Symmetry, Integrability and Geometry: Methods and Applications Deformed su(1, 1) Algebra as a Model for Quantum Oscillators

The Lie algebra $\mathfrak{su}(1,1)$ can be deformed by a reflection operator, in such a way that the positive discrete series representations of $\mathfrak{su}(1,1)$ can be extended to representations of this deformed algebra $\mathfrak{su}(1,1)_\gamma$. Just as the positive discrete series representations of $\mathfrak{su}(1,1)$ can be used to model a quantum oscillator with Meixner-Pollaczek polynomials as wave functions, the corresponding representations of $\mathfrak{su}(1,1)_\gamma$ can be utilized to construct models of a quantum oscillator. In this case, the wave functions are expressed in terms of continuous dual Hahn polynomials. We study some properties of these wave functions, and illustrate some features in plots. We also discuss some interesting limits and special cases of the obtained oscillator models.


Introduction
The su(1, 1)-model of a quantum oscillator [15] is a model that obeys the dynamics of a harmonic oscillator, but with the position and momentum operators and the Hamiltonian being elements of the Lie algebra su(1, 1) instead of the Heisenberg algebra.
There are many algebraic constructions to model a quantum oscillator. The difficulty for such models is often to determine the spectra of observables and an explicit form of their eigenfunctions. Only for some models, one can develop such a complete theory. One of these models is the qoscillator, a q-deformation of the standard quantum oscillator [9,14,18,24]. The q-oscillator has many interesting properties, both from the mathematics and physics point of view. But it also has some drawbacks, in particular the Newton-Lie (or Hamilton-Lie) equations are not satisfied.
Following this, new oscillator models were developed such that the same dynamics as in the classical or quantum case is satisfied, and in such a way that the operators corresponding to position, momentum and Hamiltonian are elements of some algebra different from the traditional Heisenberg (or oscillator) Lie algebra. In the one-dimensional case, there are three (essentially self-adjoint) operators: a position operatorq, its corresponding momentum operatorp and a (pseudo-) Hamil-tonianĤ which is the generator of time evolution. These operators should satisfy the Hamilton-Lie equations (or the compatibility of Hamilton's equations with the Heisenberg equations): in units with mass and frequency both equal to 1, and = 1. Contrary to the canonical case, the commutator [q,p] = i is not required. Apart from (1) and the self-adjointness, it is then common to require the following conditions [2]: • all operatorsq,p,Ĥ belong to some (Lie) algebra (or superalgebra) A; • the spectrum ofĤ in (unitary) representations of A is equidistant.
A very interesting model occurs for A = su(2) (or its enveloping algebra) [2][3][4]. In that case, the relevant representations are the well known su (2) representations labeled by an integer of halfinteger j. Since these representations are finite-dimensional, one is dealing with "finite oscillator models", of potential use in optical image processing [4]. Up to a constant, the HamiltonianĤ is the diagonal su(2) operator with a linear spectrum n + 1 2 (n = 0, 1, . . . , 2j). Alsoq andp have a finite spectrum, given by {−j, −j + 1, . . . , +j} [2]. The (discrete) position wave functions have been constructed, and are given by Krawtchouk functions (normalized symmetric Krawtchouk polynomials) [2], tending to the canonical wave functions in terms of Hermite polynomials when j → ∞.
In two previous papers [12,13], the su(2) model for the finite one-dimensional harmonic oscillator was extended. The underlying algebra is a deformation of su (2) with an extra reflection (or parity) operator and an additional parameter α(> −1). In the even-dimensional representations [12] (j half-integer), the spectrum of the position operator is of the form and the position wave functions could be constructed in terms of normalized Hahn (or dual Hahn) polynomials (with parameters (α, α + 1) or (α + 1, α)). In the odd-dimensional representations [13] (j integer), the spectrum of the position operator is 0, ± k(2α + k + 1), (k = 1, . . . , j).
The position (and momentum) wave functions are again Hahn polynomials (in this case with parameters (α, α) or (α + 1, α + 1)). Models of quantum oscillators with continuous spectra of position and momentum operators were constructed based on the positive discrete series representations of su(1, 1) [15]. In such a representation, labeled by a positive number a > 0, the spectrum of the position operator is R. The position wave function, when the oscillator is in the nth eigenstate of the Hamiltonian, is given by where P (λ) n (x; φ) is the Meixner-Pollaczek polynomial [16]. Many interesting properties of these su(1, 1) oscillators were described by Klimyk [15].
One type of deformation of this su(1, 1) model was offered by its q-deformation. The su q (1, 1) model was investigated in [5]. The position and momentum operators have spectra covered by a finite interval of the real line, which depends on the value of q, and the wavefunctions are given in terms of q-Meixner-Pollaczek polynomials.
In the current paper, we consider an extension of the Lie algebra su(1, 1) by a parity or reflection operator R. In this extension or deformation, the common su(1, 1) commutator [J + , J − ] = −2J 0 is replaced by [J + , J − ] = −2J 0 − γR, where γ is a (real) deformation parameter. The positive discrete series representations of su(1, 1), labeled by a positive number a, can be extended to representations of the deformed algebra su(1, 1) γ , provided γ can be written in the form γ = (2a − 1)(2c − 1) for some positive c-value (sometimes c rather than γ will be referred to as the deformation parameter). Section 2 recalls some known formulas for the su(1, 1) case, and in Section 3 the algebra su(1, 1) γ and its representations are given. The core of the paper comes in Section 4, where models for a quantum oscillator are built using su(1, 1) γ representations. Just as for the canonical oscillator, the Hamiltonian has a discrete but infinite equidistant spectrum in these models, and the position operator has spectrum R. We have managed to obtain explicit expressions for the orthonormal wave functions ψ (a,c) n (x), where a is the representation label and c the deformation label. These wave functions involve the class of so-called continuous dual Hahn polynomials [11,16]. Their properties and shapes are studied in Section 4. In Section 5 we consider some limits and special cases. For c = 1/2 (or γ = 0), our functions reduce to those studied by Klimyk [15]. Another interesting case is when c tends to +∞: then the model reduces to that of the paraboson oscillator. So for a = 1/2 and c → +∞ the model coincides with the canonical oscillator. In Section 6, we give a differential operator realization of the deformed algebra su(1, 1) γ , and in Section 7 we discuss the possibility of considering further parameters in the deformation. Our paper closes with some remarks in a concluding section.
In the su(1, 1) oscillator model, the position, momentum and Hamiltonian are chosen as follows: These operators satisfy (1). The spectrum ofĤ is thus (n + a) (n = 0, 1, 2, . . .), an equidistant and infinite spectrum. One can just as well choose the representation-dependent operator J 0 − a + 1/2 forĤ in order to get the spectrum of the standard quantum oscillator, but in fact this shift is not so relevant. A more interesting aspect is the determination of the spectrum of the position operator q and its eigenvectors. Denoting a formal eigenvector ofq, for the eigenvalue x, by the equationq v(x) = x v(x) leads by means of (5) and (6) to 2x A n (x) = (n + 1)(n + 2a)A n+1 (x) + n(n + 2a − 1)A n−1 (x).
3 The deformed algebra su(1, 1) γ and its representations The extension and deformation of the Heisenberg algebra with a reflection operator was performed in [20]. In that case, the representations of the deformed algebra correspond to representation of the Lie superalgebra osp(1|2). Recently, there has been further interest in extending algebras by reflection operators [26], or even more basically in extending differential operators by reflection operators [21,27]. It is in this context that we present an extension and deformation of su(1, 1). The universal enveloping algebra of su(1, 1) can be extended by a parity (or reflection) operator R, with action R|a, n = (−1) n |a, n in the representation H a . This means that R commutes with J 0 , anticommutes with J + and J − , and R 2 = 1. This extended algebra can be deformed by a parameter γ, leading to the definition of su(1, 1) γ .
Definition 1 Let γ be a parameter. The algebra su(1, 1) γ is a unital algebra with basis elements J 0 , J + , J − and R subject to the following relations: • R is a parity operator satisfying R 2 = 1 and • The su(1, 1) relations are deformed as follows: Clearly, for γ = 0 this is just su(1, 1) extended by a parity operator, and the positive discrete series representations H a are the same as (5) with extra action R|a, n = (−1) n |a, n . The interesting property is that also for γ = 0, the representation H a (a > 0) can be deformed in such a way that it becomes a representation for su(1, 1) γ , provided some conditions are satisfied for a. This is described in the following proposition. From now on we shall assume that γ is a given nonzero real number.
Proposition 2 Let a be a positive real number with a = 1/2 and consider the space H a with basis vectors |a, n (n = 0, 1, 2, . . .). Assume that γ can be written as Then the following action turns H a into an irreducible representation space of su(1, 1) γ .
R|a, n = (−1) n |a, n , So, for a given γ = 0 (i.e. for a fixed algebra su(1, 1) γ ), not all positive a-values are allowed as representation parameter. From (13), it follows that: The proof of Proposition 2 is essentially by direct computation: it is a simple task to verify that the actions (14)-(17) do indeed satisfy the relations (10)- (12). The conditions a > 0 and c > 0 ensure that the factors under the square roots are positive. The irreducibility follows from the fact that (J + ) m |a, n is nonzero and proportional to |a, n + m for n, m = 0, 1, 2, . . ., and that (J − ) m |a, n + m is nonzero and proportional to |a, n .
Note also that the representation given in this proposition is unitary under the star conditions Following the choice in [15] (as in Section 2) let us choose the position, momentum and Hamiltonian operators as follows: with c determined by (13). The choice of subtracting a constant c − 1 2 is just for convenience, and in fact this shift in the spectrum is not so relevant. The operators (18) satisfy (1). In the representation space H a ,Ĥ|a, n = (n + a)|a, n , therefore the spectrum ofĤ is linear and given by n + a (n = 0, 1, 2, . . .).
If we denote the formal eigenvector ofq, for the eigenvalue x, again by the equationq v(x) = x v(x) leads due to (20) to The task is now to solve (22)- (23). The solution can again be found in the context of orthogonal polynomials. However, (22)-(23) is not simply the recurrence relation of a known (normalized) set of orthogonal polynomials. Instead, we will have to combine two sets of such orthogonal polynomials. For this purpose, let us recall the continuous dual Hahn polynomials S n (x 2 ; a; b; c) in the variable x 2 , with at least two positive parameters a, b and c [1,11,16]. These polynomials of degree n (n = 0, 1, 2, . . .) in the variable x 2 are defined by [11,16]: in terms of the generalized hypergeometric series 3 F 2 of unit argument [1,6,23]. For a, c > 0 and b ≥ 0, the orthogonality relation of these polynomials reads [10,11,16]: Like all orthogonal polynomials, the continuous dual Hahn polynomials satisfy a 3-term recurrence relation. However, it is not this relation that can be used here. Instead, we shall need to make use of difference relations, given in the following proposition. Note that these difference relations appeared already in [10, eq. (2.13)]; we give a more general proof here.
Proposition 3 Continuous dual Hahn polynomials satisfy the following difference relations: Proof. Let us start with the following identity: To see that this identity holds, compare coefficients of z k in left and right hand side. Putting A = a + ix, B = a − ix, C = a + b, D = a + c, z = 1 and following (24), this can be written as (a 2 + x 2 )S n (x 2 ; a + 1, b, c) = (n + a + b)(n + a + c)S n (x 2 ; a, b, c) − S n+1 (x 2 ; a, b, c).
Since the continuous dual Hahn polynomials are symmetric in (a, b, c), (26) follows by permuting a and b. To prove (27), one can start from the following contiguous relation between 3 F 2 's: To verify that this relation is correct, one simply compares powers of z k in the left and right hand side. Finally, putting A = a + ix, B = a − ix, C = a + b, D = a + c and z = 1 in (28) yields (27). ✷ In particular, it follows from the previous proposition that S n (x 2 ; a, 0, c) = S n (x 2 ; a, 1, c) − n(n + a + c − 1)S n−1 (x 2 ; a, 1, c).
It is now rather trivial to see that (22) follows from (30), and that (23) follows from (29). So we have found a solution for the recurrence relations (22)- (23). Note that the solution is in terms of two sets of orthogonal polynomials: continuous dual Hahn polynomials with parameters (a, 0, c) for the even degree polynomials, and with parameters (a, 1, c) for the odd degree polynomials. The expression for A n (x) is a real polynomial of degree n in x. For the appropriate function Indeed, for m and n even, one writes . The extra appearance of x 2 as a factor is taken care of by |Γ(ix)| 2 x 2 = |ixΓ(ix)| 2 = |Γ(1 + ix)| 2 , and then (34) follows again from (25). Finally, for m even and n odd (or vice versa), (34) follows trivially from the fact that w(x) is an even function and A m (x)A n (x) is an odd function of x. Note that, due to the identity Γ(ix)/Γ(2ix) = 2 1−2ix √ π/Γ( 1 2 + ix), the function w can be rewritten as Similarly as in (8), the support of the orthogonal polynomials appearing in (34) is R, so the spectrum ofq is R. Rewriting w(x)A n (x) by ψ (a,c) n (x) (and rescaling the eigenvectors (21) by w(x)), we have the following: where w(x) is given in (35). The spectrum ofq is R. The functions ψ (a,c) n (x) are orthonormal: This orthonormality also implies the Dirac delta orthonormality of the formal eigenvectors: The last assertion of the theorem is standard, see e.g. [22]. This result also implies that the functions ψ (a,c) n (x) can be interpreted as position wave functions for the su(1, 1) γ oscillator model, when the oscillator is in the stationary state |a, n with energy n + a. This interpretation makes physically sense provided |ψ in terms of the Beta function. So the parameters should satisfy B(a, c) ≤ π. This is certainly satisfied when both a ≥ 1 2 and c ≥ 1 2 . It is interesting to study some plots of the wave functions ψ (a,c) n (x). First of all, note that this function is symmetric in a and c, so one can keep one parameter a fixed and let the other one c vary. In figure 1 we plot the wave functions for n = 0 and in figure 2 for n = 1, both for a = 1 and for a = 2, for some values of c. For c = 1/2 (undeformed algebra), one observes for the ground state wave function a shape similar to the Gaussian function, but with increasing variance as a increases. When c increases, the bell shapes are deformed in the middle, and the position probability decreases around the origin. The deformations for n > 0 follow a similar pattern.
In a similar way, the spectrum and eigenvalues of the momentum operatorp can be determined. Denoting the formal eigenvector ofp, for the eigenvalue p, bȳ the equationpv(p) = pv(p) leads, using (18), to a set of recurrence relations similar to (22)- (23). The solution to these equations is the same as before, up to a multiple of i. So one finds that the formal eigenvectors ofp are given bȳ where the functions ψ (a,c) n are the same as in Theorem 4. The spectrum ofp is R.
For an odd index, one uses expression (38) for ψ (a,1/2) 2n+1 (x). The computation is similar, the main difference coming from the appearance of S n (x 2 ; a, 1, 1/2) in the numerator of (38). Here, one can use The last identity (47) is again obtained by first applying a Thomae-Weber-Erdelyi transformation on the 3 F 2 [25, Appendix]: and then using [13, Eq. (40)]. A second interesting case is the limit c → +∞. Note from the action (16)-(17) that the operators have the same action on H a as the paraboson oscillator creation and annihilation operators [12, Eq. (A7)]. Under this same limit, the position and momentum operators becomê and these operators satisfy, from (12) and (18), which is the common commutator in the paraboson case [19,20,26]. Hence one can also expect the wave functions ψ tends to ∞. In order to consider this limit, note that the position operator has been divided by √ c in (49), so for its eigenvalue we should introduce a new variable by putting x = √ c ξ. Then according to (39), we need to compute the limit Let us consider the even case ψ (a,c) 2n (the odd case is similar). For the polynomial part, (37) and (24) lead to the following limit: where L (a−1) n is a Laguerre polynomial [16]. Taking care of the other factors, one finds So one finds indeed lim in terms of the paraboson wave functions [12,Eq. (A.11)]. Note that for a = 1/2 the paraboson wave functions reduce to the canonical oscillator wave functions, as the corresponding Laguerre polynomials become Hermite polynomials [12,Appendix]. So therefore, one can say that under the limit (a, c) → (1/2, +∞) the new oscillator models introduced in this paper reduce to the canonical quantum oscillator.
6 Differential operator realization of su(1, 1) γ Just as su(1, 1) has a differential operator realization, and a realization of the Hilbert space H a [15], this can be constructed for su(1, 1) γ as well. The basis vectors can be realized as monomials in a variable z: Clearly, the 'abstract' reflection operator R is realized by the concrete reflection operatorȒ with actionȒ z n = (−1) n z n .
Then it is a simple exercise to verify that the following differential operators satisfy (15)-(17) when acting on the monomials (55) In order to characterize the Hilbert space H a as a space of analytic functions, a description of the scalar product should be found in such a way that the monomials (55) are orthonormal. So far, we have not been able to construct this. Let us nevertheless mention that the position (and momentum) wave functions have a more explicit expression in this realization. Indeed, For the first sum, one finds n ψ (a,c) by using [16, (9.3.14)]. In a similar way, the second sum of (58) equals  (12). This is in fact the case; however, we shall see that it leads to 'unphysical' wave functions. Assume that we define a new deformed algebra as a unital algebra with elements J 0 , J + , J − and R subject to the relations of Definition 1, but with (12) replaced by: where as before γ = (2a − 1)(2c − 1). The actions of Proposition 2 on a representation space H a can then be generalized to J + |a, n = (n + 2a + 2b)(n + 2b + 2c) |a, n + 1 , if n is even; (n + 1)(n + 2a + 2c − 1) |a, n + 1 , if n is odd, J − |a, n = n(n + 2a + 2c − 2) |a, n − 1 , if n is even; (n + 2a + 2b − 1)(n + 2b + 2c − 1) |a, n − 1 , if n is odd, and the action of R is unchanged. We shall also assume that a, c > 0 and b ≥ 0. In this case, the expressions (18) can still be used, and thus a formal eigenvector ofq for the eigenvalue x, of the form (21), leads to: reminiscent of the more general equations (26)- (27). These equations can indeed be solved by taking Although this is a formal solution, the appearance of x 2 − b 2 as an argument of the continuous dual Hahn polynomials spoils the orthogonality relation (25), and we would not obtain R as spectrum of the position operator (but rather the values for which |x| > |b|, plus eventually some discrete points according to the orthogonality [16, (9.3.3)], depending on the values of a, b, c). Because of the unphysical nature of the corresponding eigenfunctions, we will not consider this further.

Conclusion
The one-dimensional quantum harmonic oscillator is a central problem and model in quantum mechanics. The simplest and standard model, the non-relativistic quantum harmonic oscillator in the canonical approach, has an attractive solution for its wave functions of stationary states in terms of Hermite polynomials. The dynamical algebra of this standard oscillator (or "Hermite oscillator") is the usual Heisenberg algebra. This standard model can be extended both for continuous and discrete measures (for the wave functions), and in both cases some elegant models with analytic solutions for the wave functions exist.
In the continuous case, there are two well-known ways of extending the Hermite oscillator: these are the Meixner-Pollaczek oscillator [15] and the paraboson oscillator [19]. Both extensions have the same equidistant energy spectrum, where the ground state energy is some positive value a instead of 1/2 in the case of the Hermite oscillator. The dynamical algebra is quite different though. For the Meixner-Pollaczek oscillator, the Hamiltonian together with the position and momentum operator form a basis of the Lie algebra su(1, 1). The ground state energy a corresponds to the representation label (lowest weight) of a positive discrete series representation. For the paraboson oscillator, the position and momentum operators are considered as odd generators of the Lie superalgebra osp(1|2). The ground state energy a is again a representation label for a unitary osp(1|2) representation. Note that in this case the wave functions are given in terms of (generalized) Laguerre polynomials.
By constructing models for the quantum harmonic oscillator based upon a deformation of su(1, 1), we have been able to unify the previous well-known extensions in the continuous case. The algebra su(1, 1) γ is an extension of su(1, 1) by a parity operator R, and involves a deformation parameter γ. The proposed models for the oscillator involve two parameters a and c: a is again a representation label, and c is a deformation label related to γ by γ = (2a − 1)(2c − 1). The energy spectrum is again equidistant, with ground state energy equal to a. Our main result is that the stationary wave functions of such models have elegant closed form expressions in terms of continuous dual Hahn polynomials. Some properties of these wave functions have been described in Section 4.
Clearly, the dynamical algebra for these new models is su(1, 1) γ . For c = 1/2, su(1, 1) γ becomes su(1, 1) and also the wave functions for the new models reduce to those of the Meixner-Pollaczek oscillator. For c → ∞, the algebra reduces to the paraboson algebra, and the wave functions become the known paraboson oscillator wave functions.
As far as potential applications to physical models are concerned, we can offer the argument that our proposed model has two parameters a and c (one more that the Meixner-Pollaczek oscillator and the paraboson oscillator, and two more than the Hermite oscillator). These parameters should be greater than or equal to 1/2 in order to deal with physically acceptable wave functions. Having two parameters available in a mathematical model for the quantum oscillator, opens the way to more flexibility in applications.