JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 40, NUMBER 2
FEBRUARY 1999
Superintegrability in three-dimensional Euclidean space
E. G. Kalnins and G. C. Williams
Department of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand
W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
G. S. Pogosyan
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna, Moscow Region 141980, Russia
~Received 19 May 1998; accepted for publication 1 September 1998!
Potentials for which the corresponding Schrödinger equation is maximally superintegrable in three-dimensional Euclidean space are studied. The quadratic algebra
which is associated with each of these potentials is constructed and the bound state
wave functions are computed in the separable coordinates. © 1999 American
Institute of Physics. @S0022-2488~99!02602-X#
I. INTRODUCTION
The present paper continues our study of the systems with hidden symmetry or so-called
superintegrable systems in spaces with constant curvature.
The best known systems of this kind in three-dimensional Euclidean space are the harmonic
oscillator and Kepler–Coulomb problems, which have many special properties distinct from other
spherically symmetric potentials. These include the phenomena of separation of variables for the
Hamilton–Jacobi and Schrödinger equations in more than one orthogonal coordinate system and
the existence of integrals of motion in addition to the total angular momentum L2 . In particular for
the isotropic oscillator there is the Demkov tensor D ik 5 p i p k 1 v 2 x i x k , 1 and, in the case of the
Kepler–Coulomb problem, the Pauli–Runge–Lenz vector A51/2( @ L3p# 2 @ p3L# )2r/ u r u . Both
these systems possess five functionally independent integrals of motion.2,3 The first systematic
search for all potentials for which the Schrödinger equation admits separation of variables in two
or more coordinate systems was begun by Smorodinsky and Winternitz with co-workers in Refs.
4–6 and continued by Evans in Refs. 3 and 7. They found all such systems in two- and threedimensional flat space and introduced the notion of superintegrability. In general, a physical
system in N dimensions is called minimally superintegrable if it has 2N22 integrals of motion,
and maximally superintegrable if it has 2N21 integral of motions. There are five known maximally ~and some minimally! superintegrable potentials listed in Refs. 3, 8, and 10 and investigated
from different points of view in the last decade.8–13 Note also that superintegrable potentials in
spaces of constant curvature were introduced in Refs. 14–16.
In previous articles17–19 we have looked at potentials in two-dimensional Euclidean space and
the two-dimensional sphere and hyperboloid, for which the Schrödinger equation is maximally
superintegrable. In this article we extend this study to the case of three-dimensional Euclidean
space. As previously seen in the case of two dimensions, some of these potentials ~see Table I!
admit bound state or finite solutions and it is these to which we draw attention in this article.
The basic equation that we investigate is of course the Schrödinger equation (\5m51)
S
D
1
]2
]2
1 ]2
HC52 DC1V ~ x,y,z ! C52
C1V ~ x,y,z ! C5EC.
21
21
2
2 ]x
]y
]z2
~1!
The idea is to find solutions of this equation via a separation of variables ansatz
0022-2488/99/40(2)/708/18/$15.00
708
© 1999 American Institute of Physics
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
709
TABLE I. The three-dimensional maximally superintegrable potentials.
Potential V(x,y,z)
S
1
Separating coordinates
1
1
k212 4 k222 4 k232 4
v2
V15 ~x21y21z2!1
1 2 1 2
2
x2
y
z
S
1
1
2
2
1 k12 4 k22 4
v2
V25 ~x21y214z2!1
2 1
2
2 x
y2
S
D
1
Cartesian
Cylindrical polar
Cylindrical parabolic
Cylindrical elliptic
Parabolic
1
2
2
1 k 12 4 k 22 4
V352 2
1
2 1
2
2
y2
Ax 1y 1z 2 x
a
D
Cartesian
Spherical
Cylindrical polar
Cylindrical elliptic
Sphero-conical
Oblate spheroidal
Prolate spheroidal
Ellipsoidal
D
Spheroidal-conical
Spherical
Parabolic
Prolate spheroidal II
3
C5
)
j51
c j~ u j !
for some suitable orthogonal coordinates u j ~see Table II!.
In Secs. II–IV we consider three maximally superintegrable potentials ~see Table I! and use
the Niven-type ~or Bethe20! ansatz for constructing the solution of the Schrödinger equation in
coordinates such as spheroidal, sphero-conical, and ellipsoidal ~see Table II!. In addition we
discuss the extension to the quadratic algebras that were in evidence in the case of two dimensions
and see what their implications may be.
Section V is devoted to the calculation of interbasis expansion coefficients for the V 3 potential
between spherical and parabolic bases.
II. GENERALIZED ISOTROPIC OSCILLATOR
The first potential ~see Table I! on our list of three is
F
G
1
1
1
2
2
2
v2 2 2 2
1 ~ k 12 4 ! ~ k 22 4 ! ~ k 32 4 !
1
1
,
V 1 ~ x,y,z ! 5
~ x 1y 1z ! 1
2
2
x2
y2
z2
~2!
where the constant k i > 21 . For k i 5 21 we have the ordinary isotropic oscillator potential. The
corresponding Schrödinger equation admits solutions via a separation of variables in eight coordinate systems: Cartesian, spherical, sphero-conical, cylindrical polar, cylindrical elliptic, prolate
and oblate spheroidal, and ellipsoidal. We summarize the bound state solutions in each case.
Before considering various coordinate systems we note that a basis for the symmetries of
Schrödinger’s equation with the potential ~2! consists of the six operators:
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710
Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
TABLE II. Systems of coordinate in three-dimensional Euclidean space.
Coordinate system
Coordinates
I. Cartesian
x,y,zPR
x,y,z
II. Cylindrical polar
r .0, w P @ 0,2p )
x5 r cos w, y5 r sin w, z
III. Cylindrical elliptic
zPR, e 1 , m 1 ,e 2 , m 2
~m12e1!~m22e1!
x25
,
~e22e1!
IV. Cylindrical parabolic
j ,xPR, h >0
x,y5 j h , z5 2 ( j 2 2 h 2 )
V. Spherical
r.0, u P @ 0,p # , w P @ 0,2p )
x5r cos u cos w, y5r sin u sin w, z5r cos u
VI. Prolate spheroidal
e 1 ,u 1 ,e 2 ,u 2 , w P @ 0,2p )
~m12e2!~m22e2!
y25
,z
~e12e2!
1
~u12e2!~u22e2!
x25
cos2 w,
~e12e2!
~u12e2!~u22e2! 2
y25
sin w,
~e12e2!
~u12e1!~u22e1!
z25
~e22e1!
VII. Oblate spheroidal
e 1 ,u 1 ,e 2 ,u 2 , w P @ 0,2p )
~u12e1!~u22e1!
x25
cos2 w,
~e22e1!
~u12e1!~u22e1! 2
y25
sin w,
~e22e1!
~u12e2!~u22e2!
z25
~e12e2!
VIII. Sphero-conical
r>0, e 1 , r 1 ,e 2 , r 2 ,e 3
x25r2
~r12e1!~r22e1!
,
~e12e2!~e12e3!
y25r2
~r12e2!~r22e2!
,
~e22e1!~e22e3!
z25r2
~r12e3!~r22e3!
~e32e2!~e32e1!
1
IX. Parabolic
j , h >0, w P @ 0,2p )
X. Ellipsoidal
a 1 ,u 1 ,a 2 ,u 2 ,a 3 ,u 3
x5 j h cos w, y5 j h sin w, z5 2 ( j 2 2 h 2 )
~u12a1!~u22a1!~u32a1!
x25
,
~a32a1!~a22a1!
~u12a2!~u22a2!~u32a2!
y25
,
~a12a2!~a32a2!
~u12a3!~u22a3!~u32a3!
z25
~a12a3!~a22a3!
XI. Paraboloidal
,a 3 , h 3
0, h 1 ,a 2 , h 2
~h12a3!~h22a3!~h32a3!
x25
,
~a32a2!
~h12a2!~h22a2!~h32a2!
y25
,
~a22a3!
1
z 2 5 2 ( h 1 1 h 2 1 h 3 2a 2 2a 3 )
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
M i 52D ii 2
J i j 5L 2i j 2
k 2i 2 14
x 2i
,
2H5M 1 1M 2 1M 3 ,
S D S D
k 2i 2
1 x 2j
1 x 2i 1
2
2 kj 2
2 ,
4 x 2i
4 x 2j 2
711
~3!
~4!
i, j51,2,3,
where L i j 5x i ] x j 2x j ] x i , D ii 52 ] 2x 1 v 2 x 2i is a diagonal components of the Demkov tensor1 and
i
we have the notation x 1 5x, x 2 5y, x 3 5z.
The commutators of the operators ~3! and ~4! can be closed to form a quadratic algebra as
follows:
@ M i , M j # 50,
@ M i , J jk # 50,
@ M i , J i j # 5Q i j 5Q @ i j # ,
@ J i j , J ik # 5R @ i jk # 5R,
where Q i j is totally antisymmetric and the totally antisymmetric quantity R @ i jk # is denoted by R.
Further commutators are calculated to be
@ M i , Q jk # 50,
@ M i , Q i j # 54 $ M i ,M j % 116J i j ,
@ M i , R # 54 $ M k ,J i j % 24 $ M j ,J ik % ,
@ J i j , Q i j # 54 $ M i ,J i j % 24 $ M j ,J i j % 28 ~ k 2j 21 ! M i 18 ~ k 2i 21 ! M j ,
@ J i j , Q ik # 54 $ M i ,J jk % 24 $ M j ,J ik % ,
@ J i j , R # 54 $ J i j ,J jk % 24 $ J i j ,J ik % 28 ~ k 2i 21 ! J jk 18 ~ k 2j 21 ! J ik ,
where $ A,B % 5AB1BA. The expression for the commutators of the Q and R are
@ Q i j , Q ik # 54 $ M i ,Q jk % ,
@ Q i j , R # 524 $ J i j ,Q ik % 24 $ J i j ,Q jk % .
All the commutators of the operators M i , J mn , Q pq , and R can be expressed in terms of quadratic
symmetric products of themselves. The algebra, therefore, is closed quadratically. There are relations between the symmetric products of the generators of this algebra. The exhaustive list of these
is as follows:
Q 2i j 5 83 $ J i j ,M i ,M j % 1 643 $ M i ,M j % 116v 2 J 2i j 216~ 12k 2j ! M 2i
2
2
2
2
216~ 12k 2i ! M 2j 2 128
3 v J i j 264v ~ 12k i !~ 12k j ! ,
$ Q i j ,Q ik % 5 38 $ J i j ,M i ,M k % 1 83 $ J ik ,M i ,M j % 2 83 $ J jk ,M i ,M i %
132v 2 ~ 12k 2i ! $ J i j ,J ik % 232~ 12k 2i ! M j M k 264v 2 ~ 12k 2i ! J jk ,
$ Q i j ,R % 5 38 $ J i j ,J i j ,M k % 2 83 $ J i j ,J ik ,M j % 2 83 $ J i j ,J jk ,M i % 2 643 $ J i j ,M k % 2
64
3
$ J ik ,M j %
2 643 $ J jk ,M i % 116~ 12k 2i ! $ J jk ,M j % 116~ 12k 2j ! $ J ik ,M i % 264~ 12k 2i !~ 12k 2j ! M k ,
R 2 52 43 $ J i j ,J ik ,J jk % 1 643 $ J i j ,J ik % 1 643 $ J i j ,J jk % 1 643 $ J ik ,J jk % 216~ 12k 2k ! J 2i j
2
216~ 12k 2j ! J 2ik 216~ 12k 2i ! J 2jk 1 128
3 ~ 12k k ! J i j
128
2
2
2
2
2
1 128
3 ~ 12k j ! J ik 1 3 ~ 12k i ! J jk 164~ 12k i !~ 12k j !~ 12k k ! ,
where $ A,B,C % 5ABC1CAB1BCA. Note that only five operators from ~3! and ~4! are functionally independent,7 and for all the coordinate systems that provide separable solutions for the
Schrödinger equation the operators characterizing the separation are always combinations of the
M i and J i j .
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712
Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
In the limiting case k i 5 21 , we obtain a quadratic algebra, too. In this case
R5 $ L ik , $ L i j ,L k j %% ,
Q i j 52 ~ L i j D i j 1D i j L i j ! ,
and instead of operators $ M i ,J i j ,Q i j ,R % we can consider as a basis for the symmetries the
Demkov tensor D i j , and the components of orbital momentum, L i j . In this regard we arrive at the
Lie algebra corresponding to the symmetries of the isotropic oscillator.1
Of all the coordinate systems for which separation is possible, in the case of this potential
there are only five which are not essentially a Euclidean two-space coordinate system supplemented by an additional Cartesian coordinate z. Such coordinate systems we do not consider
further here and the corresponding solutions of the Schrödinger equation and invariant algebra are
given in our previous paper17 ~see also Refs. 3 and 8!. For the remaining systems we now work out
bound state solutions and their corresponding symmetry characterization.
A. Oblate spheroidal basis
Let us consider what we call oblate spheroidal coordinates ~see Table II!. If we write these
coordinates in the form
x5x 8 cos w ,
y5x 8 sin w ,
z5y 8 ,
~5!
and put C5(x 8 ) 21/2F, the Schrödinger equation ~1! with potential ~2! assumes the form
S
F
D
G
k 21 2 14 k 22 2 41
k 32 2 41
] 2F ] 2F
]2
1
1
2
2
2
1
1
2E2
v
x
1y
1
2
2
1
2
F50.
!
~
8
8
] x 82 ] y 82
x 8 2 ]w 2 cos2 w sin2 w
4x 8 2
y 82
If we now write
F5L ~ x 8 ,y 8 ! Y ~ w ! ,
the w dependence can be extracted by requiring that
F
G
k 21 2 14 k 22 2 41
]2
2
2
Y ~ w ! 52M 2 Y ~ w ! .
]w 2 cos2 w sin2 w
~6!
The orthonormal solution of Eq. ~6! for w P @ 0,p /2# has the following form:
~ k ,k 2 !
Ym1
~ w !5
A
2 ~ 2m1k 1 1k 2 11 ! m!G ~ m1k 1 1k 2 11 !
G ~ m1k 1 11 ! G ~ m1k 2 11 !
~ k ,k 2 !
3 ~ cos w ! k 1 11/2~ sin w ! k 2 11/2P m 1
~ cos 2 w ! ,
~7!
where P (na , b ) (z) is a Jacobi polynomial and the separation constant quantizes as
M 52m1k 1 1k 2 11,
The remaining equation for the function L(x 8 ,y 8 ) is
H
F
~8!
m50,1,2,... .
k 23 2 41 M 2 2 14
]2
]2
2
2
2
1
1
2E2
v
x
1y
2
2
!
~
8
8
] x 82 ] y 82
y 82
x 82
GJ
L ~ x 8 ,y 8 ! 50.
This is exactly the equation we have already found ~see Ref. 17! in the case of two-dimensional
Euclidean space in elliptic coordinates. In terms of the original Cartesian coordinates the bound
state solutions have the form
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
n
3
L nm
~ x,y,z ! 5e 2 ~ v /2!~ x
k
2 1y 2 1z 2 !
~ x 2 1y 2 ! m1 ~ k 1 1k 2 11 ! /2z k 3 11/2
)
i51
S
713
D
x 2 1y 2
z2
1
21 , ~9!
u i 2e 1 u i 2e 2
where the u i satisfy of the system of n nonlinear equations
n
M 11
2
k 3 11
1
1
2 v 50.
u i 2e 1 u i 2e 2 jÞi u i 2 u j
(
We note that this prescription does correctly give a separable solution by noting the identity
z2
x 2 1y 2
~ u 1 2 u !~ u 2 2 u !
1
2152
.
u 2e 1 u 2e 2
~ u 2e 1 !~ u 2e 2 !
The energy E is quantized according to
E5 v ~ 2n1M 1k 3 13 ! 5 v ~ 2N1k 1 1k 2 1k 3 13 ! ,
~10!
where N5n1m is the principal quantum number.
Consider the Schrödinger equation in the spheroidal separable coordinates (u 1 ,u 2 , w ). After
the substitution C5 c 1 (u 1 ) c 2 (u 2 )Y ( w ), the separation equations are
S
H
D
d 2c~ u ! 1
2
d c ~ u ! 1 2Eu2 v 2 ~ u2e 1 !~ u2e 2 ! 1l
1
~ e 2 2e 1 ! M 2
1
1
1
1
du 2
2 u2e 1 u2e 2 du
4
~ u2e 1 !~ u2e 2 !
~ u2e 1 ! 2 ~ u2e 2 !
1
~ e 1 2e 2 !~ k 23 2 41 !
~ u2e 1 !~ u2e 2 ! 2
J
c ~ u ! 50,
~11!
where u5u 1 ,u 2 and l is the oblate spheroidal separation constant. The operator whose eigenvalue
is l is
L1 5
H
F
G J
G J F
u 2 ~ u 1 2e 1 !~ u 1 2e 2 ! ] 2
1
1
u 1 ~ u 2 2e 1 !~ u 2 2e 2 !
2
]
1
2
21
u 1 2u 2
u 1 2u 2
] u 1 2 u 1 2e 1 u 1 2e 2 ] u 1
3
H
F
]2
2
]
1
1
1 2
M 2 ~ e 1 2e 2 !
1
1
1
v
e
e
2u
u
1
!
~
2
2
1
2
4
~ u 1 2e 1 !~ u 2 2e 1 !
] u 22 2 u 2 2e 1 u 2 2e 2 ] u 2
3 ~ u 1 1u 2 2e 1 ! 1
~ k 23 2 41 !~ e 2 2e 1 !
~ u 1 2e 2 !~ u 2 2e 2 !
~ u 1 1u 2 2e 2 !
G
5J 131J 231J 121 ~ e 1 2e 2 ! M 3 2e 2 H2 ~ k 21 1k 22 1k 23 ! 1 43
~12!
with eigenvalues
n
l524e 2
(i
M 11
24e 1
u i 2e 1
n
k 11
3
(i u i32e 2 22 @~ e 1 1e 2 ! 1 ~ e 1 k 3 1e 2 M !# v 2 ~ k 21 1k 22 1k 23 ! 1 4 ,
~13!
and the second operator which characterizes the separation of variables in these coordinates is
L2 C5 ~ J 122k 21 2k 22 11 ! C52M 2 C.
~14!
To close this paragraph let us note that in the limit (e 2 2e 1 )→0 and (e 2 2e 1 )→` the oblate
spheroidal coordinates are changed into spherical and cylindrical polar coordinates, respectively.8
Correspondingly, the oblate spheroidal bases transform to spherical and cylindrical polar ones.
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714
Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
B. Prolate spheroidal basis
For prolate coordinates the description is almost exactly the same. All that is essentially
involved is the interchange of e 1 and e 2 .
C. Ellipsoidal basis
For ellipsoidal coordinates ~see Table II! we proceed as follows. We consider the Schrödinger
equation in Cartesian coordinates,
S
DF
G
~ k 21 2 41 ! ~ k 22 2 41 ! ~ k 23 2 41 !
] 2C ] 2C ] 2C
2 2
2
2
1
1
1
2E2
v
x
1y
1z
2
2
1
C50.
!
~
]x2
]y2
]z2
x2
y2
z2
If we now write
C ~ x,y,z ! 5e 2 v ~ x
2 1y 2 1z 2 ! 2k 11
1
x
y 2k 2 11 z 2k 3 11 F ~ x,y,z ! ,
the equation for F becomes
F
S
2k 2 11 ]
2k 3 11
]2
]2
] 2 2k 1 11 ]
]
]
]
1
1
1
1
] z 22 v x 1y 1z
2
2
21
]x
]y
]z
x
]x
y
]y
z
]x
]y
]z
G
D
22 v ~ k 1 1k 2 1k 3 13 ! C522EC.
To obtain the appropriate finite solutions we can make use of the identity
y2
z2
x2
~ u 1 2 u !~ u 2 2 u !~ u 3 2 u !
1
1
2152
u 2e 1 u 2e 2 u 2e 3
~ u 2e 1 !~ u 2e 2 !~ u 2e 3 !
and write
N
F ~ x,y,z ! 5
)
j51
S
D
x2
y2
z2
1
1
21 ,
u j 2e 1 u j 2e 2 u j 2e 3
~15!
where
N
k 1 11
k 2 11
k 3 11
2
1
1
1
2 v 50
u i 2e 1 u i 2e 2 u i 2e 3 jÞi u i 2 u j
(
and the energy level E is given by Eq. ~10!.
Writing the Schrödinger equation in terms of the ellipsoidal coordinates u i and using the
identities
3
E[
E 2i
(
i51 P iÞ j ~ u i 2u j !
F
~ k 21 2 41 !
x2
1
3
,
~ k 22 2 41 !
y2
~ x 2 1y 2 1z 2 ! [
1
~ k 23 2 41 !
z2
G(
3
[
i51
P~ u !
i
,
(
i51 P iÞ j ~ u i 2u j !
A~ ui!
,
P iÞ j ~ u i 2u j !
where P(u i )5(u i 2a 1 )(u i 2a 2 )(u i 2a 3 ) and @ a ik [(a i 2a k ) # and
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
A ~ u ! 5a 31a 21
~ k 21 2 41 !
u2a 1
1a 12a 32
~ k 22 2 41 !
u2a 2
1a 13a 23
~ k 21 2 41 !
u2a 3
715
,
we arrive at the following equation:
3
1
(
i51 P iÞ j ~ u i 2u j !
HA
4 P~ ui!
J
]
]
AP ~ u i ! 12Eu 2i 2 v 2 P ~ u i ! 2A ~ u i ! C50,
]ui
]ui
which, after the substitution C5 c 1 (u 1 ) c 2 (u 2 ) c 3 (u 3 ) and the introduction of the ellipsoidal
constants l 1 and l 2 , is divided into three identical differential equations
AP ~ u !
2
F
~ k 22 1 !
d
dc 1
AP ~ u ! 1 2Eu 2 2 v 2 P ~ u ! 1l 1 u2l 2 2 1 4 ~ a 3 2a 1 !~ a 2 2a 1 !
du
du 4
~ u2a 1 !
~ k 22 2 41 !
~ u2a 2 !
~ a 1 2a 2 !~ a 3 2a 2 ! 2
~ k 23 2 41 !
~ u2a 3 !
G
~ a 1 2a 3 !~ a 2 2a 3 ! c 50,
where u5u 1 ,u 2 ,u 3 . The operators that specify the eigenvalues l 1 and l 2 are
L 1 5J 121J 231J 231 ~ a 2 1a 3 ! M 1 1 ~ a 2 1a 1 ! M 3 1 ~ a 1 1a 3 ! M 2 2 ~ k 21 1k 22 1k 23 ! 1 43
and
L 2 5a 3 J 121a 2 J 131a 1 J 231a 2 a 3 M 1 1a 2 a 1 M 3 1a 1 a 3 M 2
2k 21 ~ a 3 1a 2 2a 1 ! 2k 22 ~ a 1 1a 3 2a 2 ! 2k 23 ~ a 1 1a 2 2a 3 ! 1 43 ~ a 1 1a 2 1a 3 ! ,
respectively. In terms of the zeros u j the eigenvalues of these operators are
l 1 522 @ k 2 ~ a 1 1a 3 ! 1k 1 ~ a 2 1a 3 ! 1k 3 ~ a 1 1a 2 ! 24 ~ a 1 1a 2 1a 3 !# v 22 ~ k 1 1k 2 1k 3 !
F(
N
3 ~ k 1 1k 2 1k 3 11 ! 2 29 14
i51
N
a2
N
~ k 2 11 !
~ k 1 11 !
~ k 3 11 !
1
a1
1
a3
~ u i 2a 2 ! i51 ~ u i 2a 1 ! i51 ~ u i 2a 3 !
(
(
G
~16!
and
l 2 52 21 ~ a 1 1a 2 1a 3 ! 22 v @ a 2 a 3 ~ k 1 11 ! 1a 2 a 1 ~ k 3 11 ! 1a 1 a 3 ~ k 2 11 !# 2a 1 ~ k 2 1k 3 11 ! 2
2a 2 ~ k 1 1k 3 11 ! 2 2a 3 ~ k 2 1k 1 11 ! 2
F(
N
24
i51
N
a 3a 1
N
G
~ k 2 11 !
~ k 1 11 !
~ k 3 11 !
1
a 3a 2
1
a 2a 1
.
~ u i 2a 2 ! i51
~ u i 2a 1 ! i51
~ u i 2a 3 !
(
(
~17!
D. Spherical and sphero-conical bases
For spherical-type coordinates there are two possibilities.
If we choose coordinates in Euclidean space accordingly,
x5rs 1 ,
y5rs 2 ,
z5rs 3 ,
~18!
where s 21 1s 22 1s 23 51, and put the wave function in the form
C5R ~ r ! S ~ r 1 , r 2 ! ,
~19!
where r 1 , r 2 are the spherical or sphero-conical coordinates, after separation of variables, we
arrive at two equations,
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716
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F
G
d 2 R 2 dR
J ~ J11 !
1 2E2 v 2 r 2 2
1
R50,
dr 2 r dr
r2
H
F
L 121L 231L 131 J ~ J11 ! 2
~ k 21 2 41 !
s 21
2
~ k 22 2 41 !
s 22
2
~ k 23 2 41 !
s 23
~20!
GJ
S50,
~21!
where J is the spherical separation constant.
~1! In the spherical coordinates ~see Table II! the wave function S( r 1 , r 2 ) has the separable
form
~ k ,k 2 !
S ~ q , w ! 5Z ~ q ! Y m 1
(k ,k 2 )
(w)
where Y m 1
~ w !,
is given by formula ~7!. This leads to the equation for Z:
F
G
k 23 2 41
M2
d
dZ
1
sin q
1 J ~ J11 ! 2 2 2
Z50,
sin q d q
dq
sin q cos2 q
M 52m1k 1 1k 2 11.
The solution of the above equation is ~see Ref. 8!
Z~ u !5
A
2 @ 2 ~ m1l11 ! 1k 1 1k 2 1k 3 # l!G ~ l12m1k 1 1k 2 1k 3 12 !
G ~ l1k 3 11 ! G ~ l12m121k 1 1k 2 !
~ M ,k 3 !
3 ~ cos u ! 1/21k 3 ~ sin u ! M P l
~ cos 2 u ! ,
~22!
lPN,
and for a spherical separation constant we get
J52l1M 1k 3 1 21 52l12m1k 1 1k 2 1k 3 1 23 .
~23!
~2! If we choose the sphero-conical coordinates on the sphere ~see Table II!, the solution of
the equation ~21! has the form
3
S~ r1 ,r2!5
)
)S
n
k 11/2
l 51
sl l
j51
s 21
u j 2e 1
1
s 22
u j 2e 2
1
s 23
u j 2e 3
D
,
~24!
and the spherical separation constant is quantized according to Eq. ~23! where n5l1m. This
achieves a separation of variables solution because of the identity
s 21
u j 2e 1
1
s 22
u j 2e 2
1
s 23
u j 2e 3
5
~ r 1 2 u j !~ r 2 2 u j !
~ u j 2e 1 !~ u j 2e 2 !~ u j 2e 3 !
and the Niven equations
q
k 1 11
2
k 2 11
k 3 11
1
1
1
50.
u i 2e 1 u i 2e 2 u i 2e 3 jÞi u i 2 u j
(
The functions S( r 1 , r 2 ) have the separable form
S ~ r 1 , r 2 ! 5B 1 ~ r 1 ! B 2 ~ r 2 ! ,
~25!
and the separation equations are @ P( r )5( r 2e 1 )( r 2e 2 )( r 2e 3 ) #
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
F
717
~ k 21 2 41 !
~ k 22 2 41 !
d
dB 1
AP ~ r ! AP ~ r ! 1 l2J ~ J11 ! 2
~ e 2e 2 !~ e 1 2e 3 ! 2
~ e 2e 1 !~ e 2 2e 3 !
dr
dr 4
~ r 2e 1 ! 1
~ r 2e 2 ! 2
2
~ k 23 2 41 !
~ r 2e 3 !
G
~ e 3 2e 2 !~ e 3 2e 1 ! B50,
~26!
where B5B 1 ,B 2 according as r 5 r 1 , r 2 , respectively. The sphero-conical wave functions satisfy
the eigenfunction equations
~ J 121J 131J 23! S5 @~ k 21 1k 22 1k 23 ! 2 ~ 2q121k 1 1k 2 1k 3 ! 2 2 21 # S
~27!
~ e 1 J 231e 2 J 131e 3 J 12! S
5 @ k 21 ~ e 2 1e 3 2e 1 ! 1k 22 ~ e 1 1e 3 2e 2 ! 1k 23 ~ e 1 1e 2 2e 3 ! 2 43 ~ e 1 1e 2 1e 3 ! 2l # S,
~28!
where
l52 @ k 1 ~ e 2 1e 3 ! 1k 2 ~ e 1 1e 3 ! 1k 3 ~ e 2 1e 1 ! 1e 3 k 1 k 2 1e 2 k 1 k 3 1e 1 k 3 k 2 # 1 23 ~ e 1 1e 2 1e 3 !
F
n
24 e 2 e 3
(
i51
n
G
n
k 1 11
k 2 11
k 3 11
1e 1 e 3
1e 2 e 1
.
u i 2e 1
i51 u i 2e 2
i51 u i 2e 3
(
(
~29!
Let us now go to the radial equation ~20!. This equation is very reminiscent of the radial
equation for the three-dimensional harmonic oscillator except that the orbital quantum number l is
replaced by 2l12m1k 1 1k 2 1k 3 1 23 . The orthonormal solution of the radial equation ~20!, in
terms of Laguerre polynomials L an (x), is
R nrJ~ r ! 5
A
S
D
2 v 3/2n r !
v
~ Av r ! J exp 2 r 2 L nJ11/2
~ vr2!,
r
G ~ n r 12l12m1k 1 1k 2 1k 3 13 !
2
~30!
and the energy spectrum is given by formula ~10! where the n r 50,1,2,... is the radial quantum
number and the principal quantum number now is N5(n r 1n)5(n r 1l1m).
III. GENERALIZED ANISOTROPIC OSCILLATOR
The second potential ~see Table I! is
F
G
1
1
2
2
v2 2 2
1 k 12 4 k 22 4
2
1 2 .
V 2 ~ x,y,z ! 5
~ x 1y 14z ! 1
2
2 x2
y
~31!
The corresponding Schrödinger equation has separable solutions in five coordinate systems: Cartesian coordinates, cylindrical polar coordinates, cylindrical elliptic coordinates, cylindrical parabolic coordinates, and parabolic coordinates. It is the last of these that gives interesting new
solutions. The first four coordinate systems are of cylindrical type and can be deduced from what
we already know for Euclidean two-dimensional space ~see Refs. 8 and 17!. Before considering
the bound state solutions in the case of the parabolic coordinate system we consider the quadratic
algebra of second-order symmetry operators which are associated with this potential. A basis for
these operators is
M 1 5 ] 2x 2 v 2 x 2 1
k 21 2 41
x2
,
M 2 5 ] 2y 2 v 2 y 2 2
k 22 2 14
y2
,
P5 ] 2z 24 v 2 z 2 ,
~32!
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718
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J. Math. Phys., Vol. 40, No. 2, February 1999
S D S D
L5L 2122 k 21 2
1 y2
1 x2 1
2
2 ,
2 2 k 22
4 x
4 y2 2
S
S
~33!
D
D
k 21 2 41
1
S 1 52 ~ p x L 131L 13p x ! 1z v 2 x 2 2 2 ,
2
x
k 22 2 41
1
2 2
S 2 52 ~ p y L 231L 23p y ! 1z v y 2 2 ,
2
x
~34!
where p x,y 5 ] x,y .
The relations that define the quadratic algebra are obtained by exhaustive commutation. The
nonzero commutators of the above basis are
@ M 1 , L # 5 @ L, M 2 # 5Q,
@ L, S 1 # 5 @ S 2 , L # 5B,
@ M i , S i # 5A i ,
@ P, S i # 52A i .
Further nonzero commutators with Q are
@ M i , Q # 5 @ Q, M 2 # 54 $ M 1 ,M 2 % 116v 2 L,
@ S 1 , Q # 5 @ Q, S 2 # 54 $ M 1 ,M 2 % ,
@ L, Q # 54 $ M 1 ,L % 24 $ M 2 ,L % 116~ 12k 21 ! M 1 216~ 12k 22 ! M 2 ;
nonzero commutators with A i are
@ M i , A i # 516v 2 S i ,
@ L, A 1 # 5 @ A 2 , L # 54 $ M 1 ,S 2 % 24 $ M 2 ,S 1 % ,
@ S i , A i # 5 $ M i ,M i % 22 $ M i , P % 18 v 2 ~ 12k 2i ! ,
@ P, A i # 5216v 2 S i ,
@ S i , A j # 5 $ M i ,M j % 14 v 2 L;
and nonzero commutators with B are
@ M 1 , B # 524 $ M 2 ,S 1 % ,
@ M 2 , B # 524 $ M 1 ,B % ,
@ P, B # 54 $ M 2 ,S 1 % 24 $ M 1 ,S 2 % ,
@ L, B # 524 $ L,S 1 % 14 $ L,S 2 % 216~ 12k 22 ! S 1 116~ 12k 21 ! S 2 ,
@ S 1 , B # 5 $ L,M 1 % 22 $ L, P % 24 $ S 1 ,S 2 % 24 ~ 12k 21 ! M 2 ,
@ B, S 2 # 5 $ L,M 2 % 22 $ L, P % 24 $ S 1 ,S 2 % 24 ~ 12k 22 ! M 1 .
The remaining nonzero commutators are
@ Q, A i # 524 $ M i ,A j % ,
@ Q, B # 524 $ L,A 1 % 24 $ L,A 2 % ,
@ A 1 , B # 5 $ M 1 ,Q % 24 $ S 1 ,A 2 % ,
@ A 1 , A 2 # 54 v 2 Q,
@ B, A 2 # 5 $ M 2 ,Q % 24 $ S 2 ,A 1 % .
There are also various relations among the generators of our quadratic algebra:
$ M 1 ,B % 5 $ L,A 1 % 2 $ S 1 ,Q % 24 ~ 12k 21 ! A 2 ,
$ M 2 ,B % 52 $ L,A 2 % 2 $ S 2 ,Q % 14 ~ 12k 22 ! A 1 ,
$ P,Q % 52 $ S 1 ,A 2 % 22 $ S 2 ,A 1 % ,
$ M 1 ,A 2 % 2 $ M 2 ,A 1 % 24 v 2 B50,
Q 2 5 38 $ L,M 1 ,M 2 % 18 v 2 $ L,L % 216~ 12k 21 ! M 21 216~ 12k 22 ! M 22
2
2
2
2
1 643 $ M 1 ,M 2 % 2 128
3 v L2128v ~ 12k 1 !~ 12k 2 ! ,
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
719
$ Q,A 1 % 5 38 $ M 1 ,M 2 ,S 1 % 2 83 $ M 1 ,M 1 ,S 2 % 116v 2 $ L,S 1 % 264~ 12k 21 ! S 2 ,
$ Q,A 2 % 52 38 $ M 1 ,M 2 ,S 2 % 1 83 $ M 2 ,M 2 ,S 1 % 216v 2 $ L,S 2 % 164~ 12k 22 ! S 1 ,
$ Q,B % 52 38 $ M 2 ,L,S 1 % 2 38 $ M 1 ,L,S 2 % 116~ 12k 21 ! $ M 2 ,S 2 %
116~ 12k 22 ! $ M 1 ,S 1 % 2 643 $ M 1 ,S 2 % 2 643 $ M 2 ,S 1 % ,
A 21 5 23 $ M 1 ,M 1 , P % 18 v 2 $ S 1 ,S 1 % 116v 2 ~ 12k 21 ! P232v 2 M 1 ,
$ A 1 ,A 2 % 5 34 $ M 1 ,M 2 , P % 116v 2 $ S 1 ,S 2 % 18 v 2 $ L, P % ,
$ A 1 ,B % 5 38 $ M 1 ,S 1 ,S 2 % 2 38 $ M 2 ,S 1 ,S 1 % 1 34 $ M 1 ,L, P % 1 323 $ M 1 ,M 2 % 28 ~ 12k 21 ! $ M 1 , P % 2 643 v 2 L,
$ A 2 ,B % 52 38 $ M 2 ,S 2 ,S 1 % 1 38 $ M 1 ,S 2 ,S 2 % 2 34 $ M 2 ,L, P %
2 323 $ M 1 ,M 2 % 18 ~ 12k 22 ! $ M 2 , P % 1 643 v 2 L,
B 2 5 83 $ L,S 1 ,S 2 % 1 32 $ L,L, P % 1 643 $ S 1 ,S 2 % 216~ 12k 21 ! S 22 216~ 12k 22 ! S 21
1 163 $ L,M 1 % 2 163 $ L, P % 1 323 ~ 12k 22 ! M 1 1 323 ~ 12k 21 ! M 2 216~ 12k 21 !~ 12k 22 ! P.
This completes the nonzero relations for the quadratic algebra and the associated relations among
the generators. For the last coordinate system in our list we develop the bound state solutions.
Parabolic basis
The Schrödinger equation in Cartesian coordinates with this potential has the form
S
DF
G
~ k 12 4 ! ~ k 22 4 !
] 2C ] 2C ] 2C
2 2
2
2
1
C50.
2 1
2 1
2 1 2E2 v ~ x 1y 14z ! 2
]x
]y
]z
x2
y2
2
1
2
1
If we choose the coordinates (x 8 ,y 8 , w ) according formula ~6! and the wave function C in the
form
~ k ,k 2 !
C ~ x 8 ,y 8 , w ! 5 ~ x 8 ! 21/2L ~ x 8 ,y 8 ! Y m 1
(k ,k 2 )
where Y m 1
~ w !,
m50,1,...,
( w ) is given by ~7!, then the equation for the function L(x 8 ,y 8 ) is
F
G
~ M 2 2 41 !
]2
]2
2
2
2
1
2 v ~ x 8 14y 8 ! 2
12E L ~ x 8 ,y 8 ! 50.
] x 82 ] y 82
x 82
This just the problem whose solution has been found ~see Ref. 1! in the case of two-dimensional
Euclidean space. If now we write
L ~ x,y,z ! 5e 2 ~ v /2!~ x
2 1y 2 ! 2 v z 2
~ x 2 1y 2 ! ~ k 2 /211/4! P ~ x,y ! ,
where
n
P ~ x,y ! 5
)
j51
S
x 2 1y 2
u 2j
D
12z2 u 2j ,
then the l j satisfy
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720
Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
4 ~ l11 !
ul
2
n
1
4
2
(
2 22 v u l 50
2
iÞl u l 2 u i
and energy E quantizes according to
E5 v ~ 2n1M 12 ! 5 v ~ 2N1k 1 1k 2 13 ! ,
~35!
where the principal quantum number N5n1m. This method of solution is based on the identity
x 2 1y 2
~ j 2 2 u 2 !~ h 2 1 u 2 !
2
12z2
u
5
.
u2
u2
In fact, the separation equations in j and h for solution of the Schrödinger equation
~ k ,k 2 !
C ~ j , h , w ! 5X 1 ~ j ! X 2 ~ h ! Y m 1
~w!
have the form
F
S
d2
1 d
M2
2
2 6
1
1
2E
r
2
v
r
2
1 eb
dr2 r dr
r2
DG
X ~ r ! 50,
~36!
where e 51 if r 5 j and 21 if r 5 h , and b is the parabolic separation constant. By eliminating the
energy E from Eqs. ~36! we produce the operator, the eigenvalues of which is b:
S
S
D
D
k 21 2 14 k 22 2 41
j2 ]
] h2 ] ]
j 22 h 2 ] 2
1
2 2 2 2
2
h 2
j
1v j h ~ j 2h !2 2 2
2
2
.
L5 2
j 1 h 2 h ] h ] h j ]j ]j
j h
]w 2 cos2 w sin2 w
~37!
In Cartesian coordinates the operator L can be written as
L5z
F
G S
D
k 21 2 41 k 22 2 14
]2
]2
]
]
]
2 2
2
1
1
v
x
1y
2
2
x 1y 21 ,
!
~
2
2
2 2
2
]x
]y
x
y
]z ]x
]y
and thus the parabolic basis satisfies two eigenvalue equations
LC52 ~ S 1 1S 2 ! C5 b C,
LC5 ~ k 21 1k 22 2M 2 21 ! C,
where operators L, S 1,2 are given by formulas ~33! and ~34! and the eigenvalue b is
n
b 522 ~ M 21 !
)
j51
S( D
n
u 2j
k51
u 22
.
k
~38!
IV. GENERALIZED KEPLER–COULOMB SYSTEM
The third potential we consider is
V 3 ~ x,y,z ! 52
a
Ax 2 1y 2 1z
1
2
F
G
1
1
2
2
1 k 12 4 k 22 4
1
.
2 x2
y2
The corresponding Schrödinger equation has the form
S
D
F
S
k 21 2 41 k 22 2 14
2a
] 2C ] 2C ] 2C
1
1
1
2E1
2
1 2
]x2
]y2
]z2
x2
y
Ax 2 1y 2 1z 2
~39!
DG
C50.
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
721
This equation admits separable solutions in the four coordinate systems: spherical, spheroconical,
prolate spheroidal, and parabolic.
The second-order symmetries of the corresponding Schrödinger equation are
S D
S D
S D S D
J 235L 2232 k 22 2
1 z2 1
2 ,
4 y2 2
J 125L 2122 k 22 2
J 135L 2132 k 21 2
1 z2 1
2 ,
4 x2 2
1 x2
1 y2 1
2
2
k
2
2 ,
1
4 y2
4 x2 2
~40!
S
D
k 21 2 41 k 22 2 14
1
az
2z
1 2 .
L52 @ $ p x ,L 13% 1 $ p y ,L 23% # 1 2
2
x2
y
Ax 1y 2 1z 2
~41!
These symmetry operators do not appear to close under repeated commutation. One obvious
subalgebra that is quadratically closed is that generated by the elements J 12 , J 13 , and J 23 . The
closure relations can be readily deduced from the algebra given for the first potential with the
proviso that k 3 5 21 .
A. Spherical and sphero-conical bases
If we use polar coordinates according to ~18! and write the wave function C in the separable
form C5R(r)S( r 1 , r 2 ), then the separable equations are
F
G
d 2 R 2 dR
2 a J ~ J11 !
1
1 2E1
2
R50,
dr
r dr
r
r2
H
F
L 121L 231L 131 J ~ J11 ! 2
~ k 21 2 41 !
s 21
2
~ k 22 2 41 !
s 22
GJ
~42!
~43!
S50.
~1! In the spherical coordinates, choosing the wave function S( r 1 , r 2 ) according to
~ k ,k 2 !
S ~ q , w ! 5Z ~ q ! Y m 1
(k ,k 2 )
(w)
where Y m 1
~ w !,
m50,1,2,...,
is given by formula ~7!, we go to the equation for Z:
F
G
d
dZ
M2
1
sin q
1 J ~ J11 ! 2 2 Z50,
sin q d q
dq
sin q
M 52m1k 1 1k 2 11.
The orthonormal solution of the above equation for q P @ 0,p # is
Z~ q !5
2M
Ap
S D
A
1
~ 2l12M 11 ! l!
G M 1 ~ sin q ! M C lM 11/2~ cos q ! ,
2G ~ l12M 11 !
2
~44!
where lPN and C ln (x) is a Gegenbauer polynomial.21 The spherical separation constant is given
by
J5l1M 5l12m1k 1 1k 2 11.
~45!
~2! The solution of the Schrödinger equation ~43! in sphero-conical coordinates follows from
what we have done before in Sec. II D, part ~2!. If we write S( r 1 , r 2 ) as
2
S ~ r 1 , r 2 ! 5s e3
)
l 51
n
k 11/2
sl l
)
j51
S
s 21
u j 2e 1
1
s 22
u j 2e 2
1
s 23
u j 2e 3
D
,
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722
Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
where e 50,1 then the zeros satisfy the Niven equations
n
e 1 21
k 2 11
2
k 1 11
1
1
1
50.
u i 2e 1 u i 2e 2 u i 2e 3 jÞi u i 2 u j
(
The functions S( r 1 , r 2 ) satisfy the eigenfunction equations
~ J 121J 131J 23! S5 @~ k 21 1k 22 ! 2 ~ 2q1 23 1k 1 1k 2 1 e ! 2 2 47 # S,
~46!
~ e 1 J 231e 2 J 131e 3 J 12! S5 @~ e 2 2e 1 !~ k 21 2k 22 ! 1e 3 ~ k 21 1k 22 21 ! 2 21 ~ e 1 1e 2 ! 2l # S,
~47!
where the sphero-conical separation constant l is
l522 @ k 1 ~ e 2 1e 3 ! 1k 2 ~ e 1 1e 3 ! 1 ~ e 2 21 !~ e 2 1e 1 ! 1e 3 k 1 k 2 1 ~ e 2 k 1 1e 1 k 2 !~ e 2 21 !#
F
G
n
n
n
e 1 12
k 1 11
k 2 11
3
2 ~ e 1 1e 2 1e 3 ! 24 e 2 e 3
1e 1 e 3
1e 2 e 1
.
2
i51 u i 2e 1
i51 u i 2e 2
i51 u i 2e 3
(
(
(
~48!
Finally, let us consider the radial equation ~42!. The introduction of ~45! into ~42! leads to
F
G
2 a ~ l12m1k 1 1k 2 11 !~ l12m1k 1 1k 2 12 !
d 2 R 2 dR
1
1 2E1
2
R50,
dr
r dr
r
r2
which is the radial equation for the Coulomb problem, except the orbital quantum number l is
replaced here by (l12m1k 1 1k 2 11). The bound state solution of Eq. ~43! is
R NJ ~ r ! 5
2 ~ a ! 3/2
N2
A
S D
G ~ N1J11 ! 2 a r
~ N2J21 ! ! N
J
S
e 2 a r/N
2ar
1 F 1 2N1J11;2J12;
G ~ 2J11 !
N
D
and the energy spectrum given by
E52
a2
,
N2
N5n r 1J1152m1n r 1l1k 1 1k 2 12,
n r 50,1,2,... .
B. Parabolic and prolate spheroidal bases
The remaining solutions for which separation of variables is possible can be best observed by
writing the Schrödinger equation in parabolic coordinates. If we do this and choose solutions of
the form
~ k ,k 2 !
C5S ~ j , h !~ j h ! 21/2Y m 1
(k ,k 2 )
where Y m 1
form
~ w !,
m50,1,2,...,
~49!
( w ) is given by formula ~7!, we find that the Schrödinger equation has the reduced
F
S
] 2S ] 2S
1
2
2
2
21
2 1 2E ~ j 1 h ! 1 M 2
]j
]h
4
DS
D G
1
1
2 1 2 14 a S50.
j
h
This is clearly recognizable as solvable via separation of variables in parabolic coordinates j and
h. The separable solution for the wave function S( j , h ) is
S~ j,h !5
& ~ a ! 3/2
~ j h ! 21/2 f nM1 ~ j ! f nM2 ~ h ! ,
N2
n 1,2PN,
M 52m1k 1 1k 2 11,
~50!
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Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
723
where
f nM ~ x ! 5
A
S D
2
G ~ n 1 1M 11 ! e 2 ~ a /2N ! x a x 2
n 1!
G ~ M 11 ! N
M /2
1F 1
S
2n 1 ;M 11;
ax2
N
D
~51!
and N5n 1 1n 2 1M 115n 1 1n 2 12m1k 1 1k 2 12.
It is also interesting to observe that we could contemplate an E-dependent algebra of secondorder symmetries acting on the functions H( j , h ). Indeed, a basis for such symmetries is
S
P 1 5 ] 2j 1 M 2 2
D
S
1 1
12E j 2 ,
4 j2
P 2 5 ] 2n 1 M 2 2
S
M 5 ~ j] h 2 h ] j ! 2 2 M 2 2
1
4
DS
j2
h
21
D
1 1
12E h 2 ,
4 h2
D
h2
1
2 2 .
j
2
The corresponding closure relations can be deduced from those given for the first potential.
Apart from the symbols this has the same form as was dealt with in two dimensions. If we
now regard j and h as Cartesian coordinates, separation is also possible in polar and elliptical
coordinates. The case of polar coordinates has essentially been done above. The case of elliptic
coordinates can be done by the standard prescription. This is achieved by looking for solutions of
the form
S ~ j , h ! 5e 2
A22E ~ x 2 1y 2 1z 2 !
s
~ x 2 1y 2 ! ~ 1/2!~ M 11/2!
)
j51
S
Ax 2 1y 2 1z 2 1z
u m 2e 1
1
Ax 2 1y 2 1z 2 2z
u m 2e 1
D
21 ,
where we have written the solutions in the coordinate representation. ~Recall that j 2
5 Ax 2 1y 2 1z 2 1z and h 2 5 Ax 2 1y 2 1z 2 2z.! With
j5
A
~ u 1 2e 1 !~ u 2 2e 1 !
,
~ e 2 2e 1 !
h5
A
~ u 1 2e 2 !~ u 2 2e 2 !
,
~ e 1 2e 2 !
where e 1 ,u 1 ,e 2 ,u 2 , the choice of Cartesian coordinates that is appropriate in this case is
x5
1
e 2 2e 1
AF S
e 2 2e 1
2
y5
1
e 2 2e 1
AF S
e 2 2e 1
2
z5
1
e 2 2e 1
D S
D S
FS
2
2 u 12
2
2 u 12
u 12
e 2 1e 1
2
e 2 1e 1
2
e 2 1e 1
2
DS
D GF S
D GF S
e 2 1e 1
2
2
e 2 1e 1
2
2
u 22
2
u 22
D S DG
D S DG
D S DG
u 22
2
2
2
e 2 2e 1
e 2 1e 1
1
2
2
e 2 2e 1
2
2
e 2 2e 1
2
2
cos w ,
sin w ,
2
.
This corresponds to the choice of prolate spheroidal coordinates of type II.22,8
V. INTERBASIS EXPANSION
According to the principles of quantum mechanics the solutions of the same Schrödinger
equation in the different separable coordinate systems for a given value of energy E are connected
by unitary transformations or interbasis expansions. For example, we examine here the direct
calculation of the interbasis expansion between the spherical and parabolic wave functions for
potential V 3 . We have
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724
Kalnins et al.
J. Math. Phys., Vol. 40, No. 2, February 1999
n 1 1n 2
C n 1 ,n 2 ,m ~ j , h , w ! 5
(
l50
W nl
~ k 1 ,k 2 ! C n r lm ~ r, q , w ! .
1n2m
~52!
where n r 1l5n 1 1n 2 . For calculation of the coefficients of interbasis expansion in ~52! we may
use the ‘‘asymptotic method,’’ 18,22 which is the following. Writing the parabolic wave function on
the left-hand side of ~52! in spherical coordinates (r, q , w ) accordingly,
j 2 5r ~ 11cos q ! ,
(k ,k 2 )
(w)
eliminating the function Y m 1
h 2 5r ~ 12cos q ! ,
on both sides of ~52!, and using the formula
1 F 1 ~ 2n; a ;x ! ;
G~ a !
~ 2x ! n
G ~ a 1n !
for x arbitrary large, we see that the expansion ~52! yields an equation which depends only on the
variable q. Then, by using the orthogonality relations for the functions Z lm ( q ) in the quantum
number l, we arrive at the following expression for interbasis expansions coefficients:
W nl
1n2m
~ k 1 ,k 2 ! 5 ~ 21 ! l
3
3
A
E
p
0
G ~ M 11/2!
2 n 1 1n 2 11 Ap
~ 2l12M 11 ! G ~ n 1 1n 2 1l12M 12 !~ n 1 1n 2 2l ! !l!
G ~ l12M 11 ! G ~ n 1 1M 11 ! G ~ n 1 1M 11 !~ n 1 ! ! ~ n 2 ! !
~ 11cos q ! n 1 1M ~ 12cos q ! n 2 1M C lM 11/2~ q ! sin q d q ,
M 52m1k 1 1k 2 11.
~53!
By using the Rodrigues formula for the Gegenbauer polynomials21
C ln ~ x ! 5
Ap G ~ l12l !
dl
~ 21 ! l
2 2l11/2
12x
!
~
~ 12x 2 ! l1l21/2
l! 2 l12l21 G ~ l ! G ~ l1l11/2!
dx l
and comparing ~53! with the integral representation for the Clebsch–Gordan coefficients of the Lie
group SU~2! ~Ref. 23!,
C ac ga ;b b 5 d a 1 b , g
3
E
1
21
A
~ 2c11 !~ J11 ! ! ~ J22c ! ! ~ c1 g ! !
~ 21 ! a2c1 b
2 J11
~ J22a ! ! ~ J22b ! ! ~ a2 a ! ! ~ a1 a ! ! ~ b2 b ! ! ~ b1 b ! ! ~ c2 g ! !
~ 12x ! a2 a ~ 11x ! b2 b
d c2 g
@~ 12x ! J22a ~ 11x ! J22b # dx
dx c2 g
with J5a1b1c, we obtain
W nl
a5
a5
1n2m
~ k 1 ,k 2 ! 5 ~ 21 ! n 2 C ac,aa;a1bb ,
n 1 1n 2 12m1k 1 1k 2 11
,
2
n 1 2n 2 12m1k 1 1k 2 11
,
2
c5l12m1k 1 1k 2 11,
b5
~54!
n 2 2n 1 12m1k 1 1k 2 11
.
2
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J. Math. Phys., Vol. 40, No. 2, February 1999
Kalnins et al.
725
Since the parameters in ~54! in general are not integers or half-integers, the coefficients of interbasis expansion ~51! may be considered as analytic continuation, for real values of their arguments, of the SU~2! Clebsch–Gordan coefficients. Note also that the inverse expansion of ~52!
follows from the orthonormality of SU~2! Clebsch–Gordan coefficients.
ACKNOWLEDGMENT
The authors thank P. Winternitz for very interesting discussions.
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1
2
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