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The q-Harmonic Oscillator and the Al-Salam and Carlitz polynomials

arXiv:math/9307207v1 [math.CA] 9 Jul 1993The q-Harmonic Oscillator and the Al-Salam and Carlitz polynomialsDedicated to the Memory of Professor Ya. A. Smorodinski˘ıR.

Askey† and S. K. Suslov‡Abstract. One more model of a q-harmonic oscillator based on the q-orthogonal polynomials of Al-Salam and Carlitz is discussed.

The explicitform of q-creation and q-annihilation operators, q-coherent states and ananalog of the Fourier transformation are established. A connection of thekernel of this transform with a family of self-dual biorthogonal rationalfunctions is observed.IntroductionRecent development in quantum groups has led to the so-called q-harmonic oscilla-tors ( see, for example, Refs.

[1–7] ). Presently known models of q-oscillators are closelyrelated with q-orthogonal polynomials.

The q-analogs of boson operators have been in-troduced explicitly in Refs. [3], [5] and [7], where the corresponding wave functions wereconstructed in terms of the continuous q-Hermite polynomials of Rogers [8,9], in termsof the Stieltjes–Wigert polynomials [10,11] and in terms of q-Charlier polynomials of Al-Salam and Carlitz [12], respectively.

The model related to the Rogers–Szeg¨o polynomials[13] was investigated in [1,6]. Here we introduce the explicit realization of q-creation andq-annihilation operators with the aid of another family of the Al-Salam and Carlitz poly-nomials [12] when eigenvalues of the corresponding q-Hamiltonian are unbounded.Anattempt to unify q-boson operators is also made.With a great deal of regret we dedicate this paper to the memory of Yacob A.Smorodinski˘ı, who suggested ten years ago that the special case q = 1 of this work isinteresting and admits a generalization.1.

The Al-Salam and Carlitz PolynomialsThe aim of this Letter is to show that the q-orthogonal polynomials U (a)n (x; q) studiedby Al-Salam and Carlitz are closely connected with the q-harmonic oscillator. To emphasizethese relations we use the notation uµn(x; q) = µ−nq−n(n−1)/2U (−µ)n(x; q) for the Al-Salamand Carlitz polynomials.

In our notation they can be defined by the three-term recurrencerelation of the formµqnuµn+1(x; q) + (1 −qn) uµn−1(x; q) = (x −(1 −µ)qn) uµn(x; q) ,(1)uµ0(x; q) = 1 , uµ1(x; q) = µ−1(x −1 + µ) . These polynomials are orthogonalZ 1−µuµm(x; q) uµn(x; q) dα(x) = (1 + µ)q−n(n−1)/2 (q; q)nµnδmn(2)† Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA‡ Russian Scientific Center “Kurchatov Institute”, Moscow 123182, Russia1

with respect to a positive measure dα(x), where α(x) is a step function with jumpsqk(−qµ; q)∞(q, −q/µ; q)kat the points x = qk, k = 0, 1, . .

., and jumpsµqk(−q/µ; q)∞(q, −qµ; q)kat the points x = −µqk, k = 0, 1, . .

. ( see, for example, [12,14,15] ).Here the usualnotations are(a; q)n =n−1Yk=0(1 −aqk) ,(a, b; q)n = (a; q)n(b; q)n ,(3)(a; q)∞= limn→∞(a; q)n .The orthogonality relation (2) can also be written in terms of the q-integral of Jackson,Z 1−µuµm(x; q) uµn(x; q) ˜ρ(x) dqx = (1 −q)d2n δmn ,(4)where˜ρ(x) = (qx, −µ−1qx; q)∞(q, −µ, −q/µ; q)∞; µ > 0 , 0 < q < 1(5)andd2n = q−n(n−1)/2 (q; q)nµn.

(6)For the definition of the q-integral, see [15]. The “weight function” ρ(s) = ˜ρ(x) in (5)is a solution of the Pearson-type equation ∆(σρ) = ρτ∇x1with x(s) = qs, σ(s) =(1 −qs)(µ + qs) and σ(s) + τ(s)∇x1(s) = µ.

The polynomials yn(s) = uµn(x; q) satisfy thehypergeometric-type difference equation in self-adjoint form,∆∇x1(s)σ(s) ρ(s) ∇yn(s)∇x(s)+ λn ρ(s) yn(s) = 0 ,whereλn = q3/2 q−n −1(1 −q)2 .Here ∆f(s) = f(s + 1) −f(s) = ∇f(s + 1) and x1(s) = x(s + 1/2). ( For details, see[16–19]. ) The orthogonality property (2) or (4) can be proved by using standard Sturm–Liouville-type arguments ( cf.

[16–19]).2

The explicit form of the polynomials uµn(x; q) isuµn(x; q) = 2ϕ1q−n, x−1; 0 ; q , −qµ x(7)= (−µ−1)n2ϕ1(q−n, −µx−1; 0 ; q , qx) , x = qs .It means uµn(x; q) =−µ−1n u1/µn(−µ−1x; q). In the limit q →1 it easy to see from (1) or(7) thatlimq→1 u(1−q)µn(qs; q) = 2F0(−n, −s; −; −1/µ) = cµn(s) ,(8)where cµn(x) are the Charlier polynomials.2.

Model of q-Harmonic OscillatorThe Al-Salam and Carlitz polynomials uµn(x; q) allow us to consider an interestingmodel of a q-oscillator ( cf. [7] ).

We can introduce a q-version of the wave functions ofthe harmonic oscillator asψn(s) = ˜ψn(x) = d−1n (˜ρ(x)|x|)1/2 uµn(x; q) , x = qs ,(9)where ˜ρ(x) and d2n are defined in (5) and (6), respectively. These q-wave functions satisfythe orthogonality relation(1 −q)−1Z 1−µ˜ψn(x) ˜ψm(x) |x|−1 dqx = δnm ,(10)which is equivalent to (2) and (4).The q-annihilation b and q-creation b+ operators have the following explicit formb = (1 −q)−12hµ12 q−s −p(1 −qs+1)(µq−1 + qs) q−s e∂si,(11)b+ = (1 −q)−12hµ12 q−s −e−∂sp(1 −qs+1)(µq−1 + qs) q−si,where ∂s ≡dds, eα∂sf(s) = f(s + α).

These operators are adjoint, (b+ψ, χ) = (ψ, bχ),with respect to the scalar product (10). They satisfy the q-commutation ruleb b+ −q−1b+b = 1(12)and act on the q-wave functions defined in (9) byb ψn = ˜e1/2nψn−1, b+ψn = ˜e1/2n+1ψn+1 ,(13)3

where˜en = 1 −q−n1 −q−1 .The q-Hamiltonian H = b+b acts on the wave functions (9) asHψn = ˜enψn(15)and has the following explicit formH = (1 −q)−1µq−2s + (1 −qs)(µ + qs) q1−2s−(16)µ12 q−2sp(1 −qs+1)(µq−1 + qs) e∂s −µ12 q2−2sp(1 −qs)(µq−1 + qs−1) e−∂s.By factorizing the Hamiltonian ( or the difference equation for the Al-Salam and Carlitzpolynomials ) we arrive at the explicit form (11) for the q-boson operators. The equations(13) are equivalent to the following difference-differentiation formulasµq−s−1∆uµn(x; q) = (1 −q−n) uµn−1(x; q) ,q−s∇[ ρ(s) uµn(x; q) ] = ρ(s) uµn+1(x; q) ,respectively.

Therefore, the main properties of the Al-Salam and Carlitz polynomials admita simple group-theoretical interpretation in terms of the q-Heisenberg–Weyl algebra (12).The symmetric case µ = 1 in the above formulas corresponds to the discrete q-Hermitepolynomials Hn(x; q) [12,15].3. The q-Coherent StatesFor the model of the q-oscillator under discussion, by analogy with [7] we can constructexplicitly the q-coherent states | α⟩defined byb | α⟩= α | α⟩,(17)| α⟩= fα∞Xn=0αnψn(s)(˜en!

)1/2 ,⟨α | α⟩= 1 ,where˜en! = ˜e1˜e2 .

. .

˜en = q−n(n−1)/2 (q; q)n(1 −q)n ,fα =−(1 −q) | α |2; q−1/2∞.By using (9) one can obtain| α⟩= fα (ρ|qs|)12∞Xn=0uµn(x; q) qn(n−1)/2tn(q; q)n, t = αµ12 (1 −q)12 . (18)4

With the aid of the Brenke-type generating function [12,14] for the Al-Salam and Carlitzpolynomials,∞Xn=0uµn(x; q) qn(n−1)/2tn(q; q)n= (−t , t/µ; q)∞(xt/µ; q)∞,t xµ < 1 ,(19)we arrive at the following explicit form for the q-coherent states| α⟩= fα (ρ|qs|)12−α(1 −q)1/2µ1/2, α(1 −q)1/2µ−1/2; q∞α(1 −q)1/2µ−1/2qs; q∞,(20)where ρ(s) = (q1+s, −µ−1q1+s; q)∞/(q, −µ, −q/µ; q)⊂nfty. These coherent states are notorthogonal⟨α | β⟩=(−(1 −q)α∗β ; q)∞(−(1 −q) | α |2 , −(1 −q) | β |2; q)1/2∞,where ∗denotes the complex conjugate.4.

Analog of the Fourier TransformationTo define an analog of the Fourier transform we begin, in the spirit of Wiener’sapproach to the classical Fourier transform [20] ( see also [7,21,22] ), by deriving the kernelof the formKt(x, y) =∞Xn=0tn ˜ψn(x) ˜ψn(y)(21)= (˜ρ(x)˜ρ(y)|xy|)12∞Xn=0uµn(x; q) uµn(y; q) qn(n−1)/2 (µt)n(q; q)n.The series can be summed with the aid of the bilinear generating function of Al-Salam andCarlitz [12]∞Xn=0uµ1n (x; q) uµ2n (y; q) qn(n−1)/2tn(q; q)n= (−t , t/µ1 , t/µ2; q)∞(tx/µ1 , ty/µ2; q)∞· 3ϕ2x−1 , y−1 , −qt−1qµ1(tx)−1, qµ2(ty)−1 ; q , q(22)( the 3ϕ2-series is terminating and max ( |t/µ1| , |t/µ2| ) < 1 ). The answer isKt(x, y) = (˜ρ(x)˜ρ(y)|xy|)12 (t , t , −µt ; q)∞(tx , ty ; q)∞· 3ϕ2 x−1 , y−1 , −q(µt)−1q(tx)−1 , q(ty)−1; q , q;(23)= (˜ρ(x)˜ρ(y)|xy|)12(t , t , −µ−1t ; q)∞(−µ−1tx, −µ−1ty ; q)∞· 3ϕ2 −µx−1, −µy−1, −qµt−1−qµ(tx)−1, −qµ(ty)−1 ; q, qat x = qs and at x = −µqs for s = 0, 1, .

. ., respectively.5

In view of (10) and (21),tm ˜ψm(x) = (1 −q)−1Z 1−µKt(x, y) ˜ψm(y) |y|−1dqy . (24)Letting t = i, we find that the q-wave functions (9) are eigenfunctions of the following“q-Fourier transform”,im ˜ψm(x) = (1 −q)−1Z 1−µKi(x, y) ˜ψm(y) |y|−1dqy .

(25)An easy corollary of (21) or (24) is(1 −q)−1Z 1−µKt(x, y) Kt′(x′, y) |y|−1dqy = Ktt′(x, x′) . (26)Putting t = −t′ = i, we obtain the orthogonality relation of the kernel,(1 −q)−1Z 1−µKi(x, y) K∗i (x′, y) |y|−1dqy = δxx′ ,(27)which implies the orthogonality of the rational functions (23) and leads to an inversionformula for the q-transformation (25).

In view of (8), in the limit q →1−we get one ofthe “discrete Fourier transforms” considered in [21].5. Some Biorthogonal Rational FunctionsThe rational functions (23) have appeared as the kernel of the discrete q-Fouriertransform (25).

They admit the following extension. With the aid of the bilinear generatingfunction (22) and the orthogonality property of a special case of the q-Meixner polynomials,which are dual to the polynomials (7), we obtain the biorthogonality relation,Z 1−µ2u(x, y) v(x′, y) ˜ρ(y) dqy = (1 −q) d2x δxx′ ,(28)for the 3ϕ2-rational functions of the formu(x, y) = 3ϕ2x−1 , y−1 , −qt−11qµ1(t1x)−1 , qµ2(t1y)−1 ; q, q;(29)= 3ϕ2 −µ1x−1 , −µ2y−1 , −qt2−qt2x−1 , −qt2y−1; q, qat x = qs and at x = −µ1qs for s = 0, 1, .

. ., respectively, andv(x, y) = u(x, y)|t1↔t2 ;t1t2 = µ1µ2 .

(30)6

Here,˜ρ(y) =qy , −µ−12 qy ; q∞t1µ−12 y , t2µ−12 y ; q∞;=qy , −µ−12 qy ; q∞−t−11 y , −t−12 y ; q∞at x, x′ = {qs; s = 0, 1, . .

.} and at x, x′ = {−µ1qs; s = 0, 1, .

. .

}, respectively;˜ρ(y) =qy , −µ−12 qy ; q∞t1µ−12 y , −t−11 y ; q∞;=qy , −µ−12 qy ; q∞−t−12 y , t2µ−12 y ; q∞at x, x′ = {qs, −µ1qs} and vice versa, respectively. The squared norm isd2x =q, q, −µ1, −µ2, −qµ−11 , −qµ−12 ; q∞−t1, −t2, t1µ−11 , t2µ−11 , t1µ−12 , t2µ−12 ; q∞·t1µ−11 x, t2µ−11 x; q∞qx, −µ−11 qx ; q∞|x|−1 ;=q, q, −µ1, −µ2, −qµ−11 , −qµ−12 ; q∞−t−11 , −t−12 , µ1t−11 , µ1t−12 , µ2t−11 , µ2t−12 ; q∞·−t−11 x, −t−12 x; q∞qx, −µ−11 qx ; q∞|x|−1for x = qs and for x = −µ1qs, respectively.The functions (29)–(30) are self-dual and belong to classical biorthogonal rationalfunctions [23–27].

It is interesting to compare the biorthogonality relation (28) with theorthogonality property for the big q-Jacobi polynomials [28], which live at the same ter-minating 3ϕ2-level.6. Concluding RemarksIn view of (11), it is natural to introduce operators of the forma = α(s) −β(s) e∂, a+ = α(s) −e−∂β(s)with two arbitrary functions α(s) and β(s) and to satisfy the commutation rule a a+ −qa+a = 1.

The result isα(s + 1) = qα(s) ,(1 −q)α2(s) + β2(s) −qβ2(s −1) = 1and we can choose α(s) = ε qs and β2(s) = ε2 (qs+1 −γ)(qs −δ) with (1 −q)γδε2 = 1.Since(a+ψ, χ) −(ψ, aχ) =Xs∆[β(s −1) ψ∗(s −1) χ(s)] ,7

the corresponding operators are adjoint for the two different cases considered in [7] and inthis Letter with 0 < q < 1 and q > 1, respectively.For β = constant we can trya = e∂e∂−α(s), a+ =e−∂−α(s)e−∂and obtain aa+ −qa+a = 1 −q, whenα2(s + 1) = qα2(s) , α(s + 2) = qα(s) ,which is satisfied for α = ε qs/2. This case has been considered in [5].Finally, the operatorsa = ε eγ∂α(s) + e−γ∂β(s)α(s) −β(s), a+ = ε α(s)e−γ∂+ β(s)eγ∂α(s) −β(s)obey the q-commutation rule provided that α(s)β(s) = ±1 and α(s + 2γ) = q−1α(s).Therefore, α = q−s for γ = 1/2 and ε2 = q1/2(1 −q)−1 ( cf.

[3] ).We can also introduce the operatorsa = α−1(s) −ε β−1(s) e∂, a+ = α(s) −ε e∂β(s)and obtain aa+ −qa+a = 1 −q ifα(s + 1) = qα(s) , β(s + 2) = qβ(s) ,so α = qs and β = qs/2. This leads to the Rogers–Szeg¨o polynomials [13] orthogonal onthe unit circle ( see [1,6] ).References1.

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