Superintegrability with higher integrals of motion
Ian Marquette, University of York
30 September 2009
North British Mathematical Physics Seminar and 10th anniversary
of the Edinburgh Mathematical Physics Group
e Science Institute
All quantum superintegrable systems separable in Cartesian
coordinates with a third order integral are known. Some of these
superintegrable Hamiltonians involve Painlevé transcendents. We
discuss their polynomial algebra and a relation between these
Hamiltonians and supersymmetric quantum mechanics. We discuss
how ladder operators and supersymmetry can be used to generate
new superintegrable systems with higher order integrals of motion.
Ian Marquette
Superintegrability with higher order integrals of motion
Outline
1-Introduction
2-Superintegrable classical potentials with a third order integral
3-Superintegrable quantum potentials with a third order integral
(Potentials with rational functions)
-Cubic algebras, deformed oscillator algebras, parafermionic
algebras, supersymmetric quantum mechanics
4-Potential with the fourth Painlevé transcendent
-Cubic algebra, higher order supersymmetry and third order shape
invariance
5-Higher order polynomial algebras, ladder operators and
supersymmetry
6-Concluding remarks and future research
Ian Marquette
Superintegrability with higher order integrals of motion
Introduction
We recall that in classical mechanics a Hamiltonian system (in n
dimensions) with Hamiltonian H
1
H = gik pi pk + V (~x , ~p )
2
is integrable if it allows n integrals of motion that are well defined
functions on phase space, are in involution {H, Xa }p = 0,
{Xa , Xb }p = 0, a,b=1,...,n-1 and are functionally independent.
An integrable system is superintegrable if it allows further integrals
of motion Yb (~x , ~p ), {H, Yb }p = 0, b=1,...,k that are also well
defined functions on phase space and the
integrals{H, X1 , ..., Xn−1 , Y1 , ..., Yk } are functionally independent.
Ian Marquette
Superintegrability with higher order integrals of motion
The same definitions apply in quantum mechanics but {H, Xa , Yb }
are well defined quantum mechanical operators, assumed to form
an algebraically independent set.
-The best known examples are the Kepler-Coulomb system
V (r ) = αr and the harmonic oscillator V (r ) = αr 2 .
other well known such systems are the following potentials
-Hartmann , Calogero-Moser-Sutherland , Smorodinsky-Winternitz
Superintegrable systems have applications in nuclear physics and
quantum chemistry.
Ian Marquette
Superintegrability with higher order integrals of motion
Origin of the study of superintegrable systems (closed trajectories,
accidental degeneracies)
In classical mechanics :
-Laplace-Runge-Lenz vector (Traité de mécanique céleste, 1799) of
the Kepler system
The trajectories can be obtained algebraically
-Bertrand’s theorem
In quantum mechanics :
-W.Pauli, Jr., Z. Physik 36, 336-363 (1926).
-V.Fock, Z.Phys. 98, 145-154 (1935).
-V.Bargmann, Z.Phys. 99, 576-582 (1936).
-J.M.Jauch and E.L.Hill, Phys.Rev. 57, 641-645 (1940).
-H.V.McIntosh,Am. J. phys. 27, 620-625 (1959).
Ian Marquette
Superintegrability with higher order integrals of motion
-A systematic search for superintegrable systems was started some
time ago.
J.Fris, V.Mandrosov, Ya.A.Smorodinsky, M.Uhlir and P.Winternitz,
Phys.Lett. 16, 354-356 (1965).
P.Winternitz, Ya.A.Smorodinsky, M.Uhlir and I.Fris, Yad.Fiz. 4,
625-635 (1966). (English translation in Sov. J.Nucl.Phys. 4,
444-450 (1967))
VI = α(x 2 + y 2 ) +
VIII =
β
γ
+ 2,
2
x
y
VII = α(x 2 + 4y 2 ) +
β
γ
+ 2
2
x
y
α 1
α
β
α 1
φ
φ
+ 2(
+
), VIV = + (βcos( )+γsin( ))
r r 1 + cos(φ) 1 − cos(φ)
r r
2
2
C.Daskaloyannis, J.Math.Phys. 42, 1100-1119 (2001). (Quadratic
algebras)
Ian Marquette
Superintegrability with higher order integrals of motion
A similar search for superintegrable systems in 3 dimensional
Euclidean space E3 gave a complete classification of all
”quadratically superintegrable” systems in E3 .
-A.Makarov, Kh. Valiev, Ya.A.Smorodinsky and P.Winternitz,
Nuovo Cim. A52, 1061-1084 (1967).
- N.W.Evans, Phys.Rev. A41, 5666-5676 (1990), J.Math.Phys. 32,
3369-3375 (1991).
-Many results were obtained in En , two dimensional spaces of
constant curvature, Darboux spaces, complex spaces ( J.M.Kress,
W.Miller, E.G.Kalnins, P.Winternitz, S.Post).
-Systems with magnetic field or spin were also considered
(J.Bérubé,P.Winternitz,I.Yurdusen)
-Superintegrable systems are interesting both from the
mathematical and physical point of view.
Ian Marquette
Superintegrability with higher order integrals of motion
1. The orbit lie on a n-k dimensional submanifold in phase space.
In the case of maximal superintegrability all bounded trajectories
are closed and the motion periodic. (Classical)
2. They are multiseparable (quadratic superintegrability).
(Classical/Quantum)
3. The integrals of motion form a nonabelian algebra under Lie
commutation or Poisson bracket (finite dimensional Lie algebra,
Kac-Moody algebra, polynomial algebra). (Classical/Quantum)
4. The quantum energy levels display accidental degeneracy i.e. a
degeneracy explained by higher symmetries rather than geometrical
ones. (Quantum)
5. In all known examples of maximally superintegrable systems the
Schrödinger equation is exactly solvable. It has been conjectured
that this is always the case. (Quantum) (quadratic
superintegrability in Euclidean space)
6. Quadratic superintegrability occurs for the same potentials in
the classical and quantum cases. (Classical/Quantum)
Ian Marquette
Superintegrability with higher order integrals of motion
Third order integrals
-In 1935 J. Drach (J.Drach, C.R. acad. Sci III, 200, 22-26 (1935) ;
C.R.Acad.Sci III, 599-602 (1935)).
-10 integrable classical potentials in complex Euclidean space
E2 (C). 7 reducible (M.F.Rañada, A.V.Tsiganov).
In 1998 J.Hietarinta published an article on quantum potentials
with free motion as their classical limit.
-J.Hietarinta, Phys. Lett. A246, 97-104 (1998).
-S.Gravel and P.Winternitz, J.Math.Phys. 43(12), 5902 (2002).
-S.Gravel, J.Math.Phys. 45(3), 1003-1019 (2004) (21 quantum
potentials, 8 classical potentials).
Ian Marquette
Superintegrability with higher order integrals of motion
Reducible potentials :
2
V (x, y ) = ω2 (x 2 + y 2 )
V (x, y ) = ω 2 (x 2 + y 2 ) +
V (x, y ) = ω 2 (x 2 +
b
+ yc2
x2
4y 2 ) + xb2 + yc2 .
Irreducible potentials :
2
V (x, y ) = ω2 (9x 2 + y 2 )
2
V (x, y ) = ω2 y 2 + V (x)
p
p
V (x, y ) = β12 |x| + p
β22 |y |
V (x, y ) = a2 |y | + b 2 |x|
V (x, y ) = a|y | + f (x),
where f(x) satisfies (f (x) − bx)2 f (x) = d and V(x) satisfies
15
−9V 4 (x) + 14ω 2 x 2 V 3 (x) + (16d − ω 4 x 4 )V 2 (x)+
2
3
ω2
1
( ω 6 x 6 − 2dω 2 x 2 )V (x) + cx 2 − d 2 − d x 4 − ω 8 x 8 = 0.
2
2
16
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Superintegrability with higher order integrals of motion
In these 8 cases the integrals of motion generate a cubic Poisson
algebra
{A, B}p = C ,
{A, C }p = αA2 + 2βAB + γA + δB + ǫ
{B, C }p = µA3 + νA2 − βB 2 − 2αAB + ξA − γB + ζ.
-In many cases this polynomial algebra is reducible. I.Marquette
and P.Winternitz, J.Math.Phys. 48, 012902 (2007)
Ian Marquette
Superintegrability with higher order integrals of motion
Quantum cases
Reducible potentials :
V (x, y ) =
V (x, y ) =
V (x, y ) =
ω2
2
2
2 (x + y )
2
c
ω
b
2
2
2 (x + y ) + x 2 + y 2
ω2
b
2
2
2 (4x + y ) + y 2 + cx.
Irreducible potentials :
2
2
+y
V (x, y ) = ~2 [ x 8a
+
4
V (x, y ) =
V (x, y ) =
V (x, y ) =
V (x, y ) =
V (x, y ) =
1
1
+ (x+a)
2]
(x−a)2
1
1
1
1
2
2
2
~ [ 8a4 (x + y ) + y 2 + (x+a)
2 + (x−a)2 ]
1
1
1
~2 [ 8a14 (x 2 + y 2 ) + (y +a)
2 + (y −a)2 + (x+a)2
ω2
2
2
2 (9x + y )
2
ω
~2
2
2
2 (9x + y ) + y 2
2
2
1
1
~2 [ 9x8a+y
+ (y −a)
4
2 + (y +a)2 ].
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+
1
]
(x−a)2
Superintegrability with higher order integrals of motion
Potentials with Painlevé transcendents :
V (x, y ) = ~2 (ω12 P1 (ω1 x) + ω22 P1 (ω2 y ))
V (x, y ) = ay + ~2 ω12 P1 (ω1 x)
2
1
V (x, y ) = bx + ay + (2~b) 3 P22 (( 2b
) 3 x, 0)
~2
1
1
1
V (x, y ) = ay + (2~2 b2 ) 3 (P2′ (( −4b
) 3 x, α) + P22 (( −4b
) 3 x), α)
~2p
~2 p
2
′
ω
ω
ω~ 2
V (x, y ) = ω2 (x 2 + y 2 ) + ǫ ~ω
2 P4 (
~ x, α, β) + 2 P4 (
~ x, α, β)
√
pω
~ω
+ω ~ωxP4 ( ~ x, α, β) + 3 (−α + ǫ).
P1′′ (z) = 6P12 (z) + z
P2′′ (z, α) = 2P2 (z, α)3 + zP2 (z, α) + α
′′
P4 (z) =
′
P42 (z)
2P4 (z)
+ 23 P43 (z) + 4zP42 (z) + 2(z 2 − α)P4 (z) +
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β
P4 (z) .
Superintegrability with higher order integrals of motion
I.Marquette, J.Math.Phys. 50 (1), 012101 (2009), 50 (9) 095202
(2009), I.Marquette and P.Winternitz, J. Phys. A : Math. Theor.
41, 304031 (2008).
The most general cubic algebra is
[A, B] = C
[A, C ] = αA2 + β{A, B} + γA + δB + ǫ
[B, C ] = µA3 + νA2 − βB 2 − α{A, B} + ξA − γB + ζ
.
The Casimir operator satisfies [K,A]=[K,B]=[K,C]=0 and this
implies
K = C 2 − α{A2 , B} − β{A, B 2 } + (αβ − γ){A, B} + (β 2 − δ)B 2
2
1
βν δµ
µ
+ +α2 +ξ)A2
(+βγ −2ǫ)B + A4 + (ν +µβ)A3 +(− µβ 2 +
2
3
6
3
2
1
δν
+(− µβδ +
+ αγ + 2ζ)A.
6Ian Marquette 3 Superintegrability with higher order integrals of motion
We construct a realization of the cubic algebra by means of
deformed oscillator algebras {b† , b, N} which satisfies
[N, b† ] = b† ,
[N, b] = −b,
b† b = Φ(N),
bb† = Φ(N + 1).
Φ(x) is called the ”structure function” . Φ(x) is a real function,
Φ(0) = 0 and Φ(x) > 0 for x > 0.
-C.Daskaloyannis, J.Phys.A : Math.Gen 24, L789-L794 (1991).
We have the existence of a Fock type representation (Fock basis
|n >,n=0,1,...)
N|n >= n|n >,
b|0 >= 0,
p
b † |n >= Φ(N + 1)|n + 1 >
p
b|n >= Φ(N)|n − 1 > .
We request Φ(p + 1) = 0 to have a finite-dimensional
representation.
Ian Marquette
Superintegrability with higher order integrals of motion
A = A(N),
B = b(N) + b† ρ(N) + ρ(N)b
Case β = 0 et δ 6= 0 :
A(N) =
√
γ
ǫ
δ(N + u), b(N) = −α(N + u)2 − √ (N + u) −
δ
δ
K
ζ
ǫ2
γǫ
− 3 − √ + 2)
−4δ 4δ 2
4 δ 4δ
√
ζ
ν δ
γǫ
αγ
−αǫ ξ γ 2
− −
+
)(N + u)
+ √ + √ +
+(
2δ
4 4δ 2δ 23
12
4 δ 2 δ
√
−ν δ
3αγ
γ 2 ǫα α2 ξ µδ
+(
− √ +
+
+
+ +
)(N + u)2
4
4δ
2δ
4
4
8
4 δ
√
2
γα
ν δ µδ
α2 µδ
−α
+ 1 +
−
)(N + u)3 + (
+
)(N + u)4 .
+(
2
6
4
4
8
2
2δ
ρ(N) = 1,
Φ(N) = (
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Superintegrability with higher order integrals of motion
Case β 6= 0 :
A(N) =
b(N) =
−
ρ(N) =
1
δ
β
((N + u)2 − − 2 )
2
4 β
α
1
αδ − γβ
((N + u)2 − ) +
4
4
2β 2
αδ 2 − 2γδβ + 4β 2 ǫ
1
4
4β
(N + u)2 −
1
4
1
212 3β 8 (N + u)(1 + N + u)(1 + 2(N + u))2
Φ(N) = 384µβ 10 N 10 − 1920µβ 10 N 9 + ...
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Superintegrability with higher order integrals of motion
-We define the Fock space for each energy eigenvalue. -Ultimately
we obtain Φ(x, u, E ) in terms of the energy E and the parameter u.
We have the following constraints for the structure function,
Φ(0, u, E ) = 0,
Φ(p+1, u, E ) = 0,
Φ(x) > 0 for
x = 1, 2, ..., p.
We consider the following Hamiltonian :
H=
x2 + y2
1
1
Px2 Py2
+
+ ~2 (
+
+
).
2
2
8a4
(x − a)2 (x + a)2
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Superintegrability with higher order integrals of motion
Tha Hamiltonian has two integrals :
A = Px2 − Py2 + 2~2 (
1
1
x2 − y2
+
+
)
8a4
(x − a)2 (x + a)2
1
4a2 − x 2
6(x 2 + a2 )
1
−
), Px }
B = {L, Px2 } + ~2 {y (
2
2
4a4
(x 2 − a2 )2 )
1
(x 2 − 4a2 )
2
4(x 2 + a2 )
+ ~2 {x(
−
+
), Py }.
2
4a4
x 2 − a2
(x 2 − a2 )2
[A, B] = C ,
+6
[A, C ] =
4h4
B,
a4
[B, C ] = −2~2 A3 −6~2 A2 H+8~2 H 3
~4 2
~4
~4 2
~6
~6
~8
A
+
+8
HA
−
8
H
+
2
A
−
2
H
−
6
a2
a2
a2
a4
a4
a6
K = −16~2 H 4 + 32
~4 3
~6 2
~8
~10
H
+
16
H
−
40
H
−
3
a2
a4
a6
a8
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.
Superintegrability with higher order integrals of motion
Φ(x) = (
−~8
−a2 E
1
a2 E
1
)(x
+
u
−
(
−
))(x
+
u
−
(
+ ))
a4
~2
2
~2
2
−a2 E
−a2 E
3
5
+
))(x
+
u
−
(
+ )).
~2
2
~2
2
a0 ∈ R
(x + u − (
For a = ia0 ,
E1 =
~2 (p + 2)
,
2a02
Φ1 (x) = (
~8
)x(p + 1 − x)(p + 3 − x)(p + 4 − x)
a04
where p ∈ N.
E2 = −
~2 (p)
~8
,
Φ
(x)
=
(
)x(p + 1 − x)(3 − x)(2 − x), p = 0, 1.
2
2a02
a04
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Superintegrability with higher order integrals of motion
Supersymmetric quantum mechanics
-Supersymmetry originally introduced in the context of quantum
field theory. (H.Miyazawa(1968), Y.A.Golfand et
E.P.Likhtman(1971), A.Neveu et J.H.Schwarz(1971), J.Wess et
B.Zumino(1974)).
-Supersymmetric quantum mechanics was introduced as a toy
model to study supersymmetry breaking (E.Witten (1981)).
(applications in many domain in physics)
-This method is related to earlier articles (T.F.Moutard (1875),
G.Darboux (1882), E.Schrodinger (1940), L.Infeld et T.E.Hull
(1951)).
~ d
~ d
H2 = AA† , A = √
+W (x), A† = − √
+W (x).
2 dx
2 dx
H1 0
0 0
0 A†
H=
Q=
Q† =
.
0 H2
A 0
0 0
H1 = A† A,
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Superintegrability with higher order integrals of motion
[H, Q] = [H, Q † ] = {Q, Q} = {Q † , Q † } = 0, {Q, Q † } = H.
(1)
(1)
(2)
6= 0, A† ψ0 6= 0,
Broken supersymmetry , Aψ0 6= 0, E0
(2)
E0 6= 0
(2)
En
(1)
= En
> 0,
1
(1)
(2)
Aψn ,
ψn = q
(1)
En
(1)
(1)
Unbroken supersymmetry , Aψ0 = 0, E0
(2)
E0 6= 0
(2)
En
(1)
= En+1 ,
1
(2)
(1)
ψn = q
Aψn+1 ,
(1)
En+1
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1
(1)
(2)
ψn = q
A† ψn .
(2)
En
(2)
= 0, A† ψ0 6= 0,
1
(1)
(2)
ψn+1 = q
A† ψn .
(2)
En
Superintegrability with higher order integrals of motion
For a = ia0 ,
a0 ∈ R
−1
d
1
~
−1
A† = √ (−~
+ 2 x − ~(
+
))
dx
x − ia0 x + ia0
2a0
2
~
−1
−1
1
d
+ 2 x − ~(
+
)).
A = √ (~
x − ia0 x + ia0
2a0
2 dx
H1 = A† A =
~2
3~2
~2
Px2 ~2 x 2
+
+
+
+
2
(x − ia0 )2 (x + ia0 )2
8a04
4a02
H2 = AA† =
Px2 ~2 x 2 5~2
+
+ 2.
2
8a04
4a0
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Superintegrability with higher order integrals of motion
We have an unbroken supersymmetry. The zero mode satisfies
bφ0 = 0 and the other eigenfunctions of H1 are obtained from H2
−x 2
4a2
0
2 1 e
φ0 (x) = a0 ( ) 4 2
π a0 + x 2
3
2
,
s
−1 2
1
1 1 4a
2x
φk1 +1 (x) = A ( p
( 2 ) 4 e 0 H k1 (
x)).
2a02
2k1 k1 ! 2a0 π
†
E1 =
1
(p + 2)~2
(k1 + k2 + 2)~2
=
,
2a02
2a02
E2 =
~2 (k2 − 1)
2a02
-Case a ∈ R, I.F. Marquez, J. Negro and L.M. Nieto(1998),
M.Znojil (2003).
Ian Marquette
Superintegrability with higher order integrals of motion
Fourth Painlevé transcendant
Px2 Py2
ω2 2
+
+ g1 (x) +
y
,
2
2
2
r
r
ω2 2
ω
ω
~ω ′
ω~ 2
g1 (x) =
x + ǫ P4 (
x, α, β) +
P4 (
x, α, β)+
2
2
~
2
~
r
√
ω
~ω
x, α, β) +
(−α + ǫ) ,
ω ~ωxP4 (
~
3
H=
A=
Px2 Py2
ω2 2
−
+ g1 (x) −
y
2
2
2
,
1 ω2
1 ~2
1
B = {L, Px2 } + { x 2 y − 3yg1 (x), Px } − 2 { g1xxx (x)+
2
2 2
ω 4
(
ω2 2
x − 3g1 (x))g1x (x), Py },
2
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L = xPy − yPx.
Superintegrability with higher order integrals of motion
[A, B] ≡ C
[A, C ] = 16ω 2 ~2 B
[B, C ] = −2~2 A3 − 6~2 HA2 + 8~2 H 3
+
+
ω 2 ~4
(4α2 − 20 − 6β − 8ǫα)A − 8ω 2 ~4 H
3
~5 ω 3
(−8α3 − 24α − 36αβ + 24ǫα2 + 8ǫ + 36ǫβ)
27
K = −16~2 H 4 +
−
.
4~4 ω 2
(4α2 − 8α + 4 − αβ)H 2
3
4~5 ω 3
(8α3 − 24ǫα2 + 24α + 36αβ − 8ǫ − 36ǫβ)H
27
−
4~6 ω 4
(4α − 8ǫα − 8 − 6β) .
3
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Superintegrability with higher order integrals of motion
The structure function is given by
r
E
β
1
−E 1
Φ(x) = −4ω ~ (x+u−(
+ ))(x+u−(
+ (α+3−ǫ−3i
)))
2~ω 2
2~ω 6
2
r
−E 1
−E 3 + 2ǫ α
β
+
− ))(x +u−(
+ (α+3−ǫ+3i
))) .
(x +u−(
2~ω
6
3
2~ω 6
2
2 4
-We have to distinguish two cases β < 0 et β > 0.
r
r
ǫ α
ǫ α
−β
−β
)(x + − +
)
Φ1 (x) = 4~ ω x(p +1−x)(x + − −
2 2
8
2 2
8
r
r
−β
−β
ǫ α
4 2
Φ2,3 (x) = 4~ ω x(p + 1 − x)(x ±
)(x − + ±
)
2
2
2
8
r
−β
3+ǫ α
6−ǫ α
E1 = ~ω(p +
− ), E2,3 = ~ω(p +
+ ±
).
3
3
6
6
8
4 2
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Superintegrability with higher order integrals of motion
Higher-derivative supersymmetric quantum mechanics
-The concept of higher-derivative supersymmetric quantum
mechanics was discussed in
-A.Andrianov, M.Ioffe et V.P.Spiridonov, Phys.Lett. A174, 273
(1993), A.Andrianov, F.Cannata, M.Ioffe et D.Nishnianidze,
Phys.Lett.A, 266,341-349 (2000).
H1 q † = q † (H2 + 2λ),
H1 M † = M † H2
Hi = Px2 + Vi (x)
q † = ∂ + W (x),
M † = ∂ 2 − 2h(x)∂ + b(x),
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,
q = −∂ + W (x) ,
M = ∂ 2 + 2h(x)∂ + b(x)
.
Superintegrability with higher order integrals of motion
-We can construct from these relations third order supercharges
a† = q † M (or a† = Mq † for H2 )
H1 a† = a† (H1 + 2λ) ,
-This relation can also be interpreted in terms of ladder operators
-If V1 (x, a0 ) and V2 (x, a0 ) satisfy V2 (x, a1 ) = V1 (x, a0 ) + R(a1 )
where a1 = f (a0 ) and R(a1 ) are independent of x, V1 (or V2 ) is
called a shape invariant potential
L.Gendenshtein, JETP Lett., 38, 356 (1983).
Ian Marquette
Superintegrability with higher order integrals of motion
The potentials V1 et V2 have the following form
V1,2 = ∓2h′ (x) + 4h2 (x) + 4λxh(x) + λ2 x 2 − λ
′′
,
′
d
h (x) h 2 (x)
+
+
.
W (x) = −2h(x)−λx, b(x) = −h (x)+h (x)−
2h(x) 4h2 (x) 4h2 (x)
′
2
′
h′′ (x) =
h 2 (x)
d
+6h3 (x)+8λxh2 (x)+2(λ2 x 2 −(λ+γ))h(x)+
,
2h(x)
2h(x)
We consider
h(x) =
1√
λf (z),
2
z=
√
λx,
α=1+
γ
,
λ
β=
2d
,
λ2
λ=
ω
.
~
We have f(z)=P4 (z, α, β).
Ian Marquette
Superintegrability with higher order integrals of motion
-We consider the particular case ( called reducible) d ≤ 0 with the
existence of real functions W1 and W2 such that
√
′
h (x) − −d
†
M = (∂ + W1 (x))(∂ + W2 (x)), W1,2 = −h(x) ±
.
2h(x)
-The spectrum is obtained for cases when normalizable zero modes
(0)
of the annihilation operator exist aψk = 0.
(0)
-The eigenfunctions ψk (for V2 ) are
Rx
√
′
′
ψ10 (x) = (α− −d+(W1 (x)+W2 (x))(W1 (x)−W3 (x)))e W3 (x )dx
ψ20 (x) = (W1 (x) + W2 (x))e −
ψ30 (x) = e −
and the corresponding energies
(0)
E1
= 0,
(0)
E2
=γ−
Rx
√
Ian Marquette
Rx
W2 (x ′ )dx ′
−d,
(0)
E3
W1 (x ′ )dx ′
,
,
=γ+
√
−d
.
Superintegrability with higher order integrals of motion
,
(0)
-The creation operator can also have zero modes φk
φ01 (x) = e −
Rx
W3 (x ′ )dx ′
,
Rx
′
′
φ02 (x) = (W1 (x) − W3 (x))e W1 (x )dx ,
Rx
√
′
′
φ03 (x) = (−2 −d+(W1 (x)−W3 (x))(W1 (x)+W2 (x)))e W2 (x )dx
(0)
E1
= −2λ,
(0)
E2
= γ − 2λ −
√
−d,
(0)
E3
= γ − 2λ +
√
−d
.
.
-We can have three, two or one infinite sequence of levels.
-When a potential has only one infinite sequence of levels, it can
also allow a singlet state or a doublet states
a† ψ(x) = aψ(x) = 0,
Ian Marquette
(a† )2 ψ(x) = aψ(x) = 0.
Superintegrability with higher order integrals of motion
Two examples
Case α = 5, β = −8, f (z) =
V (x, y ) =
4z(2z 2 −1)(2z 2 +3)
(2z 2 +1)(4z 4 +3)
and ǫ = −1.
ω2 2
192~4 ω 2 x 2
16~2 ω 2 x 2
(x + y 2 ) −
+
2
(4ω 2 x 4 + 3~2 )2 4ω 2 x 4 + 3~2
30
25
20
15
10
5
5
-5
Ian Marquette
Superintegrability with higher order integrals of motion
We obtain
φ(x) = 4~4 ω 2 x(p + 1 − x)(x + 4)(x + 2),
E = ~ω(p + 3)
φ(x) = 4~4 ω 2 x(p+1−x)(p−3−x)(p−1−x), E = ~ω(1−p), p = 0, 1
φ(x) = 4~4 ω 2 x(p + 1 − x)(x − 3)(x − 2),
Ian Marquette
E = ~ω(p − 1), p = 0, 1
Superintegrability with higher order integrals of motion
ψn (x) = Nn (a† )n e
χ1 (x) = C1 e
−ωx 2
2~
−ωx 2
2~
(−9~3 + 18~2 ωx 2 + 12~ω 2 x 4 + 8ω 3 x 6 )
(3~2 + 4ω 2 x 4 )
(~ + 2ωx 2 )
,
(3~2 + 4ω 2 x 4 )
aχ1 (x) = 0,
χ2 (x) = C2 e
a† χ1 (x) = χ2 (x),
−ωx 2
2~
x(3~ + 2ωx 2 )
(3~2 + 4ω 2 x 4 )
a† χ2 (x) = 0
The eigenfunctions are
r
ω
Hk (
y ), E = ~ω(n + k + 3),
ψn,k = φn (x)e
~
r
2
ω
− ωy
2~
Hm1 (
φm1 = χ1 (x)e
y ) Em1 = ~ω(m1 − 1) ,
~
r
2
ω
− ωy
φm2 = χ2 (x)e 2~ Hm2 (
y ), Em2 = ~ωm2 .
~
2
− ωy
2~
Ian Marquette
Superintegrability with higher order integrals of motion
2z
Case α = −1, β = − 32
9 , f (z) = − 3 −
V (x, y ) =
2z 2 −3
z(2z 2 +3)
and ǫ = 1.
24~3 ω
4~2 ω
ω2 1 2
( x + y 2) −
+
2 9
(2ωx 2 + 3~)2 (2ωx 2 + 3~)
3
2
1
5
-5
-1
Ian Marquette
Superintegrability with higher order integrals of motion
We obtain
1
5
φ(x) = 4~4 ω 2 x(p + 1 − x)(x + )(x + ),
3
3
5
E = ~ω(p + ),
3
1
4
φ(x) = 4~4 ω 2 x(p + 1 − x)(x − )(x + ),
3
3
4
5
φ(x) = 4~4 ω 2 x(p + 1 − x)(x − )(x − ),
3
3
4
E = ~ω(p + ),
3
Ian Marquette
E = ~ω(p + 1).
Superintegrability with higher order integrals of motion
We have the following eigenfunctions
† n1
ψn1 ,k1 = Nn1 k1 (a ) e
−ωx 2
6~
r
(−45~2 + 4ω 2 x 4 ) − ωy 2
ω
x
e 2~ Hk1 (
y ),
(3~ + 2ωx 2 )
~
5
E1 = ~ω(n1 + k1 + ),
3
ψn2 ,k2
r
−ωx 2
2
6~
ω
e
† n2
− ωy
2~
H k2 (
= Nn2 k2 (a )
e
y ),
2
(3~ + 2ωx )
~
E2 = ~ω(n2 + k2 + 1),
† n3
ψn3 ,k3 = Nn3 k3 (a ) e
−ωx 2
6~
r
(9~2 − 12~ωx 2 − 4ω 2 x 4 ) − ωy 2
ω
e 2~ Hk3 (
y ),
2
(3~ + 2ωx )
~
4
E3 = ~ω(n3 + k3 + ).
3
Ian Marquette
Superintegrability with higher order integrals of motion
Higher order polynomial algebras, ladder operators and
SUSY
V.A.Dulock and H.V.McIntosh (1965) ; A.Cisneros and
H.V.McIntosh (1970) ; J.M.Lyman and P.K.Aravind (1993),
R.D.Mota,V.D.Granados,A.Queijeiro and J.Garcia (2001, 2002,
2003).
-I.Marquette, arxiv :0908.4399, arxiv :0908.4432 (2009)
-Higher order integrals and polynomial algebras can be constructed
from creation and annihilation operators.
[Hx , A†x ] = λx A†x ,
[Hy , A†y ] = λy A†y
.
The following polynomial are integrals of motion of (H = Hx + Hy )
m †n
n
I1 = A†m
x Ay − Ax Ay ,
m †n
n
I2 = A†m
x Ay + Ax Ay ,
Ian Marquette
mλx − nλy = 0.
Superintegrability with higher order integrals of motion
[Ax , A†x ] = Q(Hx + λx ) − Q(Hx ), [Ay , A†y ] = S(Hy + λy ) − S(Hy ),
-The integrals (A, B and C) do not close in a cubic algebra for the
two following potentials
1
1
V (x, y ) = ~2 [ 8a14 (x 2 + y 2 ) + y12 + (x+a)
2 + (x−a)2 ],
V (x, y ) = ~2 [ 8a14 (x 2 + y 2 ) +
1
(y +a)2
+
1
(y −a)2
+
1
(x+a)2
+
1
].
(x−a)2
We obtain for these two cases a quintic and a seventh order
algebras.
[A, I1 ] = I2 ,
[A, I2 ] = δI1 ,
[I1 , I2 ] = mA7 + nA6 + µA5 + νA4
+αA3 + βA2 + γA + ǫ .
-The Casimir operators and realizations in terms of deformed
oscillator algebras were obtained
Ian Marquette
Superintegrability with higher order integrals of motion
The caged anisotropic harmonic oscillator was studied
V =
λ1 λ2
ω2 2 2
(k x + m2 y 2 ) + 2 + 2 .
2
x
y
-The ladder operators can be used to obtain the integrals of
motion and the polynomial algebras
Px2 Py2
+
+ g1 (x) + g2 (y ) ,
2
2
r
r
ω12 2 ~ω1 ǫ1 ′
ω1
ω1
ω1 ~ 2
g1 (x) =
x +
f (
x) +
f (
x)
2
2 1
~
2 1
~
r
p
ω1
~ω1
+ω1 ~ω1 xf1 (
x) +
(−α1 + ǫ1 ) ,
~
3
r
r
ω2
ω2
ω22 2 ~ω2 ǫ2 ′
ω2 ~ 2
g2 (y ) =
y +
f2 (
y) +
f2 (
y)
2
2
~
2
~
r
p
ω2
~ω2
+ω2 ~ω2 yf2 (
y) +
(−α2 + ǫ2 ) ,
~
3
H=
Ian Marquette
Superintegrability with higher order integrals of motion
Mielnik showed that the factorization is not necessarily unique in
supersymmetric quantum mechanics. He obtain superpartners of
the harmonic oscillator written in terms of the error function.
B.Mielnik, J.Math.Phys. 25 (12) 1984.
-Supersymmetric quantum mechanics can be used to generate new
superintegrable Hamiltonians.
I.Marquette, arxiv :0908.1246 (2009)
- We obtain systems with higher order integrals and new systems
involving Painlevé transcendents
Ian Marquette
Superintegrability with higher order integrals of motion
Concluding remarks
The study of superintegrable potentials with higher order integrals
is important.
-These systems are rare from a mathematical point of view but
possess many properties and could have applications in nuclear
physics, atomic physics, quantum chemistry and condensed matter.
-Superintegrable systems are related to many algebraic structures
(finite dimensional Lie algebras, Kac Moody algebras, polynomial
algebras, deformed oscillator algebras, polynomial superalgebras).
-Higher order integrals are useful in classical and quantum
mechanics (classical :trajectories, quantum :energy
spectrum,relate eigenbases and eigenvalues of one symmetry
operator to those of another)
Ian Marquette
Superintegrability with higher order integrals of motion
-Some of them involve one of the Painlevé transcendents.
Singlets, doublets, additional degeneracies not
obtained,E = α1 + α2 p in all cases, equidistant energy spectrum
-There is a relation between superintegrability and supersymmetry
-Superintegrability and ladder operators
-The classification of superintegrable systems with second order
and third order remain to be completed ( polar(F.Tremblay , sixth
Painlevé), parabolic, elliptic ) , in other spaces ?
-The method using the cubic algebra can be applied to systems
with a second and a third order integrals
Ian Marquette
Superintegrability with higher order integrals of motion
-General seventh order polynomial algebras, generalization in three
dimensions
-Only particular cases of third order supersymmetry were
investigated
-Systems with higher order ladder operators (classification ?),
(C.P.Boyer and W.Miller Jr., J.Math.Phys. 15, 9 (1974).)
-Cases involving the first and the second Painlevé transcendents
remain to be solved (SUSYQM ?)
-Classical case (ladder operators and integrals of motion)
Ian Marquette
Superintegrability with higher order integrals of motion