Acta Polytechnica


doi:10.14311/AP.2016.56.0166
Acta Polytechnica 56(3):166–172, 2016 © Czech Technical University in Prague, 2016

available online at http://ojs.cvut.cz/ojs/index.php/ap

A SUPERINTEGRABLE MODEL WITH REFLECTIONS ON S3
AND THE RANK TWO BANNAI-ITO ALGEBRA

Hendrik De Biea, ∗, Vincent X. Genestb, Jean-Michel Lemayc,
Luc Vinetc

a Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Galglaan 2,
9000 Ghent, Belgium

b Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA
02139, USA

c Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal, QC,
Canada, H3C 3J7

∗ corresponding author: Hendrik.DeBie@UGent.be

Abstract. A quantum superintegrable model with reflections on the three-sphere is presented. Its
symmetry algebra is identified with the rank-two Bannai-Ito algebra. It is shown that the Hamiltonian
of the system can be constructed from the tensor product of four representations of the superalgebra
osp(1|2) and that the superintegrability is naturally understood in that setting. The exact separated
solutions are obtained through the Fischer decomposition and a Cauchy-Kovalevskaia extension theorem.

Keywords: Bannai-Ito algebra; Cauchy-Kovalevskaia extension; quantum superintegrable model.

1. Introduction
Superintegrability shares an intimate connection with exact solvability. For classical systems, this connection
is fully understood while it remains an empirical observation for general quantum systems. The study of
superintegrable models has proved fruitful in understanding symmetries and their algebraic description, and has
also contributed to the theory of special functions. A quantum system in n dimensions with Hamiltonian H
is said to be maximally superintegrable if it possesses 2n− 1 algebraically independent constants of motion
c1,c2, . . . ,c2n−1 commuting with H, that is [H,ci] = 0 for i = 1, . . . , 2n− 1, where one of these constants is
the Hamiltonian itself. Such a system is further said to be superintegrable of order l if the maximum order in
momenta of the constants of motion (except H) is l.

One of the important quantum superintegrable models is the so-called generic three-parameter system on
the two-sphere [12], whose symmetries generate the Racah algebra which characterizes the Wilson and Racah
polynomials sitting atop the Askey scheme [1]. All two-dimensional second order superintegrable models of the
form H = ∆ + V where ∆ denotes the Laplace-Beltrami operator have been classified [12] and can be obtained
from the generic three-parameter model through contractions and specializations [11]. A similar model with
four parameters defined on the three-sphere has also been introduced and its connection to bivariate Wilson and
Racah polynomials has been established [10].

Recently, superintegrable models defined by Hamiltonians involving reflection operators have been the subject
of several investigations [2–5, 9]. One of the interesting features of these models is their connection to less
known bispectral orthogonal polynomials referred to as −1 polynomials. Many efforts have been deployed
to characterize these polynomials, which can be organized in a tableau similar to the Askey one [13–19]. Of
particular relevance to the present paper is the Laplace-Dunkl equation on the two-sphere studied in [6, 7],
which has the rank-one Bannai-Ito algebra as its symmetry algebra [17]. This Bannai-Ito algebra encodes the
bispectrality of the Bannai-Ito polynomials which depend on four parameters and stand at the highest level
of the hierarchy of −1 orthogonal polynomials. As such, this Laplace-Dunkl system on the two sphere can be
thought of as a generalization with reflection operators of the generic three-parameter model (without reflections)
on the two-sphere which is recovered when wavefunctions with definite parities are considered. The goal of this
paper is to introduce a novel quantum superintegrable model with reflections on the three-sphere which similarly
embodies the generic four-parameter model introduced and studied in [10].

The paper is divided as follows. In Section 2, we introduce a superintegrable model with four-parameters on
the three-sphere and exhibit its symmetries explicitly. In Section 3, it is shown how the Hamiltonian of the
model can be constructed from four realizations of the superalgebra osp(1|2). Moreover, the symmetry algebra
is characterized and is seen to correspond to a rank-two generalization of the Bannai-Ito algebra. In Section 4,
the structure of the space of polynomial solutions is exhibited using a Fischer decomposition and an explicit

166

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vol. 56 no. 3/2016 A Superintegrable Model With Reflections on S3

basis for the eigenfunctions is constructed with the help of a Cauchy-Kovalevskaia extension theorem. Some
concluding remarks are offered in Section 5.

2. A superintegrable model on S3
Let s1,s2,s3,s4 be the Cartesian coordinates of a four-dimensional Euclidian space and take the restriction to
the embedded three-sphere: s21 + s22 + s23 + s24 = 1. Consider the system with four parameters µ1,µ2,µ3,µ4 with
µi ≥ 0 for i = 1, 2, 3, 4 governed by the Hamiltonian

H =
∑

1≤i<j≤4
J2ij +

4∑
i=1

µi
s2i

(µi −Ri), (2.1)

where

Jij =
1
i
(si∂sj −sj∂si ), Rif(si) = f(−si), (2.2)

are the angular momentum operators and reflection operators, respectively. The six quantities

Ljk =
(

1
2

+ µjRj + µkRk +
(
iJjk + µj

sk
sj
Rj −µk

sj
sk
Rk

) k∏
l=j+1

Rl

)
RjRk, 1 ≤ j < k ≤ 4, (2.3)

can easily be verified to commute with H on the 3-sphere and are thus conserved. It can be shown that any four
of the Ljk are algebraically independent. Hence H defines a maximally superintegrable system of first order.
There are also four more conserved quantities of the form

MA =
(

1 +
∑
i∈A

µiRi +
∑
j<k
j,k∈A

(
iJjk + sk

µj
sj
Rj −sj

µk
sk
Rk

) k∏
l=j+1

Rl

) ∏
i∈A

Ri, (2.4)

where A = {1, 2, 3},{1, 2, 4},{1, 3, 4} or {2, 3, 4}. Furthermore, a direct computation yields

[H,Ri] = 0, i = 1, 2, 3, 4. (2.5)

The reflections are thus discrete symmetries of the system.

3. Algebraic construction from osp(1|2)
The superalgebra osp(1|2) can be presented with five generators x,D,E, |x|2 and D2 with the following defining
relations:

{x,x} = 2|x|2, {D,D} = 2D2, {x,D} = 2E, [D,E] = D, [D, |x|2] = 2x,
[E,x] = x, [D2,x] = 2D, [D2,E] = 2D2, [D2, |x|2] = 4E, [E, |x|2] = 2|x|2, (3.1)

where [a,b] = ab− ba is the commutator and {a,b} = ab + ba is the anti-commutator. One can realize four
mutually commuting copies of this superalgebra by taking

Di = ∂si −
µi
si
Ri, D

2
i = DiDi, xi = si, |xi|

2 = s2i , Ei = si∂si +
1
2
, (3.2)

where i = 1, 2, 3, 4. Each superalgebra possesses a sCasimir element given by

Si =
1
2
(
[Di,xi] − 1

)
, (3.3)

which anticommutes with the odd generators

{Si,Di} = {Si,xi} = 0, (3.4)

and thus commutes with the even generators

[Si,Ei] = [Si, |xi|2] = [Si,D2i ] = 0. (3.5)

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Hendrik De Bie, Vincent X. Genest, Jean-Michel Lemay, Luc Vinet Acta Polytechnica

It is immediate to verify that in the realization (3.2), the reflection Ri verifies the same commutation relations
as the sCasimir

[Ri,Ei] = [Ri, |xi|2] = [Ri,D2i ] = {Ri,Di} = {Ri,xi} = 0. (3.6)

This implies that one can construct a Casimir operator of the form

Qi = SiRi. (3.7)

It is straightforward to verify that Qi indeed commutes with every generator. These four realizations of osp(1|2)
can act as building blocks for many other realizations. Let [n] = {1, 2, . . . ,n} and A ⊂ [4]. The operators given
by

DA =
∑
i∈A

(
Di

sup A∏
j=i+1

Rj

)
, D2A = DADA, xA =

∑
i∈A

(
si

sup A∏
j=i+1

Rj

)
, |xA|2 =

∑
i∈A

s2i , EA =
∑
i∈A

Ei, (3.8)

verify the commutation relations (3.1) for any A ⊂ [4] and thus form new realizations of osp(1|2). These result
from the repeated application of the coproduct of osp(1|2) (see [20]). Moreover, for any A the sCasimir and the
Casimir operators are also similarly defined:

SA =
1
2
(
[DA,xA] − 1

)
, QA = SA

∏
i∈A

Ri. (3.9)

One can directly check that

Qi = µi, Qjk = Ljk, QB = MB, (3.10)

where Qjk denotes QA with A = {j,k} and B is any 3-subset of [4]. Another explicit computation gives

S2[4] −S[4] −
3
4

=
∑

1≤i<j≤4
J2ij + (s

2
1 + s

2
2 + s

2
3 + s

2
4)

4∑
i=1

µi
s2i

(µi −Ri). (3.11)

However, since |x[4]|2 = s21 + s22 + s23 + s24 commutes with S[4] and all the Casimirs, it is central in the algebra
generated by the Casimirs and can thus be treated as a constant. Taking |x[4]|2 = 1, it is straightforward by
comparing (3.11) and (2.1) that

S2[4] −S[4] −
3
4

= H. (3.12)

Hence, a quadratic combination of the sCasimir of four copies of osp(1|2) yields the Hamiltonian of the
superintegrable model presented in Section 1 and the intermediate Casimirs are its symmetries. Indeed, it can
be checked that [QA,H] = 0 for A ⊂ [4]. The symmetry algebra has the following structure relations

{QA,QB} = Q(A∪B)\(A∩B) + 2QA∩BQA∪B + 2QA\(A∩B)QB\(A∩B), (3.13)

where A,B ⊂ [4] and Q∅ = −1/2 as prescribed by the definitions (3.8) and (3.9). This algebra has already been
studied in [8] and is interpreted as a rank 2 Bannai-Ito algebra. To see this, we remark that the Casimirs with
A ⊂ [3] generate the (rank 1) Bannai-Ito algebra. Let K1 = Q12,K2 = Q23 and K3 = Q13. The recurrence
relations (3.13) can then be rewritten as

{K1,K2} = K3 + ω3, {K2,K3} = K1 + ω1, {K3,K1} = K2 + ω2, (3.14)

where ω1,ω2,ω3 are central elements given by

ω1 = 2Q3Q123 + 2Q1Q2, ω2 = 2Q1Q123 + 2Q2Q3, ω3 = 2Q2Q123 + 2Q1Q3. (3.15)

This corresponds to the Bannai-Ito algebra introduced in [17] which appears in a corresponding superintegrable
model with reflections on S2 as its symmetry algebra [6].

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4. Wavefunctions
To obtain the solutions to the equation Hψ = λψ let us first introduce the gauge transformation

z → z̃ ≡ G(~s)−1zG(~s), G(~s) =
4∏
i=1
|si|µi, (4.1)

where z is any operator and ~s ≡ (s1,s2,s3,s4). Under this transformation, the generators of osp(1|2) in the
realization (3.2) become

D̃i = ∂si +
µi
si

(1 −Ri), D̃2i = D̃iD̃i, x̃i = xi = si, |x̃i|
2 = s2i ,

Ẽi = si∂si + γi, R̃i = Ri, S̃i = −µiRi, Q̃i = µi, (4.2)

where

γA =
∑
i∈A

(
µi +

1
2

)
. (4.3)

These operators also verify (3.1) and correspond to the realization of osp(1|2) (or equivalently sl−1(2)) arising
in the one-dimensional parabose oscillator [2]. Furthermore, the construction (3.8) can be reproduced with
this transformed realization to obtain operators of the form D̃A, x̃A, ẼA, S̃A and Q̃A and is trivially seen to be
equivalent to the gauge transformation of the corresponding operators. Hence, we can obtain eigenvalues and
eigenfunctions of H by finding eigenfunctions of S̃[4]. Note that since S̃[4] commutes with P = R1R2R3R4, this
is equivalent to finding eigenfunctions of Q̃[4].

We thus aim to obtain polynomial eigenfunctions of S̃[4]. To do so, let us first introduce Pm(Rn), the space
of homogeneous polynomials of degree m in the variables s1,s2, . . . ,sn. We define Km(Rn) the kernel space of
degree m as

Km(Rn) = ker D̃[n] ∩Pm(Rn). (4.4)

When n = 4, this is an eigenspace of S̃[4]. Indeed, take ψm ∈Km(R4) and compute

S̃[4]ψ̃m =
1
2

(D̃[4]x̃[4] − x̃[4]D̃[4] − 1)ψ̃m =
1
2

(D̃[4]x̃[4] − 1)ψ̃m =
1
2

(D̃[4]x̃[4] + x̃[4]D̃[4] − 1)ψ̃m

=
1
2
(
{x̃[4],D̃[4]}− 1

)
ψ̃m =

1
2

(2Ẽ[4] − 1)ψ̃m =
( 4∑
i=1

si∂si + γ[4] −
1
2

)
ψ̃m, (4.5)

where we used the property D̃[4]ψ̃m = 0 and the commutation relations (3.1). Since ψ̃m is a homogeneous
polynomial of degree m, it is an eigenfunction of the Euler operator :

∑4
i=1 si∂siψ̃m = mψ̃m. This implies

S̃[4]ψ̃m =
(
m + γ[4] −

1
2

)
ψ̃m (4.6)

and shows that Km(R4) is an eigenspace of S̃[4]. We use two results in order to construct explicitly the
eigenfunctions. First, the space of homogeneous polynomials Pm(Rn) admits a decomposition in terms of the
kernel spaces. This is called the Fischer decomposition and can be cast as

Pm(Rn) =
m⊕
j=0

x̃
j
[n]Km−j(R

n). (4.7)

Second, we use the Cauchy-Kovalevskaia isomorphism (CK-map) between the space of m-homogeneous polyno-
mials in n− 1 variables and the kernel space of degree m in n variables :

CKµnsn : Pm(R
n−1) →Km(Rn). (4.8)

One can compute the CK-map explicitly. To compute CKµ4s4 , take p(s1,s2,s3) ∈Pm(R
3) and let

CKµ4s4
[
p(s1,s2,s3)

]
=

m∑
α=0

sα4 pα(s1,s2,s3), (4.9)

169



Hendrik De Bie, Vincent X. Genest, Jean-Michel Lemay, Luc Vinet Acta Polytechnica

where pα(s1,s2,s3) ∈Pm−α(R3) and p0(s1,s2,s3) ≡ p(s1,s2,s3). Demand that

D̃[4]

m∑
α=0

sα4 pα(s1,s2,s3) = 0 (4.10)

to fix and compute the coefficients pα(s1,s2,s3). A straightforward calculation yields

CKµ4s4 =
∞∑
i=0

(−1)i(s4)2i

i!(γ4)i(2)2i
D̃2i[4] +

∞∑
i=0

(−1)i+1(s4)2i+1

i!(γ4)i+1(2)2i+1
D̃2i+1[4] , (4.11)

where (a)i = a(a + 1) . . . (a + n− 1) denotes the Pochhammer symbol. Similarly, one obtains

CKµnsn =
∞∑
i=0

(−1)i(sn)2i

i!(γn)i(2)2i
D̃2i[n] +

∞∑
i=0

(−1)i+1(sn)2i+1

i!(γn)i+1(2)2i+1
D̃2i+1[n] , (4.12)

for n = 2, 3, 4. Now, iterating the Fischer decomposition (4.7) and the CK-map (4.8), the eigenspace Km(R4)
can be expressed as

Km(R4) ∼= CKµ4s4

[ m⊕
j2=0

x̃
m−j2
[3] CK

µ3
s3

[ j2⊕
j1=0

x̃
j2−j1
[2] CK

µ2
s2

[
Pj1 (R)

]]]
. (4.13)

This means that we can explicitly construct a basis of eigenfunctions {ψ̃(m)j1,j2,j3 (~s)}j1+j2+j3=m of Km(R
4) with

ψ̃
(m)
j1,j2,j3

(~s) = CKµ4s4
[
x̃
j3
[3]CK

µ3
s3

[
x̃
j2
[2]CK

µ2
s2

[sj11 ]
]]
. (4.14)

This calculation can be carried straightforwardly with the help of the identities with ψ̃m ∈Km(Rn)

D̃2α[n]x̃
2β
[n]ψ̃m = 2

2α(−β)α(1 −m−β −γ[n])αx
2β−2α
[n] ψ̃m,

D̃2α+1[n] x̃
2β
[n]ψ̃m = 2

2αβ(1 −β)α(1 −m−β −γ[n])αx
2β−2α−1
[n] ψ̃m,

D̃2α[n]x̃
2β+1
[n] ψ̃m = 2

2α(−β)α(−m−β −γ[n])αx
2β+1−2α
[n] ψ̃m,

D̃2α+1[n] x̃
2β+1
[n] ψ̃m = 2

2α+1(−β)α(m + β + γ[n])(1 −m−β −γ[n])αx
2β−2α
[n] ψ̃,

(4.15)

which follows from (3.1). The result can be presented in terms of the Jacobi polynomials P (α,β)n (x), defined
as [1]

P (α,β)n (x) =
(α + 1)n

n! 2
F1

(
−n, n + α + β + 1

α + 1

∣∣∣1 −x
2

)
, (4.16)

with the help of the identity:

(x + y)nP (α,β)n
(x−y
x + y

)
=

(α + 1)n
n!

xn2F1

(
−n, −n−β

α + 1

∣∣∣− y
x

)
. (4.17)

One obtains

ψ̃
(m)
j1,j2,j3

(~s) = Pj1,j2,j3 (~s)Qj1,j2 (s1,s2,s3)Rj1 (s1,s2), (4.18)

where

Pj1,j2,j3 (~s) =
c!

(γ4)c
(s21 + s

2
2 + s

2
3 + s

2
4)
c

×


P

(γ4−1,j1+j2+γ[3]−1)
c

(
s21+s

2
2+s

2
3−s

2
4

s21+s
2
2+s

2
3+s

2
4

)
− s4x̃[3]

s21+s
2
2+s

2
3+s

2
4
P

(γ4,j1+j2+γ[3])
c−1

(
s21+s

2
2+s

2
3−s

2
4

s21+s
2
2+s

2
3+s

2
4

)
if j3 = 2c,

x̃[3]P
(γ4−1,j1+j2+γ[3])
c

(
s21+s

2
2+s

2
3−s

2
4

s21+s
2
2+s

2
3+s

2
4

)
−s3

j1+j2+c+γ[3]
c+γ4

P
(γ4,j1+j2+γ[3]−1)
c

(
s21+s

2
2+s

2
3−s

2
4

s21+s
2
2+s

2
3+s

2
4

)
if j3 = 2c + 1,

Qj1,j2 (s1,s2,s3) =
b!

(γ3)b
(s21 + s

2
2 + s

2
3)
b

×


P

(γ3−1,j1+γ[2]−1)
b

(
s21+s

2
2−s

2
3

s21+s
2
2+s

2
3

)
− s3x̃[2]

s21+s
2
2+s

2
3
P

(γ3,j1+γ[2])
b−1

(
s21+s

2
2−s

2
3

s21+s
2
2+s

2
3

)
if j2 = 2b,

x̃[2]P
(γ3−1,j1+γ[2])
b

(
s21+s

2
2−s

2
3

s21+s
2
2+s

2
3

)
−s2

j1+b+γ[2]
b+γ3

P
(γ3,j1+γ[2]−1)
b

(
s21+s

2
2−s

2
3

s21+s
2
2+s

2
3

)
if j2 = 2b + 1,

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vol. 56 no. 3/2016 A Superintegrable Model With Reflections on S3

Rj1 (s1,s2) =
a!

(γ2)a
(s21 + s

2
2)
a ×



P

(γ2−1,γ1−1)
a

(
s21−s

2
2

s21+s
2
2

)
− s1s2

s21+s
2
2
P

(γ2,γ1)
a−1

(
s21−s

2
2

s21+s
2
2

)
if j1 = 2a,

s1P
(γ2−1,γ1)
a

(
s21−s

2
2

s21+s
2
2

)
−s2 a+γ1a+γ2 P

(γ2,γ1−1)
a

(
s21−s

2
2

s21+s
2
2

)
if j1 = 2a + 1.

Note that the expressions for Pj1,j2,j3 (s1,s2,s3,s4) and Qj1,j2 (s1,s2,s3) contain the operators x̃[3] and x̃[2]
respectively. Recalling the expressions (4.2) and (3.8), it can be seen that these operators only contain variables
si and reflection operators Ri. These reflections conveniently account for signs occuring in the solutions without
having to give a different expression for every parity combination of the parameters j1,j2 and j3.

By effecting the reverse gauge transformation, we thus obtain a basis for the eigenspace of the operator S[4]
given by

ψ
(m)
j1,j2,j3

(~s) = ψ̃(m)j1,j2,j3 (~s)G(~s), (4.19)

where m = 0, 1, . . . and j1 + j2 + j3 = m. With the help of (4.6), they obey the relation

S[4]ψ
(m)
j1,j2,j3

(~s) =
(
m + γ[4] −

1
2

)
ψ

(m)
j1,j2,j3

(~s). (4.20)

Recalling (3.12), this also implies

Hψ
(m)
j1,j2,j3

(~s) = (m + γ[4])(m + γ[4] − 2)ψ
(m)
j1,j2,j3

(~s). (4.21)

Finally, we can normalize these eigenfunctions as

Ψ(m)j1,j2,j3 (~s) =
η1η2η3√

2
ψ

(m)
j1,j2,j3

(~s), (4.22)

where

η1 = γ2

√
Γ(a + γ[2])

a!Γ(a + γ1)Γ(a + γ2)
×

{
1 if j1 = 2a,√

a+γ2
a+γ1

if j1 = 2a + 1,
(4.23)

η2 = γ3

√
Γ(b + j1 + γ[3])

b!Γ(b + γ3)Γ(b + j1 + γ[2])
×

{
1 if j2 = 2b,√

b+γ3
b+j1+γ[2]

if j2 = 2b + 1,
(4.24)

η3 = γ4

√
Γ(c + j1 + j2 + γ[4])

c!Γ(c + γ4)Γ(c + j1 + j2 + γ[3])
×

{
1 if j3 = 2c,√

c+γ4
c+j1+j2+γ[3]

if j3 = 2c + 1,
(4.25)

so that ∫
S3

Ψ(m)†j1,j2,j3 (~s)Ψ
(n)
k1,k2,k3

(~s) d~s = δn,mδj1,k1δj2,k2. (4.26)

This can be verified directly from the orthogonality relation of the Jacobi polynomials [1].

5. Conclusion
To sum up, we have introduced a new quantum superintegrable model with reflections on the three-sphere. Its
symmetries were given explicitly and were shown to realize a rank-two Bannai-Ito algebra. It was observed that
the model can be constructed through the combination of four independent realizations of the superalgebra
osp(1|2). A quadratic expression in the total sCasimir operator was found to coincide with the Hamiltonian
while the intermediate Casimir operators were seen to coincide with its symmetries. The exact solutions
have been obtained by using a Cauchy-Kovalevskaia extension theorem. We did not find many occurences of
this remarkably simple technique in the superintegrability literature and we trust it could find many other
applications. Furthermore, an interesting feature of this model is the appearance in a scalar model of the rank 2
Bannai-Ito algebra which arose as a particular case in the analysis of the Dirac-Dunkl equation [8]. One expects
that the bivariate Bannai-Ito polynomials will arise as overlaps between wavefunctions of this model separated
in different hyperspherical coordinate systems. These polynomials have never been identified so far and we aim
to study this question in the near future.

Acknowledgements
The research of HDB is supported by the Fund for Scientific Research-Flanders (FWO-V), project “Construction of
algebra realisations using Dirac-operators”, grant G.0116.13N. VXG holds a postdoctoral fellowship from the Natural
Science and Engineering Research Council of Canada (NSERC). JML holds a scholarship from the Fonds de recherche du
Québec – Nature et technologies (FRQNT). The research of LV is supported in part by NSERC.

171



Hendrik De Bie, Vincent X. Genest, Jean-Michel Lemay, Luc Vinet Acta Polytechnica

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172


	Acta Polytechnica 56(3):166–172, 2016
	1 Introduction
	2 A superintegrable model on S3
	3 Algebraic construction from osp(1|2)
	4 Wavefunctions
	5 Conclusion
	Acknowledgements
	References