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The q-Harmonic Oscillator and an Analog of the Charlier polynomials

arXiv:math/9307206v1 [math.CA] 9 Jul 1993The q-Harmonic Oscillator and an Analog of the Charlier polynomialsR. Askey† and S. K. Suslov‡Abstract.

A model of a q-harmonic oscillator based on q-Charlier poly-nomials of Al-Salam and Carlitz is discussed. Simple explicit realizationof q-creation and q-annihilation operators, q-coherent states and an ana-log of the Fourier transformation are found.

A connection of the kernel ofthis transform with biorthogonal rational functions is observed.Models of q-harmonic oscillators are being developed in connection with quantumgroups and their various applications ( see, for example, Refs. [1–5]).

The q-analogs ofboson operators were introduced explicitly in Refs. [1,3] and [5], where the correspondingwave functions were found in terms of the continuous q-Hermite polynomials of Rogers [6,7]and in terms of the Stieltjes–Wigert polynomials [8,9], respectively.

Here we introduceone more explicit realization of q-creation and q-annihilation operators with the aid ofq-Charlier polynomials of Al-Salam and Carlitz [10].The q-orthogonal polynomials V an studied by Al-Salam and Carlitz may be consideredas a q-version of the Charlier polynomials cµn(s) ( see, for example, [11,12] ). To emphasizethis analogy we use the notation cµn(x | q) for the Al-Salam and Carlitz polynomials.

Inour notation they can be defined by the three-term recurrence relationµq−n−1cµn+1(x | q) + (1 −qn)q−ncµn−1(x | q) =(µ + q)q−n−1 −xcµn(x | q) ,(1)cµ0(x | q) = 1 , cµ1(x | q) = µ−1(µ + q −qx) . These polynomials are orthogonal∞Xs=0cµm(q−s | q) cµn(q−s | q) ρ(s)q−s = (q; q)nµnδmn(2)with respect to a positive measureρ(s) = (µ; q)∞µsqs2(q, µ; q)s; 0 < µ, q < 1 ,(3)where the usual notations (see [13]) are(a; q)n =n−1Yk=0(1 −aqk) ,(a, b; q)n = (a; q)n(b; q)n ,(4)(a; q)∞= limn→∞(a; q)n .† Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA‡ Russian Scientific Center “Kurchatov Institute”, Moscow 123182, Russia1

The weight function (3) is a solution of the Pearson equation ∆(σρ) = ρτ∇x1 (for details,see [12,14]) with x(s) = q−s, σ(s) = (1 −q−s)(µ −q1−s), σ(s) + τ(s)∇x1(s) = µ. Theexplicit form of the polynomials cµn(x | q) iscµn(x | q) = 2ϕ0(q−n, x; q, q1+n/µ), x = q−s . (5)For the definition of the basic hypergeometric function 2ϕ0 , see [13].

In the limit q →1 iteasy to see from (1) or (5) thatlimq→1 c(1−q)µn(q−s | q) = 2F0(−n, −s; −1/µ) = cµn(s) . (6)This justifies our notation for the Al-Salam and Carlitz polynomials.The polynomials cµn(x | q) give us the possibility to introduce a new model of a q-oscillator.

We can define a q-version of the wave functions of harmonic oscillator asψn(s) = d−1n q−s2 ρ12 (s) cµn(q−s | q) ,(7)where d2n = (q; q)n/µn. These q-wave functions satisfy the orthogonality relation∞Xs=0ψn(s) ψm(s) = δnm .

(8)The q-annihilation a and q-creation a+ operators have the following explicit forma =(1 −q)−12hµ12 qs −p(1 −qs+1)(1 −µqs)e∂si,(9)a+ =(1 −q)−12hµ12 qs −e−∂sp(1 −qs+1)(1 −µqs)i,where ∂s ≡dds, eα∂sf(s) = f(s + α). They satisfy the q-commutational ruleaa+ −qa+a = 1(10)and act on the q-wave functions defined in (7) byaψn = e12nψn−1, a+ψn = e12n+1ψn+1 ,(11)whereen = 1 −qn1 −q .In this model of the q-oscillator equations (11) are equivalent to difference-differentiationformulasµqs∆cµn(x | q) = (qn −1) cµn−1(x | q) ,qs∇[ρ(s)cµn(x | q)] = ρ(s) cµn+1(x | q) ,2

respectively. Here ∆f(s) = ∇f(s + 1) = f(s + 1) −f(s) and x = q−s.

In view of (6) thefunctions ψn(s) converge in the limit q →1−to the wave functions of the discrete modelof the linear harmonic oscillator considered in [15].The q-Hamiltonian H = a+a acts on the wave functions (7) asHψn = enψn(12)and has the following explicit formH = (1 −q)−1µq2s + (1 −qs)(1 −µqs−1)−(13)µ12 qsp(1 −qs+1)(1 −µqs)e∂s −µ12 qs−1p(1 −qs)(1 −µqs−1e−∂s.By factorizing the Hamiltonian ( or the difference equation for the Al-Salam and Carlitzpolynomials ) we arrive at the explicit form (9) for the q-boson operators.Since a+a = H, the relation (10) can be written in the equivalent form[a, a+] = 1 −(1 −q)H ≡qN . (14)The operatorN =1log q log[1 −(1 −q)H](15)can be considered as the number operator, since[a, N] = a,[N, a+] = a+ .

(16)From these relations one can obtain the equations (11) and the spectrum (12) of theq-Hamiltonian in abstract form. The q-wave functions areψn(s) = cna+n ψ0(s) ,a ψ0(s) = 0 ,where cn = (en!

)−1/2 and en! = e1e2 .

. .

en.For the model of the q-oscillator under discussion we can construct explicitly q-coherentstates and an analog of the Fourier transformation. For the coherent states | α⟩defined bya | α⟩= α | α⟩,⟨α|α⟩= 1 ,| α⟩= fα∞Xn=0αnψn(s)(en!

)1/2 ,(17)fα =(1 −q) | α|2; q 12∞,(1 −q) | α|2 < 1we can write| α⟩= fαρq−s 12∞Xn=0tn(q; q)ncµn(x | q) , t = αµ12 (1 −q)12 . (18)3

With the aid of the generating function [10]∞Xn=0tn(q; q)ncµn(x | q) = (qtx/µ; q)∞(t , qt/µ; q)∞, |t| < 1, |qt/µ| < 1(19)we arrive at the following explicit form for the q-coherent states| α⟩= fαρq−s 12α(1 −q)1/2µ−1/2q1−s; q∞α(1 −q)1/2µ1/2, αq(1 −q)1/2µ−1/2; q∞,(20)where ρ = (q; q)−1∞(qs+1 , µqs; q)∞µsqs2 . These coherent states are not orthogonal⟨α | β⟩=(1 −q) | α |2 , (1 −q) | β |2; q1/2∞((1 −q)α∗β ; q)∞,where ∗denotes the complex conjugate.To define an analog of the Fourier transform we can consider, following Wiener’sapproach to the classical Fourier transform [16] ( see also [17,18] ), the kernel of the formKt(s, p) =∞Xn=0tnψn(s) ψn(p)(21)=ρ(s)ρ(p)q−s−p 12∞Xn=0(µt)n(q; q)ncµn(q−s | q) cµn(q−p | q) .The series can be summed with the aid of the bilinear generating function by Al-Salamand Carlitz [10]∞Xn=0cµ1n (x | q) cµ2n (y | q)tn(q; q)n= (qtx/µ1 , qty/µ2; q)∞(t , qt/µ1 , qt/µ2; q)∞· 3ϕ2x , y , tqtx/µ1, qty/µ2; q, q2tµ1µ2.

(22)The answer isKt(s, p) =ρ(s)ρ(p)q−s−p 12 (tq1−s , tq1−p ; q)∞(qt , qt , µt; q)∞· 3ϕ2 q−s , q−p , µttq1−s , tq1−p ; q, q2tµ. (23)The q-wave functions (7) are eigenfunctions of the “discrete q-Fourier transform”,imψm(s) =∞Xp=0Ki(s, p) ψm(p) .

(24)4

The orthogonality relation of the kernel,∞Xp=0Ki(s, p) K∗i (s′, p) = δss′ ,(25)implies the orthogonality of the rational functions (23). In view of (6) in the limit q →1−we get the “discrete Fourier transform” considered in [17].Similarly, with the aid of the bilinear generating function (22) and the orthogonalityproperty of the Wall polynomials, which are dual to the polynomials (5), one can obtainthe biorthogonality relation∞Xs=0um(s) vn(s) ρ(s)q−s = d2nδmn(26)withρ(s) =µ2t1 , µ2t2 ; qs(q, µ2 ; q)sµs1qsandd2n = (t1, t2 ; q)∞(µ1, µ2 ; q)∞·(q, µ1 ; q)nµ1t1 , µ1t2 ; qnµ−n2for the 3ϕ2-rational functions of the formum(s) = 3ϕ2q−m , q−s , t1t1µ1q1−m , t1µ2q1−s ; q, q2t2;(27)vn(s) = un(s)|t1↔t2 ;t1t2 = µ1µ2 .These functions are self-dual.They belong to classical biorthogonal rational functions[19,20].We have considered here the explicit form of q-boson operators which satisfy thecommutational rule (10) when 0 < q < 1.

The case q > 1 is also interesting. It leads toanother family of Al-Salam and Carlitz polynomials.It is evident that for the models of the q-oscillator under discussion one can readilyconstruct dynamical symmetry group SUq(1, 1) [4] and write an explicit realization forirreducible representations |j, m⟩q = ψj+m(s)ψj−m(s′) of the groupSUq(2) [1,2].5

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