiv:0710.0843. [2] Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., Riglioni D., A maximally superintegrable deformation of the N -dimensional quantum Kepler–Coulomb system, J. Phys. Conf. Ser. 474 (2013), 012008, 9 pages, arXiv:1310.6554. [3] Ballesteros Á., Herranz F.J., Santander M., Sanz-Gil T., Maximal superintegrability on N -dimensional curved spaces, J. Phys. A: Math. Gen. 36 (2003), L93–L99, math-ph/0211012. [4] B laszak M., Marciniak K., On reciprocal equivalence of Stäckel systems, Stud. Appl. Math. 129 (2012), 26–50, arXiv:1201.0446. [5] B laszak M., Marciniak K., Domański Z., Separable quantizations of Stäckel systems, Ann. Physics 371 (2016), 460–477, arXiv:1501.00576. [6] B laszak M., Sergyeyev A., Maximal superintegrability of Benenti systems, J. Phys. A: Math. Gen. 38 (2005), L1–L5, nlin.SI/0412018. [7] B laszak M., Sergyeyev A., Natural coordinates for a class of Benenti systems, Phys. Lett. A 365 (2007), 28–33, nlin.SI/0604022. [8] B laszak M., Sergyeyev A., Generalized Stäckel systems, Phys. Lett. A 375 (2011), 2617–2623. [9] Boyer C.P., Kalnins E.G., Miller Jr. W., Stäckel-equivalent integrable Hamiltonian systems, SIAM J. Math. Anal. 17 (1986), 778–797. [10] Crampin M., Sarlet W., A class of nonconservative Lagrangian systems on Riemannian manifolds, J. Math. Phys. 42 (2001), 4313–4326. [11] Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with inte- grals quadratic in momenta on a two-dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055. [12] Gonera C., Isochronic potentials and new family of superintegrable systems, J. Phys. A: Math. Gen. 37 (2004), 4085–4095. [13] Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., Coupling-constant metamorphosis and duality be- tween integrable Hamiltonian systems, Phys. Rev. Lett. 53 (1984), 1707–1710. [14] 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. [15] Kalnins E.G., Kress J.M., Miller Jr. W., Pogosyan G.S., Nondegenerate superintegrable systems in n- dimensional Euclidean spaces, Phys. Atomic Nuclei 70 (2007), 545–553. [16] 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. http://dx.doi.org/10.2991/jnmp.2008.15.s3.5 http://arxiv.org/abs/0710.0843 http://dx.doi.org/10.1088/1742-6596/474/1/012008 http://arxiv.org/abs/1310.6554 http://dx.doi.org/10.1088/0305-4470/36/7/101 http://arxiv.org/abs/math-ph/0211012 http://dx.doi.org/10.1111/j.1467-9590.2011.00544.x http://arxiv.org/abs/1201.0446 http://dx.doi.org/10.1016/j.aop.2016.06.007 http://arxiv.org/abs/1501.00576 http://dx.doi.org/10.1088/0305-4470/38/1/L01 http://arxiv.org/abs/nlin.SI/0412018 http://dx.doi.org/10.1016/j.physleta.2007.01.001 http://arxiv.org/abs/nlin.SI/0604022 http://dx.doi.org/10.1016/j.physleta.2011.05.046 http://dx.doi.org/10.1137/0517057 http://dx.doi.org/10.1137/0517057 http://dx.doi.org/10.1063/1.1388030 http://dx.doi.org/10.1063/1.1388030 http://dx.doi.org/10.1063/1.2192967 http://arxiv.org/abs/math-ph/0412055 http://dx.doi.org/10.1088/0305-4470/37/13/011 http://dx.doi.org/10.1103/PhysRevLett.53.1707 http://dx.doi.org/10.1063/1.2337849 http://dx.doi.org/10.1134/S1063778807030143 http://dx.doi.org/10.1088/0305-4470/34/22/311 http://arxiv.org/abs/math-ph/0102006 Classical and Quantum Superintegrability of Stäckel Systems 23 [17] Kalnins E.G., Miller W., Post S., Models for quadratic algebras associated with second order superintegrable systems in 2D, SIGMA 4 (2008), 008, 21 pages, arXiv:0801.2848. [18] Marciniak K., B laszak M., Flat coordinates of flat Stäckel systems, Appl. Math. Comput. 268 (2015), 706– 716, arXiv:1406.2117. [19] 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. [20] Schouten J.A., Ricci-calculus. An introduction to tensor analysis and its geometrical applications, Grundlehren der Mathematischen Wissenschaften, Vol. 10, 2nd ed., Springer-Verlag, Berlin – Göttingen – Heidelberg, 1954. [21] Sergyeyev A., Exact solvability of superintegrable Benenti systems, J. Math. Phys. 48 (2007), 052114, 11 pages, nlin.SI/0701015. [22] Sergyeyev A., B laszak M., Generalized Stäckel transform and reciprocal transformations for finite- dimensional integrable systems, J. Phys. A: Math. Theor. 41 (2008), 105205, 20 pages, arXiv:0706.1473. [23] Sklyanin E.K., Separation of variables – new trends, Progr. Theoret. Phys. Suppl. (1995), 35–60, solv-int/9504001. [24] Stäckel P., Über die Integration der Hamilton–Jacobischen Differential Gleichung Mittelst Separation der Variabeln, Habilitationsschrift, Halle, 1891. http://dx.doi.org/10.3842/SIGMA.2008.008 http://arxiv.org/abs/0801.2848 http://dx.doi.org/10.1016/j.amc.2015.06.099 http://arxiv.org/abs/1406.2117 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/10.1007/978-3-662-12927-2 http://dx.doi.org/10.1063/1.2738829 http://arxiv.org/abs/nlin.SI/0701015 http://dx.doi.org/10.1088/1751-8113/41/10/105205 http://arxiv.org/abs/0706.1473 http://dx.doi.org/10.1143/PTPS.118.35 http://arxiv.org/abs/solv-int/9504001 1 Introduction 2 A class of flat and constant curvature Stäckel systems 3 Maximally superintegrable flat and constant curvature Stäckel systems 4 Stäckel transforms preserving maximal superintegrability 5 Stäckel transform of maximally superintegrable Stäckel systems 6 Quantization of maximally superintegrable Stäckel systems References
id	nasplib_isofts_kiev_ua-123456789-148607
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2025-11-29T13:26:49Z
publishDate	2017
publisher	Інститут математики НАН України
record_format	dspace
spelling	Błaszak, M. Marciniak, K. 2019-02-18T16:31:01Z 2019-02-18T16:31:01Z 2017 Classical and Quantum Superintegrability of Stäckel Systems / M. Błaszak, // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70H06; 70H20; 81S05; 53B20 DOI:10.3842/SIGMA.2017.008 https://nasplib.isofts.kiev.ua/handle/123456789/148607 In this paper we discuss maximal superintegrability of both classical and quantum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal superintegrability and we apply this condition to our class of Stäckel 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Classical and Quantum Superintegrability of Stäckel Systems Article published earlier
spellingShingle	Classical and Quantum Superintegrability of Stäckel Systems Błaszak, M. Marciniak, K.
title	Classical and Quantum Superintegrability of Stäckel Systems
title_full	Classical and Quantum Superintegrability of Stäckel Systems
title_fullStr	Classical and Quantum Superintegrability of Stäckel Systems
title_full_unstemmed	Classical and Quantum Superintegrability of Stäckel Systems
title_short	Classical and Quantum Superintegrability of Stäckel Systems
title_sort	classical and quantum superintegrability of stäckel systems
url	https://nasplib.isofts.kiev.ua/handle/123456789/148607
work_keys_str_mv	AT błaszakm classicalandquantumsuperintegrabilityofstackelsystems AT marciniakk classicalandquantumsuperintegrabilityofstackelsystems

Classical and Quantum Superintegrability of Stäckel Systems

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