Journal of Applied Mathematics and Physics, 2014, 2, 996999 Published Online October 2014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2014.211113 How to cite this paper: Sasaki, Y. (2014) Degeneration of the Superintegrable System with Potentials Described by the Sixth Painlevé Transcendents. Journal of Applied Mathematics and Physics, 2, 996999. http://dx.doi.org/10.4236/jamp.2014.211113 Degeneration of the Superintegrable System with Potentials Described by the Sixth Painlevé Transcendents Yoshikatsu Sasaki Department of Mathematics, Hiroshima University, HigashiHiroshima, Japan Email: sasakiyo@hiroshimau.ac.jp Received Ju ly 2014 Abstract This article concerns the quantum superintegrable system obtained by Tremblay and Winternitz, which allows the separation of variables in polar coordinates and possesses three conserved quantities with the potential described by the sixth Painlevé equation. The degeneration proce dure from the sixth Painlvé equation to the fifth one yields another new superintegrable system; however, the Hermitian nature is broken. Keywords Superintegrable System, Painlevé Equat i on, Degenerati on 1. Introduction 1.1. Superintegrable Systems Separating in Polar Coordinates Consider the quantum superintegrable system which allows the separation of variables in polar coordinates and possesses three conserved quantities as follows: , , { } ( ) { } ( ) { } 3 121122 3 , ,, k lm klm klm YAL ppgpgp ++ = = ++ ∑ xx , where () () 12 ,cos, sinxx rr θθ = =x , , , , . Here , , , are arbitrary functions, and ’s are real constants. Note that the Hermitian nature of the operators causes the anticommutator and the parity. In the followings, we use such the notation as for brevity. Tremblay and Winternitz [1 ] classified the cases where the above system is superintegrable, i.e. it allows the
Y. Sasaki third conserved quantity , and obtained which is written by the solution of the sixth Painlevé equation. If , , , { } ( ) 22 1222 2rr Hr rSr θ θ −− =−∂+∂+∂ + , , { } {} {} {}{} { } ( ) 2 31 121122 22 12 ,cossin ,sincos , 11 1 ,cossin,,, rr YL pGGrpGGrp GGr rr r θθ θ θθθ θ θθ =+ −++ =−∂∂−∂+∂ +∂ , ()( )( ) 22 1 ,2 cosGG r'S' r θβθθ θ == − , ( )( )() 0 sin2 cosTT' βθβ θθθ =+− + , ; . The commutation is reduced to ()( ) { } ( ) { } 20 0 sin4cos6sin4cos43 sincos 82cossin0. T''''TTT'T T'T' T' T'T θθθθθ θβθ θθβθ ′′′ ′′′′ − −++++− +−+ = By change of variables s.t. ()( ) ( ) ( ) { } ( ) 22 21 1 tan2112, 4121 8 tttTwt tt θ ββ = −−=+++−− , the above equation is reduced to FVII . Here, FVII is a 4th order ODE defined by ( )()( )( )()( ) { } ( ) 2 22 0 2 21621 124 18212 212 1421 80, ttw''''ttt wtt w'wt wwAttwt w' ww' A ′′′′′ ′′′′ −−−−−−−+− +−−−−− + += with an independent variable , a dependent variable , constant parameters and (See [1] [2]). This equation can be integrated twice, and reduced to SDI.a [1] [3] [4] with constants of integration and : ( )()()( ) 22 222 2 0 234 14 40t tww'tw' ww' tw' wAw'Atw' wBwA ′′ ′′ − −−−+−++−++= . Then, by the Bäcklund correspondence ()()()( ) { } () () { } ( )() () () { } ( )( ) { } ( )()() { } () () { } ( )()( )( ) 2 2 2 22 0 22 1 222 01 11 1 41111212 88 11 1211121, 88 11 1 411111, 44 t wtttuuu tu' uuttutu tutuu t wtttuuu'uuttu tutu ttu θ θθ θ θθ ∞ ∞ =−−−−−−+Θ −+− − −−−+ −−− ′=−−−±−−+ −−− −− SDI.a is reduced to the sixth Painlevé equation [3]: ()() { } () () { } ( )()() { } () () ( ) () () 2 2 22 222 222 01 111111 111 2 1122121112, t uuuu t u'ttu t u' uuutttt ututtut θθ θθ ∞ ′′ = +−+− −+−+− +−−−−+− −−−−−
Y. Sasaki where the correspondence of the parameters is given by , () ()( )()( ) ()( )()() ()() { } 2 2222 2222222 001101 210 2 2 2222 2222 222 31040101 2 3 30 2, 4, 4, 4, 32, 4. tt t tt t AAA AA B AA θθθ θθθθθθ θθθθθθ θθθ ∞∞∞ ∞ ∞∞ =Θ+++=Θ− −=Θ− − =Θ−−= Θ+−+Θ−+ = + FVII is reduced to the sixth Painlevé equation with because of the symmetry of FVII . 1.2. Degeneration Scheme of the Painlevé Equations Six Painlevé equations are the nonlinear ODEs which define the special functions containing Gauss’ hypergeo metric function, Bessel functions, Airy functions, etc., and yields elliptic functions and trigonometric functions as the autonomous limits [5] [6]. Solutions to the Painlevé equations are called the Painlevé transcendents. So, the Painlevé transcendents are the ancestors of all classical special functions satisfying ODEs. And, all of the Painlevé equations are derived from the sixth Painlevé equation by some limitation which is called the degene ration scheme [5] [7]. For example, the fifth Painlevé equation: ( ) {} ( ) { } () () ( ) ( ) () () 2 222 2 0 2 12 11122 12 11 uuuu'u' tutuu utuu u θθ ηκ η ∞ ′′ =+−−+ −+− + ++−+− is derived from the sixth Painlevé equation as follows: replace by ( ) 0 ,1;, ,,ut ε θθ ηεκηε ∞ + +− , and then take limitation . In this article, we liftup the degeneration scheme of the Painlevé equation to the superintegrable system, and get the system with potential described by the fifth Painlevé transcendents. The degenerated system should break one or more rules for classification set up by Tremblay and Winternitz [1]. 2. Results Theorem 1. By change of variables , , the superintegrable system obtained by Tremblay and Winternitz [1] is reduced into the system ( ) ( ) ( ) 22 21 22 1 2rr s HrrssSs r −− =−∂+∂+−−∂+ , ( ) ( ) ( ) 2 2 12 s X ssSs=−−−∂+ , ( )() ( ) { } {} {} 11 2 2 212 1, ,, sx xx Y issgg−=−−−∂∂+∂ +∂ , where , , ( ) 2 1 2 1rs x ss s sr −+ − ∂ =∂−∂ − . Moreover, FVII is reduced to FVII , i.e. if solves FVII , then solves FVII . Theorem 2. By the degeneration scheme from the six Painlevé equation to the fifth Painlevé equation, the system is reduced into the one ( ) ( ) 22 21 22 2 rr s HrrisS sr −− =−∂+ ∂+∂+ , ,
Y. Sasaki ( ) ( ) { } {}{} 11 2 2 212 ,,, sxxx Y iisgg−=−∂ ∂+∂+∂ , where , , . Theorem 3. By change of the independent variable and , the system is reduced into the one () 22 1222 2rr Hr rSr σ σ −− =−∂+∂+∂+ , , ( ) { } {}{} 11 2 22 12 ,,, xxx Yig g σ −=− ∂∂+∂+∂ , where , , , . Each theorem is obtained by a straightforward computation. 3. Discussion The degeneration scheme broke the reality of the coordinates, which is not a surprising conclusion. The fact says that, if the assumption of the Tremblay and Winternitz [1] is made loos er, then another superintegrable system may appear. So, the author thinks that the assumption of the Hermitian nature is too strong to get the sixth Pain levé equation with fullparameter or other Painlevé equations. Marquette and Winternitz [8] also obtained other superintegrable systems with potentials described by the first, second and fourth Painlevé equations. But it is uncertain if the system above degenerates into the system obtained in [8]. Acknowledgements The author thanks Professor R. Conte for introducing him the superintegrable systems obtained by Winternitz school, and for thoughtfulness in his stay at Ecole Normale Supérieure de Cachan. This research is partially supported by “Strategic Fostering Program for Young Researchers Engaged in Nat ural Sciences toward the Establishment of the Sustainable Society” of Hiroshima University funded by “Institu tional Program for Young Researcher Overseas Visits” of Japan Society for the Promotion of Science. References [1] Tremblay, F. and Winternitz, P. (2010) Th i rd Order Superintegrable Systems Separating in Polar Coordinates. Journal of Physics A: Mathematical and Theoretical, 43, Article ID: 175206 (17p). http://dx.doi.org/10.1088/17518113/43/17/175206 [2] Cosgr ove , C.M. (2006) Higher Order Painlevé Equations in the Polynomial Class II. Bureau Symbol P1. Studies in Applied Mathematics, 116, 321413 . http://dx.doi.org/10.1111/j.14679590.2006.00346.x [3] Cosgr ove , C.M. and Scoufis, G. (1993) Painlevé Classification of a Class of Differential Equations of the Second Or der and Second Degree. Stud. Appl. Math., 88, 2587. [4] Conte, R., Grundland, A. M. and Musette, M. (2006 ) A Reduction of the Resonant ThreeWave Interaction to the Ge neric Sixth Painlevé Equation. Journal of Physics A: Mathematical and General, 39, 1211512127. http://dx.doi.org/10.1088/03054470/39/39/S07 [5] Conte, R. and Musette, M. (2008) The Painlevé Handbook. Springer Science+Business Media B.V., Dordrecht. [6] Ince, E.L. (1956 ) Ordinary Differential Equations. Dover Publ., Inc., New York. [7] Okamoto, K. (1986) Isomonodromic Deformation and Painlevé Equations, and the Garnier Syst e m. J. Fac. Sci. Univ. Tokyo, Sect. IA, 33, 575618. [8] Marqu ette, I. and Winternitz, P. (2008) Superintegrable Systems with Third Order Integrals of Motion. Journal of Physics A: Mathematical and General, 41, Article ID: 304031 (10p). http://dx.doi.org/10.1088/17518113/41/30/304031
