---

AN ANALOG OF THE FOURIER TRANSFORMATION

arXiv:math/9307205v1 [math.CA] 9 Jul 1993AN ANALOG OF THE FOURIER TRANSFORMATIONFOR A q-HARMONIC OSCILLATORR. Askey,1 N. M. Atakishiyev,2 and S. K. Suslov31Dept.

of Math., U. of Wis., Madison, WI 53706, USA2Physics Institute, Baku 370143, Azerbaijan3Kurchatov Institute of Atomic Energy, Moscow 123182, RussiaABSTRACTA q-version of the Fourier transformation and some of its properties are discussed.INTRODUCTIONModels of q-harmonic oscillators are being developed in connection with quan-tum groups and their various applications (see, for example, Refs. [M], [Bi], [AS1],and [AS2]).

For a complete correspondence with the quantum-mechanical oscillatorproblem, these models need an analog of the Fourier transformation that relates thecoordinate and momentum spaces. In the present work we fill this gap for one ofthe models, the one based on the continuous q-Hermite polynomials [M], [AS1] when−1 < q < 1.In Section I we assemble all those formulas from [AS1], which are necessary forthe subsequent exposition.

In Section II we discuss the relation between the Mehlerbilinear generating function for Hermite polynomials and the kernel exp ihpxof theFourier transformation that connects the coordinate x and momentum p spaces [W].We used the bilinear formula of L. J. Rogers to obtain a reproducing kernel andan analogue of the Fourier transform in the setting determined by the continuousq-Hermite polynomials of Rogers. Some properties of the q-Fourier transformationare discussed in Sections III-IV.In the following we take 0 < q < 1, although most of the formulas remain correctwhen −1 < q < 0.

The limiting case q →0 is of some mathematical interest.1. q-Hermite functions.

The continuous q-Hermite polynomials were intro-duced by Rogers [R]. They can be defined by the three term recurrence relation2xHn(x | q) = Hn+1(x | q) + (1 −qn)Hn−1(x | q),(1.1)H0(x | q) = 1, H1(x | q) = 2x.

They are orthogonal on −1 ≤x = cos θ ≤1 withrespect to a positive measure ρ(x)1

ρ(x) = 4 sin θ(qe2iθ, qe−2iθ; q)∞(1.2)= 4p1 −x2∞Yj=1[1 −2(2x2 −1)qj + q2j]where the usual notations (see [GR]) are(a; q)∞=∞Yn=0(1 −aqn) ,(1.3)(a, b; q)∞= (a; q)∞(b; q)∞.A q-wave function for this q-harmonic oscillator was introduced explicitly in [M],[AS1] asψn(x) = αd−1n ρ1/2(x)Hn(x | q),α =1 −q21/4,(1.4)where dn > 0 andd−2n= 12π (qn+1; q)∞. (1.5)These q-wave functions (1.4) satisfyZ 1−1ψn(x)ψm(x) dx = α2δnm .

(1.6)See [AI] for a proof of this orthogonality relation.q-annihilation b and q-creation b+ operators were introduced explicitly in [M]and [AS1]. They satisfy the commutation rulebb+ −q−1b+b = 1(1.7)and act on the q-wave functions defined in (1.4) bybψn(x) = ˜e1/2n ψn−1(x),b+ψn(x) = ˜e1/2n+1ψn+1(x)(1.8)where˜en = 1 −q−n1 −q−1 = q1−nen .

(1.9)In the limit when q →1−the functions ψn(α2ξ) converge to the classical wavefunctionsΨn(ξ) =1π1/4(2nn! )1/2 Hn(ξ)e−ξ2/2(1.10)2

of the linear harmonic oscillator in the coordinate representation.2. An analog of the Fourier transformation.

In proving that the Hermitefunctions (1.10) are complete in the space L2 over (−∞, ∞) and that a Fourier trans-form of any function from L2 belongs to the same space, the important role is playedby the bilinear generating function (or the Poisson kernel) [W]Kt(ξ, η) =∞Xn=0tnΨn(ξ)Ψn(η)= [π(1 −t2)]−1/2 exp4ξηt −(ξ2 + η2)(1 + t2)2(1 −t2). (2.1)Since in the limit γ →∞the continuously differentiable functionδ(x, γ) =γ√π e−γ2x2becomes the Dirac delta function δ(x), from (2.1) in the limit t →1−follows thecompleteness property of the system (1.10),∞Xn=0Ψn(ξ)Ψn(η) = δ(ξ −η) .

(2.2)On another hand, setting t = i reduces the right-hand side of (2.1) to the kernelof the Fourier transformation, i.e.Ki(ξ, η) =1√2π eiξη .Actually this idea provides the possibility of finding the Fourier transformation fordifference analogs of the harmonic oscillator and its q-generalizations, when a prioriit is not clear how one can define the explicit form of the kernels and how thesetransformations can look like. As first examples we considered Kravchuk and Charlierfunctions [AAS].

Here we shall give the necessary formulas for the q-Hermite functions(1.4).Let us define a q-analog of (2.1) asKt(x, y; q) = α−2∞Xn=0tnψn(x)ψn(y) . (2.3)Substituting the formulas (1.4) and (1.5) into (2.3), we getKt(x, y; q) = (q; q)∞2πρ1/2(x)ρ1/2(y)∞Xn=0tn(q; q)nHn(x|q)Hn(y|q).

(2.4)3

This series (the Poisson kernel) was summed as an infinite product by Rogers [R],and the value is stated in [AI]. A very simple proof was given by Bressoud [Br].

Theformula is∞Xn=0tn(q; q)nHn(cos θ|q)Hn(cos ϕ|q) == (t2; q)∞(tei(θ+ϕ), tei(θ−ϕ), te−i(θ+ϕ), te−i(θ−ϕ); q)−1∞,(2.5)so we can write (2.4) asKt(x, y; q) =(q, t2; q)∞ρ1/2(x)ρ1/2(y)2π(tei(θ+ϕ), tei(θ−ϕ), te−i(θ+ϕ), te−i(θ−ϕ); q)∞. (2.6)In complete analogy with the case of the harmonic oscillator,limt→1−Kt(x, y; q) = α−2∞Xn=0ψn(x)ψn(y) = δ(x −y) .

(2.7)This is even easier to prove than in the classical case, since any set of polynomialsorthogonal on a finite interval is complete in L2, and this is equivalent to (2.7).Formula (2.7) follows fromtnψn(x) =Z 1−1Kt(x, y; q)ψn(y) dy,|t| < 1 ,(2.8)which is obvious by integration, which is justified by uniform convergence. An easycorollary of (2.3) isZ 1−1Kt(x, y; q)Kτ(y, x′; q) dy = Ktτ(x, x′; q), |t| < 1, |τ| < 1 .

(2.9)Not only is the limit t →1−interesting, t →i is also interesting. The kernel inthis case is the special case of (2.6) when t = i, i.e.Ki(x, y; q) = 4π (q2; q2)∞[sin θ sin ϕ(qe2iθ, qe−2iθ, qe2iϕ, qe−2iϕ; q)∞]1/2(iei(θ+ϕ), ie−i(θ+ϕ), iei(θ−ϕ), ie−i(θ−ϕ); q)∞,(2.10)x = cos θ, y = cos ϕ.The classical Fourier transform has a kernel which is bounded, and the fourthpower of it is the identity.

The fourth power of the q-Fourier transform,Fq[ψ](x) = limr→1−Z 1−1Kir(x, y; q)ψ(y) dy ,(2.11)4

is also the identity, from (2.8) with t →i and i4 = 1. Since [−1, 1] is compact, thekernel Ki(x, y; q) can not be bounded and have F 4q = I.

It is not, and the singularitycomes from the first term in the four products in the denominator of (2.10). Thesingular part is just the value of (2.10) when q = 0, i.e.−(sin θ sin ϕ)1/2π cos(θ + ϕ) cos(θ −ϕ) .

(2.12)Thus this q-Fourier transform looks more like a weighted Hilbert transform than itlooks like the classical Fourier transform.An explicit form of the transformation(2.11) isFq[ψ](x) = PvZ 1−1Ki(x, y; q)ψ(y) dy(2.13)+ i2hk(x)ψ(p1 −x2) + k(−x)ψ(−p1 −x2)i,wherek(x) =x(x2(1 −x2))1/4 ∞Yk=11 + 4ix√1 −x2qk −q2k1 −4ix√1 −x2qk −q2k!1/2and Pv denotes Cauchy’s principal value integral.The inverse of this transformation follows fromlimr,r′→1−Z 1−1Kir(x, y; q)K∗ir′(y, x′; q) dy = δ(x −x′) ,(2.14)where ∗denotes the complex conjugate. Formula (2.14) follows from (2.9) when weobserve thatK∗ir(y, x; q) = K−ir(y, x; q) .

(2.15)Another form of (2.14) islimr,r′→1−Z 1−1dy K∗ir′(x, y; q)Z 1−1Kir(y, z; q)f(z) dz = f(x) . (2.16)Indeed, by (2.9) and (2.15) we getZ 1−1dy K∗ir′(x, y; q)Z 1−1Kir(y, z; q)f(z) dz=Z 1−1Krr′(x, z; q)f(z) dz −→f(x)with r, r′ →1−.5

By using the change of variables x = α2ξ and y = α2η, where α2 = 1−q21/2,from (2.3) in the limit q →1−we can writeα2Kt(α2ξ, α2η; q) =∞Xn=0tnψn(α2ξ)ψn(α2η)−→∞Xn=0Ψn(ξ)Ψn(η) = Kt(ξ, η) ,orlimq→1−α2Kt(α2ξ, α2η; q) = Kt(ξ, η) . (2.17)With the aid of the same consideration, from (2.11) we getFq[ψ](x) = limr→1−Z 1/α2−1/α2 α2Kir(α2ξ, α2η; q)ψ(α2η) dη−→1√2πZ ∞−∞eiξηf(η) dη = F[f](ξ) ,where f(η) = limq→1−ψ(α2η).

Therefore, the classical Fourier transformation can berealized as a limiting case of the q-Fourier transform.3. “Momentum and position” operators.

As is well known from quantummechanics, the kernel of the classical Fourier transformation is also an eigenfunctionof the momentum and position operators, i.e.QξKi(ξ, η) = ξKi(ξ, η) ,(3.1)PξKi(ξ, η) = i−1 ddξ1√2π eiξη= ηKi(ξ, η) .We can retain this property in the case of the q-Fourier transformation, if we choosethe corresponding operators for the kernel (2.10) asQ =√1 −q2qN/2b + b+qN/2,(3.2)P =√1 −q2iqN/2b −b+qN/2,where N = log[1 −(1 −q−1)b+b]/ log q−1 is the “particle number” operator. In fact,the action of the operator Q on the kernel (2.10) givesα2QxKi(x, y; q) =√1 −q2qNx/2bx + b+x qNx/2 ∞Xn=0inψn(x)ψn(y)(3.3)=√1 −q2∞Xn=0in he1/2n ψn−1(x) + e1/2n+1ψn+1(x)iψn(y) .6

From the three-term recurrence relation (1.1) for the continuous q-Hermite polyno-mials, it follows thate1/2n+1ψn+1(x) + e1/2n ψn−1(x) =2√1 −q x ψn(x) . (3.4)Therefore for the “position” operator Q we have, indeed,QxKi(x, y; q) = xKi(x, y; q) .

(3.5)Exactly in the same manner, one can obtain directly from definitions (3.2) andexpansion (2.3) for t = i, thatPxKi(x, y; q) = QyKi(x, y; q) ,(3.6)and, consequently,PxKi(x, y; q) = yKi(x, y; q) . (3.7)EquationsQxKt(x, y; q) = xKt(x, y; q) ,(3.8)PxKt(x, y; q) = 2ty −(1 + t2)xi(1 −t2)Kt(x, y; q)are an extension of (3.5) and (3.7).The commutation rule of the operators (3.2) is[Q, P] = i 1 −q2qN(3.9)and, therefore, the Hamiltonian of the q-oscillator,˜H = b+b = 1 −q−N1 −q−1 ,(3.10)has the following form˜H(P, Q) = i −2(1 −q)−1[Q, P]2q−1[Q, P](3.11)in terms of these momentum and position operators.The equationP 2xKi(x, p; q) = p2Ki(x, p; q)(3.12)may be considered as an equation of motion for a“q-free particle”.7

4. Some properties of the q-Fourier transformation.

The kernel Ki(ξ, η)corresponds to the classical Fourier transformationf(ξ) =1√2πZ ∞−∞eiξηg(η) dη = F[g](ξ) ,(4.1)which has well-known properties [W]. With the aid of the kernel (2.6) we defined aq-version of the Fourier transformation byψ(x) = limr→1−Z 1−1Kir(x, y)ϕ(y) dy = Fq[ϕ](x) .

(4.2)We can establish the simple properties of this generalization.The orthogonalityproperty (2.14) of the kernel (2.6) results in the inversion formulaϕ(y) =limr′→1−Z 1−1K∗ir′(x, y)ψ(x) dx ,(4.3)as well as in the relation ∥ϕ∥2 = ∥ψ∥2.Analogs of the propertiesi−1 ddξ F[g](ξ) = F[ηg](ξ) ,(4.4)iFdgdη(ξ) = ξF[g](ξ) ,have the formsPxFq[ϕ](x) = Fq[yϕ](x) ,(4.5)Fq[Pyϕ](x) = −xFq[ϕ](x) .Moreover, the following properties of the Fourier transform,F[f](ξ + ξ0) = eiξ0PξF[f](ξ) = F[eiξ0ηf](ξ) ,(4.6)F[f(η −η0)](ξ) = F[e−iη0Pηf](ξ) = eiξη0F[f](ξ) ,admit a generalizationKi(x0, Px)Fq[ϕ](x) = Fq[Ki(x0, y)ϕ](x) ,(4.7)Fq[Ki(y0, −Py)ϕ](x) = Ki(x, y0)Fq[ϕ](x) .8

To define a q-version of the convolution ϕ ∗ψ let us retain the propertyFq[φ ∗ψ] = Fq[φ] · Fq[ψ] ,(4.8)orlimr→1−Z 1−1Kir(x, z)(ϕ ∗ψ)(z) dz =(4.9)limr′,r′′→1−Z 1−1Z 1−1Kir′(x, y)Kir′′(x, y′)ϕ(y)ψ(y′) dy dy′ .Using (2.14) and (4.9) we arrive at the definition(ϕ ∗ψ)(z) =limr,r′,r′′→1−Z 1−1Z 1−1dy dy′ ϕ(y)ψ(y′)(4.10)Z 1−1dx Kir′(x, y)Kir′′(x, y′)K∗ir(x, z) .The usual properties,ϕ ∗ψ = ψ ∗ϕ ,(ϕ ∗ψ) ∗χ = ϕ ∗(ψ ∗χ) ,are valid.In the limit q →1−we can writeα4Z 1−1Kir′(x, y)Kir′′(x, y′)K∗ir(x, z) dx(4.11)−→1(2π)3/2Z ∞−∞eiξ(η+η′−ζ) dξ =1√2π δ(η + η′ −ζ) ,and, therefore,(ϕ ∗ψ)(z) −→1√2πZ ∞−∞f(η)g(ζ −η) dη .We think the transformation (4.2) deserves a more detailed consideration.REFERENCES[AAS] R. Askey, N. M. Atakishiyev and S. K. Suslov – Fourier transformationsfor difference analogs of the harmonic oscillator. Proceedings of the XV9

Workshop on High Energy Physics and Field Theory, Protvino, Russia,6–10 July 1992. [AI] R. Askey and M. E. H. Ismail – A generalization of the ultraspherical polyno-mials.

Studies in Pure Mathematics (P. Erd¨os, ed. ), Birkh¨auser, Boston,Massachusetts, pp.

55–78, 1983. [AS1] N. M. Atakishiyev and S. K. Suslov – Difference analogs of the harmonicoscillator.Theoretical and Mathematical Physics, Vol.

85, No. 1, pp.1055–1062, 1990.

[AS2] N. M. Atakishiyev and S. K. Suslov – A realization of the q-harmonicoscillator. Theoretical and Mathematical Physics, Vol.

87, No. 1, pp.

442–444, 1991. [Bi] L. C. Biedenharn – The quantum group SUq(2) and a q-analogue of theboson operators.

J. Phys. A: Math.

Gen., Vol. 22, No.

18, pp. L873–L878,1989.

[Br] D. M. Bressoud – A simple proof of Mehler’s formula for q-Hermite polyno-mials, Indiana Univ. Math.

J., Vol. 29, pp.

577–580, 1980. [GR] G. Gasper and M. Rahman – Basic Hypergeometric Series.CambridgeUniversity Press, Cambridge, 1990.

[M] A. J. Macfarlane – On q-analogues of the quantum harmonic oscillator andthe quantum group SU(2)q. J. Phys. A: Math.

Gen., Vol. 22, No.

21, pp.4581–4588, 1989. [R] L. J. Rogers – Second memoir on the expansion of certain infinite products.Proc.

London Math. Soc., Vol.

25, pp. 318–343, 1894.

[W] N. Wiener – The Fourier Integral and Certain of Its Applications. CambridgeUniversity Press, Cambridge, 1933.53706, USARussia10

---

출처: arXiv:9307.205 • 원문 보기