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% Dynamical algebra of the $q$-deformed three-dimensional oscillator
% J. Van der Jeugt
% J. Math. Phys. 34 (1993), 1799-1806
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\begin{center}
{\LARGE Dynamical algebra of the $q$-deformed three-dimensional
oscillator}\\[2cm]
J. Van der Jeugt$^{a)}$ \\[.5cm]
{\em Vakgroep Toegepaste Wiskunde en Informatica,\\
Universiteit Gent, Krijgslaan 281--S9, B-9000 Gent, Belgium}\\
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\vspace{3cm}
\begin{abstract}
The $q$-deformed three-dimensional harmonic oscillator is defined
in terms of the $q$-bosons corresponding to the spherical components of
a nondeformed three-dimensional oscillator. It is shown that the
dynamical algebra is ${\rm sp}_q(6,R)$.
Two important subalgebra chains are identified~:
${\rm sp}_q(6,R)\supset {\rm su}_q(3) \supset {\rm so}_q(3)$ and
${\rm sp}_q(6,R)\supset {\rm sp}_{q^2}(2,R) \oplus {\rm so}_q(3)$.
The basis states of the $q$-deformed oscillator are classified
according to these subalgebras. Finally, the Hamiltonian eigenvalues
are discussed.
\end{abstract}
\vspace{1cm}
\noindent
PACS numbers~: 02.20, 03.65F
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\noindent-----------------------------------\\
$^{a)}$ {\footnotesize Research Associate of N.F.W.O.
(National Fund for Scientific Research of Belgium)}
\newpage
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\section{Introduction}

Since the introduction of quantum groups and quantum enveloping
algebras$^{1-3}$, these new mathematical structures are finding
applications in various branches in physics$^4$.
For the simplest quantum enveloping algebra ${\rm su}_q(2)$ it
was shown that its generators could be realized in terms
of so-called $q$-deformed boson creation and annihilation
operators$^{5,6}$. This realization was not only useful to
describe the quantum enveloping algebra, it also provided an
easy tool to study the representations of ${\rm su}_q(2)$.
The $q$-analogue of the one-dimensional harmonic oscillator is
thus described by the $q$-boson algebra, with operators
$a$, $a^\d$ (the annihilation and creation operators) and $N$
(the number operator) satisfying
\beq
a a^\d - q^{\pm 1} a^\d a = q^{\mp N}, \qquad
[N,a^\d]=a^\d, \qquad [N,a]=-a .
\eeq

The one-dimensional harmonic oscillator is of pedagogic interest
as it is the standard example in quantum mechanics. Whether
$q$-deformations of harmonic oscillator algebras have applications
in physics remains to be studied, although some investigations
have already confirmed their usefulness in simple models$^{7-10}$.
The three-dimensional harmonic oscillator is not only of pedagogic
interest, it also plays an important role in the phenomenology of
the nuclear shell model$^{11}$ and the diatomic molecule$^{12}$.
In the present paper, we shall study the $q$-deformation of the
three-dimensional harmonic oscillator.

There is some ambiguity in how to define the $q$-bosons for
the $q$-deformed three-dimensional oscillator~: one can deform
the Cartesian boson creation and annihilation operators, or one
can deform the spherical tensor components. Since the quantum
deformation of ${\rm su}(2)$ is defined in terms of the spherical
tensor basis rather than in terms of the Cartesian basis, we
have chosen to define the $q$-deformed three-dimensional oscillator
in terms of $q$-deformed spherical tensor bosons.

For the nondeformed case, it is well known that the full dynamical
algebra of the three-dimensional oscillator is the noncompact
${\rm sp}(6,R)$ algebra.
All basis states of the three-dimensional oscillator are
classified in two irreducible (but infinite-dimensional) discrete
series representations of ${\rm sp}(6,R)$ $^{13,14}$.
In order to describe these basis states, it is helpful to make use
of the subalgebra chains
${\rm sp}(6,R) \supset {\rm su}(3) \supset {\rm so}(3)$, where
${\rm so}(3)$ labels the total angular momentum, or
${\rm sp}(6,R) \supset {\rm sp}(2,R) \oplus {\rm so}(3)$, where
the ${\rm sp}(2,R)\simeq {\rm su}(1,1)$ generators are often referred
to as the quasispin operators. For the latter chain,
${\rm sp}(2,R)$ and ${\rm so}(3)$  are complementary algebras$^{13,15-17}$,
a notion which has been extended to other situations and which is
useful in many explicit calculations.

In the present paper, we first show that the basis states of the
$q$-deformed three-dimensional oscillator can be classified in
two infinite-dimensional representations of the quantum enveloping
algebra ${\rm sp}_q(6,R)$, thus
showing that ${\rm sp}_q(6,R)$ is the dynamical algebra. Next,
the basis states are described in the chain of quantum enveloping
algebras ${\rm sp}_q(6,R) \supset {\rm su}_q(3) \supset {\rm so}_q(3)$.
For this purpose, we make use of a recent identification of an
${\rm so}_q(3)$ subalgebra in the quantum enveloping algebra
${\rm su}_q(3)$. Next, a triple of operators is shown to commute
with the ${\rm so}_q(3)$ generators; these operators satisfy
the defining relation of ${\rm sp}_{q^2}(2,R) \simeq {\rm su}_{q^2}(1,1)$.
Thus, a second subalgebra chain,
${\rm sp}(6,R)_q \supset {\rm sp}_{q^2}(2,R) \oplus {\rm so}_q(3)$ is
identified. It is remarkable that the deformation label for ${\rm sp}(2,R)$
is $q^2$ rather than $q$.
The orthonormal basis states in both chains are easily related to
each other.
In a final section, we discuss the eigenvalues of the
Hamiltonian. Here, the Hamiltonian is defined similarly as for the
$q$-deformed one-dimensional harmonic oscillator$^5$. It should
be noted that this form is taken by analogy to the nondeformed case,
and it remains a matter of speculation whether this operator
has a similar physical meaning as its nondeformed counterpart.

The quantum enveloping algebras appearing in this paper are in fact
the $q$-deformations of the enveloping algebras of the classical
Lie algebras. These $q$-deformations can be endowed with a coproduct,
a counit and an antipode, leading to a Hopf algebra structure.
For the chains considered here, we are dealing with subalgebras
as $q$-deformed enveloping algebras; it will be noted that such
substructures are in general not subalgebras when considered as
Hopf algebras. This seems to be the situation when one is dealing
with the $q$-analogue of nonregular subalgebras of Lie
algebras$^{18-20}$.
%
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\section{Definitions}

The $q$-deformed three-dimensional oscillator is defined in terms of
three independent $q$-boson triples $(b_i, b_i^\d, N_i)$ where $i$
takes three values $i=+,0,-$, and
\beq
b_i b_i^\d - q^{\pm 1} b_i^\d b_i = q^{\mp N_i}, \qquad
[N_i,b_i^\d]=b_i^\d, \qquad [N_i,b_i]=-b_i .
\label{21}
\eeq
The remaining commutation relations are all zero. In~(\ref{21}), $q$ is
a generic real or complex number. For such a system,
the vacuum state $|0\rangle$ is defined by $b_i|0\rangle=0$ for
$i=+,0,-$. All (orthonormal) basis states are of the form
\beq
|n_+ n_0 n_- \rangle =
{ (b_+^\d)^{n_+} (b_0^\d)^{n_0} (b_-^\d)^{n_-}
\over \sqrt{[n_+]![n_0]![n_-]!}  } |0\rangle ,
\label{22}
\eeq
where $[x]\equiv[x]_q$ is the $q$-number $(q^x-q^{-x})/(q-q^{-1})$ in
the basis $q$ and $[x]!=[x][x-1]\cdots[1]$.
In this paper we use the convention that
the basis will not  be written explicitly as an index for the $q$-number
when the basis is equal to $q$; if the basis is different from $q$, e.g.\
$q^2$, then it will be retained.
The following actions are valid~:
\beq
 \begin{array}{l}
 N_i|n_+ n_0 n_- \rangle =n_i|n_+ n_0 n_- \rangle\\[3mm]
 b_+|n_+ n_0 n_- \rangle =\sqrt{[n_+]}|n_+-1 n_0 n_- \rangle\\[3mm]
 b_+^\d|n_+ n_0 n_- \rangle =\sqrt{[n_++1]}|n_++1 n_0 n_- \rangle
 \end{array}
\label{23}
\eeq
and similarly for $b_0^{(\d)}$ and $b_-^{(\d)}$. We also introduce
the operator $N=N_+ + N_0 + N_-$ with eigenvalue $n=n_+ + n_0 + n_-$.

To show that the basis states (\ref{22}) can be classified in representations
of ${\rm sp}_q(6,R)$, it is sufficient to note that the quantum enveloping
algebra ${\rm sp}_q(6,R)$ can be realized by means of the present $q$-boson
operators. Since a quantum enveloping algebra is generated by the Chevalley
generators, it is enough to give a realization of these. In the
present situation, we can put~:
\beq
 \begin{array}{lll}
 e_1=b_+^\d b_0, & f_1=b_0^\d b_+, & h_1=N_+-N_0, \\[3mm]
 e_2=b_0^\d b_-, & f_2=b_-^\d b_0, & h_2=N_0-N_-, \\[3mm]
 e_3=(b_-^\d)^2/[2], & f_3= -(b_-)^2/[2], & h_3=N_-+{1\over 2} .
 \end{array}
\label{24}
\eeq
One can verify that $[h_i,e_j]=a_{ji}e_j$ ($i,j=1,2,3$), where $A=(a_{ij})$
is the Cartan matrix for $C_3={\rm sp}(6)$. We also have,
\beq
 [e_i,f_j]=\delta_{ij} [h_i]_{q^{d_i}},
\eeq
where $D={\rm diag}(d_1,d_2,d_3) = {\rm diag}(1,1,2)$ symmetrizes $A$,
i.e.~$a_{ij}d_j=a_{ji}d_i$. The remaining relations to be satisfied
are the Serre relations$^{2,3}$, but these are easy to check here since
we are dealing with $q$-boson operator realizations. Thus, (\ref{24})
are the generators for ${\rm sp}_q(6)$. Since the last generating
triple $(e_3,f_3,h_3)$ satisfies the noncompact conjugation relation,
it is appropriate to speak of ${\rm sp}_q(6,R)$.

From the generators (\ref{24}) and the explicit basis states (\ref{22})
it follows that all basis states belong to one of two (irreducible)
infinite-dimensional representations of ${\rm sp}_q(6,R)$. The first
representation consists of those states (\ref{22}) with $n$ even; the
second representation consists of those states (\ref{22}) with $n$ odd.

So far, there is little difference between the $q$-deformed and the
nondeformed three-dimensional oscillator. The main difference will
come from the identification of the subalgebras in the $q$-deformed
case.

\section{The chain ${\rm sp}_q(6,R) \supset {\rm su}_q(3) \supset
 {\rm so}_q(3)$}

It is immediately evident that the subalgebra generated by
$e_1$, $e_2$, $f_1$, $f_2$, $h_1$ and $h_2$ is ${\rm su}_q(3)$,
as part of the Cartan matrix for ${\rm sp}(6)$ yields the Cartan
matrix for ${\rm su}(3)$ and the relations remain valid.
In fact, it has been shown$^{18}$ that the basis states (\ref{22})
with $n_++n_0+n_-=n$ are the basis vectors for the totally symmetric
representation $\{n,0,0\}$ of ${\rm su}_q(3)$. This result describes
in fact the decomposition of the ${\rm sp}_q(6,R)$ representation
to irreducible ${\rm su}_q(3)$ representations.

The difficult part is to identify a subalgebra of type ${\rm so}_q(3)
\simeq {\rm su}_q(2)$ in ${\rm su}_q(3)$. This problem has been
addressed in ref.~18, and further discussed in refs.~19 and 20. The
solution given there is of the following form$^{18}$~:
\bea
L_0 &=& N_+ - N_- , \nn \\
L_{+1}&=&q^{N_- - {1\over 2}N_0} \sqrt{[2N_+]\over[N_+]} b^\d_+ b_0  +
  b^\d_0b_- q^{N_+ - {1\over 2}N_0} \sqrt{[2N_-]\over[N_-]} , \label{31}\\
L_{-1}&=&b^\d_0b_+ q^{N_- - {1\over 2}N_0} \sqrt{[2N_+]\over [N_+]}  +
   q^{N_+ - {1\over 2}N_0} \sqrt{[2N_-]\over[N_-]} b^\d_-b_0 . \nn
\eea
These expressions are no longer invariant under the transformation
$q\rightarrow q^{-1}$. Therefore, a second solution exists by
replacing $q$ by $q^{-1}$ in (\ref{31}); this has been discussed
in ref.~19.
The operators (\ref{31}) satisfy
\beq
[L_0,L_{\pm 1}]=\pm L_{\pm 1}, \qquad [L_{+1},L_{-1}]=[2L_0].
\label{32}
\eeq
Moreover, in the limit $q\rightarrow 1$, the operators (\ref{31}) coincide
with the classical ${\rm so}(3)$ generators of ${\rm su}(3)$. Therefore,
(\ref{31}) can be identified as the ${\rm so}_q(3)$
subalgebra of ${\rm su}_q(3)$. The Casimir operator for the algebra (\ref{32})
is well known and is given by~:
\beq
C_L= L_{+1}L_{-1} + [L_0][L_0-1] = L_{-1}L_{+1} + [L_0][L_0+1] .
\label{33}
\eeq

The basis states (\ref{22}) can be transformed to orthonormal
${\rm so}_q(3)$ basis vectors. This reduction, and the corresponding
transformation coefficients, has been determined in ref.~18. For fixed
$n$, the basis states (\ref{22}) are transformed to states $v(n,l,m)$,
where $l=n,n-2,\ldots,1$ or 0, and $m=-l,-l+1,\ldots,l$, with $l$ and
$m$ the ${\rm so}_q(3)$ labels.
The result can be written in the following form$^{18}$~:
\bea
\lefteqn{v(n,l,m)= q^{-\{l(l+1)+(m-1)(m+2n)\}/4}
\left( [n+l]!![n-l]!![l+m]![l-m]![2l+1]/[n+l+1]!\right)^{1/2} } \nn\\
&\times& \sum_{x=\max(0,m)}^{\lfloor(l+m)/2\rfloor} \sum_{y=0}^{(n-l)/2}
(-1)^y q^{x(n-{1\over 2})-y(l+{3\over 2})}
\left( [2x]!![2y]!![2x-2m]!![l+m-2x]![n-l-2y]!! \right)^{-1} \nn\\
&\times& \left([m+n-2x-2y]![2x+2y]!![2x+2y-2m]!!\right)^{1/2}
|x+y,n+m-2x-2y,x+y-m\rangle . \nn \\
\label{34}
\eea
Herein, $[2t]!!$ stands for $[2t][2t-2]\cdots[2]$.
These vectors are genuine
orthonormal $q$-generalised angular momentum states~:
\beq
\begin{array}{l}
L_0 v(n,l,m) = m \;v(n,l,m) , \\[3mm]
L_{\pm 1} v(n,l,m) = \sqrt{[l\mp m][l\pm m+1]} \;v(n,l,m\pm 1).
\end{array}
\label{35}
\eeq
In the following section, we shall deduce a simpler expression for
the states (\ref{34}), making use of a second quantum subalgebra chain.

\section{The chain ${\rm sp}_q(6,R) \supset {\rm sp}_{q^2}(2,R)
\oplus {\rm so}_q(3)$}

In ref.~18, an operator $s$ was introduced which was shown to commute
with the operators $L_i$ ($i=+1,0,-1$). Here, we shall relabel this
operator by $S_{+1}$~:
\beq
S_{+1}={1\over[2]}\left( (b_0^\d)^2 q^{N_++N_-+1}
 - \sqrt{ {[2N_+]\over[N_+]}{[2N_-]\over[N_-]} } b_+^\d b_-^\d
 q^{-N_0-{1\over 2}} \right).
\label{41}
\eeq
It can be verified by direct action on the states (\ref{22}) that
$[L_i,S_{+1}]=0$ for $i=+1,0,-1$. Similarly, one can now define
the conjugate of (\ref{41})~:
\beq
S_{-1}={1\over[2]}\left( (b_0)^2 q^{N_++N_-+1}
 - b_+ b_-\sqrt{ {[2N_+]\over[N_+]}{[2N_-]\over[N_-]} }
 q^{-N_0-{1\over 2}} \right).
\label{42}
\eeq
Again, this operator satisfies $[L_i,S_{-1}]=0$ for $i=+1,0,-1$, and
is thus another ${\rm so}_q(3)$ scalar. It is now interesting to
calculate the commutator of $S_{+1}$ and $S_{-1}$. By direct action
on the states (\ref{22}), one finds~:
\beq
[S_{+1},S_{-1}] = [-2N_+-2N_0-2N_--3]/[2].
\label{43}
\eeq
Introducing the operator
\beq
S_0={1\over 2}(N_++N_0+N_-+{3\over 2})={1\over 2}(N+{3\over 2}),
\label{44}
\eeq
it follows that the operators $S_i$ satisfy the following relations~:
\bea
&&[S_0,S_{\pm 1}]= \pm S_{\pm 1} , \label{45}\\
&&[S_{+1},S_{-1}] = -[2S_0]_{q^2} , \label{46}
\eea
where $[x]_{q^2}=(q^{2x}-q^{-2x})/(q^2-q^{-2})$.
These are the defining relations$^{21}$ for the quantum enveloping algebra
${\rm su}_{q^2}(1,1)$; for obvious reasons, this algebra will be
denoted as here ${\rm sp}_{q^2}(2,R)$. It is rather surprising that the
deformation number for this algebra is $q^2$ rather than $q$.
Since $[L_i,S_j]=0$ for all $i,j=+1,0,-1$, we are dealing with two
commuting quantum subalgebras, and hence we have identified the
quantum subalgebra ${\rm sp}_{q^2}(2,R)\oplus {\rm so}_q(3)$. The Casimir
operator for ${\rm sp}_{q^2}(2,R)$ is again well known and reads here~:
\beq
C_S=S_{+1}S_{-1}-[S_0]_{q^2}[S_0-1]_{q^2}
 =S_{-1}S_{+1}-[S_0]_{q^2}[S_0+1]_{q^2} .
\label{47}
\eeq

We shall now construct the basis vectors in this second chain of
quantum subalgebras. Let
\beq
u(0,l,l)=|l,0,0\rangle = {1\over\sqrt{[l]!}} (b_+^\d)^l |0\rangle,
\qquad (l=0,1,2,\ldots).
\label{48}
\eeq
It is easy to verify that $L_0 u(0,l,l)=l \;u(0,l,l)$ and
$C_Lu(0,l,l)=[l][l+1] u(0,l,l)$.  From the expression for
$S_0$ one deduces that
\beq
S_0 u(0,l,l)= {1\over 2}(l+{3\over 2}) u(0,l,l);
\label{49}
\eeq
therefore, we shall denote
\beq
s={1\over 2}(l+{3\over 2}).
\label{410}
\eeq
Since $S_{-1} u(0,l,l)=0$, it follows from (\ref{47}) that
$C_S u(0,l,l)=-[s]_{q^2}[s-1]_{q^2} u(0,l,l)$. Next, we define~:
\beq
u(0,l,m) = \sqrt{[l+m]!\over [2l]![l-m]!} (L_{-1})^{l-m} u(0,l,l),
\label{411}
\eeq
where $m=l,l-1,\ldots,-l$. These are genuine orthonormal
${\rm so}_q(3)$ states as one can verify that~:
\beq
\begin{array}{l}
L_0 u(0,l,m) = m \;u(0,l,m) , \\[3mm]
L_{\pm 1} u(0,l,m) = \sqrt{[l\mp m][l\pm m+1]} \;u(0,l,m\pm 1).
\end{array}
\label{412}
\eeq
Moreover, since all $S_i$ ($i=+1,0,-1$) commute with $L_j$
($j=+1,0,-1$), it follows that
\beq
\begin{array}{l}
S_0 u(0,l,m)=s \;u(0,l,m) ,\\[3mm]
C_S u(0,l,m)= - [s]_{q^2}[s-1]_{q^2} \;u(0,l,m) ,
\end{array}
\label{413}
\eeq
where $s$ is given by (\ref{410}). Finally, we define~:
\beq
u(r,l,m) = {\cal N}_{rlm} (S_{+1})^r u(0,l,m),
\qquad (r=0,1,2,\ldots),
\label{414}
\eeq
where ${\cal N}_{rlm}$ is some normalization constant to be determined
later. From the commutation relations (\ref{45}) one deduces that
$S_0S_{+1}^r = S_{+1}^r (S_0+r)$, leading to
\beq
S_0 u(r,l,m) = (s+r) u(r,l,m) = {1\over 2} (l+2r+{3\over 2}) u(r,l,m) .
\label{415}
\eeq
From the definition (\ref{414}) and the relation (\ref{46}),
it follows that
\beq
\begin{array}{l}
S_{+1} u(r,l,m) = \alpha(r,l,m) u(r+1,l,m) ,\\[3mm]
S_{-1} u(r,l,m) = \beta (r,l,m) u(r-1,l,m) ,
\end{array}
\label{416}
\eeq
where $\alpha$ and $\beta$ are functions to be determined.
Calculating the action of $S_{+1}S_{-1}$ and $S_{-1}S_{+1}$, and
comparing with the action of $C_S$, one can deduce that
\beq
\alpha(r,l,m)=\sqrt{[r+1]_{q^2}[r+2s]_{q^2}},\qquad
\beta(r,l,m)=\sqrt{[r]_{q^2}[r+2s-1]_{q^2}}.
\label{417}
\eeq
Thus, the following relations hold~:
\beq
\begin{array}{l}
S_{+1} u(r,l,m) = \sqrt{[r+1]_{q^2}[r+2s]_{q^2}} u(r+1,l,m),\\[3mm]
S_{-1} u(r,l,m) = \sqrt{[r]_{q^2}[r+2s-1]_{q^2}} u(r-1,l,m),\\[3mm]
S_0 u(r,l,m) = (r+s) u(r,l,m),
\end{array}
\label{418}
\eeq
where $s$ is given by (\ref{410}). One can now determine the
normalization factor~:
\bea
\langle u(r,l,m)|u(r,l,m)\rangle &=& {\cal N}_{rlm}^2
\langle u(0,l,m)| S_{-1}^r S_{+1}^r u(0,l,m)\rangle \nn\\
&=&{\cal N}_{rlm}^2 [r]_{q^2}! { [r+2s-1]_{q^2}!\over [2s-1]_{q^2}!} \equiv 1.
\label{419}
\eea
Combining (\ref{48}), (\ref{411}) and (\ref{414}), and using
$2s=l+3/2$, one finds~:
\bea
\lefteqn{u(r,l,m) = } \nn\\
&&\left( {[l+{1\over 2}]_{q^2}!\over [r]_{q^2}![l+r+{1\over 2}]_{q^2}!}
{[l+m]!\over[2l]![l-m]!}\right)^{1/2} S_{+1}^rL_{-1}^{l-m}
{(b_+^\d)^l\over\sqrt{[l]!}} |0\rangle .
\label{420}
\eea
Herein, $r=0,1,2,\ldots\;$, $l=0,1,2,\ldots\;$, $m=l,l-1,\ldots,-l$, and
for an integer $t$, $[t+{1\over 2}]_{q^2}!$ stands for
$$
[t+{1\over 2}]_{q^2}[t-{1\over 2}]_{q^2}\cdots[{3\over 2}]_{q^2}
[{1\over 2}]_{q^2} = [2t+1][2t-1]\cdots[3][1]/[2]^t .
$$
The $L_i$ action on these states is the same as in (\ref{412}), and the
$S_i$ action has been given in (\ref{418}). Thus the ${\rm so}_q(3)$
labels are $(l,m)$, and the ${\rm sp}_{q^2}(2,R)$ labels are $(s,r+s)$ with
$s=(l+3/2)/2$. Because of this connection between the ${\rm sp}_{q^2}(2,R)$
label and the ${\rm so}_q(3)$ label, the two quantum subalgebras can
be called complementary subalgebras$^{13-17}$. Moreover, note
that (\ref{44}) yields $2S_0=N+{3\over 2}$, implying that
$N u(r,l,m)= (l+2r) u(r,l,m)$. Therefore, the states (\ref{34}) and
(\ref{420}) are identical, with
\beq
v(n,l,m) = u(r,l,m)\qquad\hbox{for}\qquad n=l+2r.
\label{421}
\eeq
This completes the study of quantum subalgebra chains for the dynamical
algebra of the $q$-deformed oscillator, and the classification of
basis states according to these subalgebras.

\section{The Hamiltonian}

In this final section, we briefly describe some considerations
about the Hamiltonian eigenvalues for the $q$-deformed three-dimensional
oscillator. Again, there is some ambiguity in the definition of
the Hamiltonian. If one defines $H=N_++N_0+N_-+{3\over 2}$,
implying $H=2S_0$, then the eigenvalues are exactly the same as
in the nondeformed case. However, following ref.~5, one
can use a different approach, and define~:
\beq
{\cal H}={1\over 2} (b_+b_+^\d + b_+^\d b_+ + b_0b_0^\d + b_0^\d b_0
+ b_-b_-^\d + b_-^\d b_-).
\label{H}
\eeq
This Hamiltonian is diagonal in the basis (\ref{22}), but is no longer
diagonal in the basis $v(n,l,m)$. In fact, since ${\cal H}$ still
commutes with $N$ and with $L_0$, its action takes the form
$$
{\cal H} v(n,l,m) = \sum_{l'} C_{ll'} v(n,l',m).
$$
Diagonalizing ${\cal H}$ in this basis gives of course the same
eigenvalues as determining the eigenvalues of ${\cal H}$ in the
basis $|n_+,n_0,n_-\rangle$ with fixed $n=n_++n_0+n_-$, i.e.~:
\beq
{1\over 2}([n_+]+[n_++1]+[n_0]+[n_0+1]+[n_-]+[n_-+1]).
\label{eigval}
\eeq
Consider now all the states with $n$ quanta; there are $(n+1)(n+2)/2$
such states. In the nondeformed case, all these states have the
same Hamiltonian eigenvalue $n+3/2$. In the case $q\ne 1$, it follows
from (\ref{eigval}) that there are $p(3,n)$ different eigenvalues
for ${\cal H}$, where $p(3,n)$ is the number of partitions of $n$ into
at most 3 parts. For each such partition $n=p_1+p_2+p_3$
($0\leq p_i\leq n$) the corresponding number of states is 6 if
all $p_i$ are distinct, 3 if two $p_i$ are equal and 1 if all $p_i$
are equal. For $q$ real, all the different eigenvalues are greater than
the nondeformed eigenvalue $n+3/2$. As we have already mentioned in
the introduction, whether these eigenvalues are related to
energy levels of some quantum mechanical object is a matter of
speculation. Nevertheless, the techniques developed in this paper
show how many of the classical concepts can be introduced for
the $q$-deformed three-dimensional oscillator, and we hope that
both these techniques and the present results will prove to be useful
in the study of deformations of realistic models in physics.

\section*{Acknowledgements}
It is a pleasure to thank Dr.\ C.~Quesne (ULB, Brussels) for
stimulating discussions.
%
\newpage
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\end{document}