Journal of Applied Mathematics and Physics, 2014, 2, 996-999
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, 996-999.
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, Higashi-Hiroshima, Japan
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:
( )
( )
22
12
1,
2
Hp pVr
θ = ++
,
( )
2
3
2XL S θ = +
,
{ }
( )
{ }
( )
{ }
3 121122
3
, ,,
k lm
klm
klm
YAL ppgpgp
++ =
= ++
∑
xx
,
where
() ()
12
,cos, sinxx rr θθ = =x
,
kk
pix=− ∂∂
( )
1, 2 j=
,
312 21
Lxp xpi θ =−=− ∂∂
,
()( )()
2
,VrRr Sr θθ = +
,
{ }
,A BABBA= +
. Here
( )
Rr
,
( )
S θ
,
( )
1
gx
,
( )
2
gx
are arbitrary functions,
and
klm
A
’s are real constants. Note that the Hermitian nature of the operators causes the anti-commutator
{ }
,
and the parity. In the followings, we use such the notation as
θ
θ ∂ =∂∂
for brevity.
Tremblay and Winternitz [1 ] classified the cases where the above system is superintegrable, i.e. it allows the

Y. Sasaki
997
third conserved quantity
Y
, and obtained
( )
S θ
which is written by the solution of the sixth Painlevé equation.
If
()
2
VS r θ =
,
( )
TT θ =
,
( )
TS θ
′=
,
{ }
( )
22 1222
2rr
Hr rSr
θ
θ
−−
=−∂+∂+∂ +

,
( )
22
2XS
θ
θ =− ∂+
,
{ }
{} {}
{}{} { }
( )
2
31 121122
22 12
,cossin ,sincos ,
11 1
,cossin,,,
rr
YL pGGrpGGrp
GGr
rr r
θθ θ
θθθ θ
θθ
=+ −++
=−∂∂−∂+∂ +∂
()( )
11
,G Gr θ βθ = =
,
()( )( )
22 1
,2 cosGG r'S'
r
θβθθ θ == −
,
( )( )()
0
sin2 cosTT' βθβ θθθ =+− +
,
( )
01 2
cos sin βθβθ βθ = +
;
12
, : const ββ
.
The commutation
[]
,0HY =
is reduced to
()( ) { }
( ) { }
20
0
sin4cos6sin4cos43 sincos
82cossin0.
T''''TTT'T T'T'
T' T'T
θθθθθ θβθ
θθβθ
′′′ ′′′′
− −++++−
+−+ =

By change of variables
() ( ) ( ) ()
,,T twt θθ 
s.t.
()( )
( )
( )
{ }
( )
22
21
1
tan2112, 4121
8
tttTwt tt θ ββ = −−=+++−−
,
the above equation is reduced to F-VII
()
2
1
12 ,0
4
β −− 
. Here, F-VII
( )
02
,AA
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
t
, a dependent variable
( )
w wt=
, constant parameters
0
A
and
2
A
(See [1]
[2]). This equation can be integrated twice, and reduced to SD-I.a [1] [3] [4] with constants of integration
3
B
and
4
A
:
( )()()( )
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 −+−
− −−−+ −−−
′=−−−±−−+ −−− −−
SD-I.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
998
where the correspondence of the parameters is given by
1 θ
∞∞
Θ= +
,
() ()( )()( )
()( )()() ()()
{ }
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−+
= +
F-VII
()
2
1
12 ,0
4
β −− 
is reduced to the sixth Painlevé equation with
22
10
θθ =
because of the symmetry of
F-VII
( )
02
,AA
.
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
( )
01
,;,,,
t
ut θ θθθ
∞
by
( )
0
,1;, ,,ut ε θθ ηεκηε
∞
+ +−
, and then take limitation
0 ε →
.
In this article, we lift-up 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
( )
1t ss= −
,
()
()
0
11
4
w vAss=+−
, 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
( )
1s S ssT=− −∂
,
( )
1
s
ss
θ
∂= −−∂
,
1
12
1
rs xss
sr
+
∂ =∂−∂
−
( )
2
1
2
1rs
x
ss
s
sr
−+
−
∂ =∂−∂
−
.
Moreover, F-VII
( )
0,0A
is reduced to F-VII
( )
2
00
,4 AA−
, i.e. if
( )
w wt=
solves F-VII
( )
0,0A
, then
( )
v vs=
solves F-VII
( )
2
00
,4AA−
.
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
−−

=−∂+ ∂+∂+


,
( ) ( )
2
2
2
s
XisS s=− ∂+ 
,

Y. Sasaki
999
( )
( )
{ }
{}{}
11 2
2
212
,,,
sxxx
Y iisgg−=−∂ ∂+∂+∂
,
where
s
SisT= ∂
,
s
is
θ
∂=∂
,
1
22
rs xsr
∂=− ∂− ∂
2
22
rs xii
sr
∂ =−∂−∂
.
Theorem 3. By change of the independent variable
( )
expsi σ =
and
() ()
,cos, sinxy rr σσ =
, the system is
reduced into the one
()
22 1222
2rr
Hr rSr
σ
σ
−−

=−∂+∂+∂+


,
( )
22 2XS
σ
σ =− ∂+
,
( )
{ }
{}{}
11 2
22 12
,,,
xxx
Yig g
σ
−=− ∂∂+∂+∂
,
where
ST
σ
= ∂
,
s
is
σ
∂=∂
,
1
22 xy
xi∂ =−∂+ ∂
,
2
22 yx
xi∂=−∂− ∂
.
Each theorem is obtained by a straight-forward 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 full-parameter 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.
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