representations of the gegenbauer

REPRESENTATIONS OF THE GEGENBAUER OSCILLATORALGEBRA AND THE OVERCOMPLETENESS OF SEQUENCES

OF NONLINEAR COHERENT STATES

ABDESSATAR BARHOUMI*

Abstract. The main purpose of this paper is to investigate a generalizedoscillator algebra, naturally associated to the Gegenbauer polynomials anda related class of nonlinear coherent vectors. We derive their overcomplete-ness relation, in so doing, the partition of unity in terms of the eigenstatesof the sequences of coherent vectors is established. It turns out that, inmaking up such relation, the positivity of the obtained measure is intimatelyconnected, up to a conjecture, to the parameter appearing in the beta distri-bution. An example of complex hypercontractivity property for an “ultras-pherical” Hamiltonian is developed to illustrate our theory.

1. Introduction

Coherent states representation [36] are ubiquitous in the mathematical physicsliterature. Yet there seems to be a lack of general theory in the context of repre-sentations of nonlinear coherent states. This paper is an attempt to partially fillthis gap.

The original coherent states based on the Heisenberg-Weyl group has beenextended for a number of Lie groups with square integrable representations [4],and they have many applications in quantum mechanics [24]. In particular, theyare used as bases of coherent states path integrals [24] or dynamical wavepacketsfor describing the quantum systems in semiclassical approximations [31]. Thisframework has been given in a general and elegant mathematical form throughthe work of Perelomov [37].

Many definitions of coherent states exist. The first one, so-called Klauder-Gazeau type [23], defines the coherent states as the eigenstates of the annihilationoperator A for each individual oscillator mode of the electromagnetic field; namely,a system of coherent states is defined to be a set Ω(z); z ∈ X of quantum statesin some interacting Fock space Γ [3, 39], parameterized by some set X such that:(i) AΩ(z) = z Ω(z), ∀z ∈ X, (ii) the map z 7→ Ω(z) is smooth, and (iii) the systemis overcomplete; i.e. ∫

X

|Ω(z)〉〈Ω(z)| ν(dz) = I. (1.1)

2000 Mathematics Subject Classification. Primary 46L53; Secondary 33D45, 44A15.Key words and phrases. Bargmann transform, beta-type distribution, coherent state, Gegen-

bauer oscillator algebra, hypercontractivity, one-mode interacting Fock space, orthogonal poly-nomials, overcompleteness relation, Poisson kernel.

* Research supported by “The Bilateral Polish-Tunisian Governmental Co-operation 07-09.”269

Serials Publications www.serialspublications.com

270 ABDESSATAR BARHOUMI

Physicists usually call property (1.1) completeness relation.A second definition of coherent states for oscillators, so-called Perelomov-type

[37], assumes the existence of a Weyl-type unitary operators

W (z) = exp(zA† − zA)

acting on some interacting Fock space Γ. The coherent states parameterized by zare given by

Ω(z) = W (z)Φ0, z ∈ X.

Here the annihilation operator A and the creation operator A† are mutually adjointand Φ0 is the Fock vacuum.

A third definition is based on the Heisenberg uncertainty relation with theposition operator Q and momentum operator P given, as usual, by

Q =i

2(A† + A) and P =

i

2(A† −A),

the coherent states defined above have the minimum-uncertainty value ∆Q∆P =1/2 and maintain this relation in time (temporal stability of coherent states) [5].

Let us stress that coherent states have two important properties. First, theyare not orthogonal to each other with respect to the positive measure in (1.1).Second, they provide a resolution of the identity, i.e., they form an overcompleteset of states in the interacting Fock space. In fact it is well known that theyform a “highly overcomplete” set in the sense that there are much smaller subsetsof coherent states which are also overcomplete. Using them one can express anarbitrary state as a line integral of coherent states [40].

It is therefore clear that knowing that a set of coherent states is overcomplete isnot only of theoretical interest; it is also of practical interest in the sense that weare encouraged to search for resolutions of the identity which will make possiblethe expansion of an arbitrary state in terms of these coherent states. Sometimesit is not easy to find a resolution of the identity and weaker concepts are alsosufficient (for example the concept of frames in wavelets). But it is clear that aprerequisite for going down that route is the question of completeness.

In the present paper, considering only the first of the above mentioned defini-tions, we are concerned by a class of nonlinear coherent states: the Gegenbauercoherent states. These coherent states exhibit nonclassical features and can berealized physically as the stationary states of center-of-mass motion of a trappedion [33].

The questions considered here are interrelated by four basic notions: orthogonalpolynomials, oscillator algebra, overcompleteness relation and, finally, hypercon-tractivity.

For the Gegenbauer orthogonal polynomials system on real line, we construct anappropriate oscillator algebra such that these polynomials make up the eigenfunc-tions system of the associated oscillator hamiltonian. The Gegenbauer oscillatoralgebras are free algebras generated by four operators:

• the annihilation operator A and the creation operator A†, which are mu-tually adjoint,

ON GEGENBAUER COHERENT STATE REPRESENTATION 271

• the self-adjoint preservation operator N , and the unity I,satisfying the following commutation relations:

[N,A] = −A, [N, A†] = A†, A A† = Θ(N), A†A = Θ(N + I), (1.2)

where Θ is a real analytical function. By definition, we qualify nonlinear coherentstates those associated with oscillator algebra characterized by (1.2) with Θ beinga nonlinear (non constant) real function.

Notice that, when Θ(s) = σs + τ , we recover the commutation relations ofthe usual harmonic oscillator algebra. When Θ(s) = s(σs + τ), we recover thecommutation relations of the generalized harmonic oscillator algebra of Meixner-type. A detailed study of this class is in progress and will appear in a forthcomingpaper. In the Gegenbauer class, the function Θ is given by

Θ(s) =s(s− 1 + 2β)

4(s + β)(s− 1 + β), s ≥ 1, β > −1

2,

see Section 2 and Section 3 below.Our approach aims at generalizing the pioneering work of Bargmann [11] for

the usual harmonic oscillator. It is well-known that the classical Segal-Bargmanntransform in Gaussian analysis yields a unitary map of the L2 space of the Gaussianmeasure on R onto the space of L2 holomorphic functions of the Gaussian measureon C, see [11, 12, 22, 26]. Later on, based on the work by Accardi-Bozejko [1],Asai [6] has extended the Segal-Bargmann transform to non-Gaussian cases. Thecrucial point is the introduction of a coherent vector as a kernel function in such away that a transformed function, which is a holomorphic function on a certain do-main, becomes a power series expression. Along this line, Asai-Kubo-Kuo [9] haveconsidered the case of the Poisson measure compared with the case of the Gauss-ian measure. More recently, Asai [7, 8] has constructed a Hilbert space of analyticL2 functions with respect to a more general family and give examples includingLaguerre, Meixner and Meixner-Pollaczek polynomials. However, the Gegenbauerpolynomials, and the overcompleteness property are beyond their scope. To ourknowledge, these issues have not been systematically (and intrinsically) addressedin the literature before. In fact, the representations of coherent states are usuallydeveloped in Bosonic Bargmann Fock model. To our opinion this is rather un-natural, and the construction presented in this paper is directly connected to thebeta-type distribution.

The present paper is organized as follows. In Section 2, we consider the Hilbertspace of square integrable functions L2(R, µβ) in which the normalized Gegenbauerpolynomial system constitutes an orthonormal basis. Using the Poisson kernel, wedefine the generalized Fourier transform for this system of polynomials, which al-lows introducing the position and momentum operators. Then, considering thegiven L2-space as a realization of the Fock space, the creation and annihilationoperators can be standardly constructed. Together with the standard number op-erator in this Fock space, they satisfy commutation relations that generalize theHeisenberg relations and generate a Lie algebra that we naturally call the Gegen-bauer oscillator algebra Aβ . In Section 3, following [1] we explicit an equivalentirreducible unitary representation of Aβ on the basis of an adapted one-mode in-teracting Fock space. Sections 4, 5 and 6 are devoted to the true subject of this


paper. We shall first define a class of nonlinear coherent vectors and we studythe associated Bargmann representations. Secondly, we shall derive their over-completeness relation. It turns out that the constructed measure determining apartition of unity for these family of coherent vectors is not always positive. Weprove that this is intimately connected, up to a conjecture, to the parameter ap-pearing in the beta distribution. In particular, we found the result proved byBozejko in a private communication [17] asserting that the answer is negative forthe arcsine case. Finally, in Section 6, to illustrate our main results, we give aspecific example of complex hypercontractivity property for an “ultraspherical”Hamiltonian.

2. The Gegenbauer Oscillator Algebra

2.1. Gegenbauer polynomials. Let µβ be the beta-type distribution with pa-rameter β > − 1

2 given by

µβ(dx) =1√π

Γ(β + 1)Γ(β + 1

2 )(1− x2)β− 1

2 χ]−1,1[(x)dx, (2.1)

where Γ is the gamma Euler function and χ]−1,1[ is the indicator function of ]−1, 1[.The following three important distributions are particular cases of (2.1).

• µ 12(dx) =

12

χ]−1,1[(x) dx : the uniform distribution;

• µ0(dx) =1π

1√1− x2

χ]−1,1[(x) dx : the arcsine distribution;

• µ1(dx) =2π

√1− x2 χ]−1,1[(x) dx : the famous Wigner semi-circle distri-

bution occurring in random matrices and free probability.As easily seen, µβ has finite moments of all orders and the linear span of mono-

mials xn, n = 0, 1, · · · , is dense in L2(µβ) := L2(R, µβ). The inner product isdefined by

〈f, g〉L2(µβ) :=∫

Rf(x)g(x)µβ(dx) , f, g ∈ L2(µβ).

From [10, 28], we recall the following useful background. The Gram-Schmidtprocedure applied to xn, n = 0, 1, 2, · · · gives a complete system Pn,β∞n=0 oforthogonal polynomials such that Pn,β is of degree n with leading coefficient 1 andgiven explicitly by

Pn,β(x) =Γ(β)n!

2nΓ(n + β)Gn,β(x) , n ≥ 0,

where Gn,β is the classical ultraspherical Gegenbauer polynomial [32] defined by

Gn,β(x) =(−1)n

2n

Γ(β + 12 )Γ(n + 2β)

Γ(2β)Γ(n + β) + 12

(1− x2)12−β

n!Dn

x

[(1− x2)n+β− 1

2

].

From the general theory of orthogonal polynomials, it is well known, (see [21]),that there exist two sequences of real numbers αn,β ∈ R, ωn,β ≥ 0, so-called theSzego-Jacobi parameters such that the following relations hold: for n ≥ 0,

(x− αn,β)Pn,β(x) = Pn+1,β(x) + ωn,βPn−1,β(x), (2.2)


〈Pn,β , Pm,β〉L2(µβ) = (ω1,β ω2,β · · ·ωn,β) δn,m,

where δn,m is the Kronecker symbol and by convention

ω0,β = 1 , P−1,β = 0 , P0,β = 1.

Explicitly, we have

ωn,β =n(n− 1 + 2β)

4(n + β)(n− 1 + β), n ≥ 1 (2.3)

and, from the fact µβ is a symmetric measure,

αn,β = 0, n ≥ 0.

2.2. Gegenbauer oscillator algebra. Denote Q0,β(x) = 1 and

Qn,β(x) = (ω1,β ω2,β · · ·ωn,β)−12 Pn,β(x) , n ≥ 1.

Then Qn,β∞n=0 is a complete orthonormal system in L2(µβ) and the recurrencerelation (2.2) becomes

xQn,β(x) = Λn,βQn+1,β(x) + Λn−1,βQn−1,β(x) , n ≥ 1 (2.4)

withΛn,β := (ωn+1,β)

12 , n ≥ 0.

The relation (2.4) indicates a manner by which the position operator Xβ on L2(µβ),acts on the basis elements Qn,β∞n=0:

(XβQn,β)(x) := xQn,β(x) = Λn,βQn+1,β(x) + Λn−1,βQn−1,β(x) , n ≥ 0. (2.5)

Now we want to define the momentum operator Pβ on L2(µβ). For this end weuse the Poisson kernel κz

β ∈ L2(µβ)⊗ L2(µβ) defined by

κzβ(x, y) :=

∞∑n=0

znQn,β(x)Qn,β(y) , z ∈ C.

We define the integral kernel operator Kβ,z on L2(µβ) by

(Kβ,zφ)(y) :=∫

Rκz

β(x, y)φ(x)µβ(dx).

It is noteworthy that Kβ,i is a unitary operator on L2(µβ) (see [41, 42, 43]) and

(Kβ,i)−1 = [Kβ,i]

∗ = Kβ,−i. (2.6)

The unitary operators Kβ,i and Kβ,−i are called the generalized Fourier transformand inverse Fourier transform, respectively. We shall denote Kβ,i simply by Kβ .

We define the momentum operator Pβ on L2(µβ) by

Pβ = K−1β XβKβ .

The energy operator is then defined by

Hβ = X2β + P2

β . (2.7)

Proposition 2.1. The operators Xβ, Pβ and Hβ act on the basis elements ofL2(µβ) by

(1) XβQn,β = Λn,βQn+1,β + Λn−1,βQn−1,β;


(2) PβQn,β = i [−Λn,βQn+1,β + Λn−1,βQn−1,β ];(3) HβQn,β = 2 (ωn,β + ωn+1,β) Qn,β.

Proof. The statement (1) follows directly from the definition of Xβ . We easilyverify that

Kβ(z)Qn,β = znQn,β , z ∈ C, n ≥ 0.

Thus, by using (2.6), one calculate

PβQn,β = K−1β [Xβ (inQn,β)]

= inK−1β [Λn,βQn+1,β + Λn−1,βQn−1,β ]

= i [−Λn,βQn+1,β + Λn−1,βQn−1,β ] .

This proves (2). The identity (3) follows immediately from (1), (2) and (2.7). ¤

It is noteworthy that relation (3) tells us that the basis vectors are eigenfunctionsof the self-adjoint operator Hβ .

Definition 2.2. The creation and annihilation operators are defined as follows

a†β =12

(Xβ + iPβ) , aβ =12

(Xβ − iPβ) .

Proposition 2.3.(a†β

)∗= aβ , a†βQn,β = Λn,βQn+1,β , aβQn,β = Λn−1,βQn−1,β ;

[aβ ,a†β

]= Θβ (nβ + iβ)−Θβ (nβ) ; (2.8)

[nβ ,a†β

]= a†β , [nβ ,aβ ] = −aβ , (2.9)

where nβ is the standard number operator acting on basis vectors by

nβQn,β = nQn,β , n ≥ 0,

iβ is the identity operator on L2(µβ) and Θβ is the function given by

Θβ(s) =s(s− 1 + 2β)

4(s + β)(s− 1 + β). (2.10)

Proof. A straightforward verification. ¤

The right hand side of (2.8) is uniquely determined by the spectral theorem.In fact, (2.8) can be regarded as a generalization of the usual boson Fock CCR.The label ∗ means the conjugation operation and operators are represented by thesame notations as their closures.

Definition 2.4. The Lie algebra generated by the operators a†β , aβ , nβ , iβ withcommutation relations (2.8) and (2.9) is called the Gegenbauer oscillator algebraand will be denoted Aβ .


3. One-mode Interacting Fock Space Representation

In this Section we shall give a unitary equivalent irreducible Fock representationof the algebra Aβ . Let us consider the Hilbert space K to be the complex numberswhich, in physical language, corresponds to a 1-particle space in zero space-timedimension. In this case, for each n ∈ N, also K⊗n is 1-dimensional, so we identifyit to the multiples of a number vector denoted by Φ+(n). The pre-scalar producton K⊗n can only have the form:

〈z, w〉⊗n := λn z w, z, w ∈ C,

where the λn’s are positive numbers.According to our setting, we define the sequence λβ = λn,β∞n=0 by

λn,β := ω0,β ω1,β · · ·ωn,β , n ≥ 0.

From (2.3), λn,β can be rewritten explicitly

λn,β =βΓ(n + 2β)Γ(β)2n!

22n(n + β)Γ(2β)Γ(n + β)2, n ≥ 0. (3.1)

Definition 3.1. The beta-type one-mode interacting Fock space, denoted Γβ , isthe Hilbert space given by taking quotient and completing the orthogonal directsum ∞⊕

n=0

(K⊗n, 〈 · , · 〉⊗n,β

)

where 〈 · , · 〉⊗n,β ≡ 〈 · , · 〉⊗n with the choice (3.1).

Denote Φ0 = Φ+(0) the vacuum vector and, for n ≥ 1, Φn = Φ+(n). For two

elements Φ =∞∑

n=0

anΦn, Ψ =∞∑

n=0

bnΦn in Γβ , we have

〈Φ, Ψ〉Γβ=

∞∑n=0

λn,βanbn. (3.2)

The creation operator is the densely defined operator A†β on Γβ satisfying

A†β : Φn 7−→ Φn+1, n ≥ 0.

The annihilation operator Aβ is given, according to the scalar produces (3.2) asthe adjoint of A†β , by AβΦ0 = 0 and

AβΦn+1 =λn+1,β

λn,βΦn, n ≥ 0. (3.3)

Hence, for any n ≥ 0,

AβA†β(Φn) =λn+1,β

λn,βΦn, A†βAβ(Φn) =

λn,β

λn−1,βΦn

and the following relations arise

AβA†β =λNβ+Iβ

λNβ

, A†βAβ =λNβ

λNβ−Iβ

, (3.4)


where Iβ is the identity operator on Γβ , Nβ is the number operator satisfyingNβΦn = nΦn, n ≥ 0, and the right hand side of (3.4) is uniquely determined bythe spectral theorem.

Proposition 3.2. The Lie algebra generated by A†β, Aβ, Nβ, Iβ gives rise to aunitary equivalent irreducible representation of the Gegenbauer oscillator algebraAβ.

Proof. From the relations (2.3), (3.1) and (3.4) we read, for any n ≥ 0,

[Aβ , A†β ]Φn =(

λn+1,β

λn,β− λn,β

λn−1,β

)Φn

= (ωn+1,β − ωn,β) Φn

= [Θβ(Nβ + Iβ)−Θβ(Nβ)] Φn,

where the last equality is in the sense of functional calculus and Θβ is the functiondefined in (2.10). In a similar way we found

[Nβ , A†β ]Φn = A†βΦn, [Nβ , Aβ ]Φn = −AβΦn, ∀n ≥ 0.

The irreducibility property is obvious from the completeness of the family Φn∞n=0

in Γβ . ¤

As it is expected, we want the Fock space representation of the Gegenbaueroscillator algebra to be unitary equivalent to the one in the L2(µβ)-space. Thisquestion is answered by Accardi-Bozejko isomorphism [1].

Now we state the result of Accardi and Bozejko : there exists a unitary isomor-phism Uβ : Γβ −→ L2(µβ) satisfying the following relations:

(1) UβΦ0 = 1;

(2) Uβ A†β U−1β Qn,β = Λn,βQn+1,β ;

(3) Uβ (Aβ + A†β) U−1β Qn,β = x Qn,β .

From the above relations (1)-(3) we observe that Accardi-Bozejko isomorphism isuniquely determined by the correspondences

Φ0 7−→ Q0,β ;

Φn 7−→ (Λ0,β Λ1,β · · ·Λn−1,β) Qn,β , n ≥ 1. (3.5)Moreover, the unitary equivalence between the two representations of the Gegen-bauer oscillator algebra is obtained through the formulas

Uβ A†β U−1β = a†β , Uβ Aβ U−1

β = aβ , Uβ Nβ U−1β = nβ .

4. Bargmann Space and Holomorphic Representation

Let X be a connected open subset of C and νβ

be an absolutely continuouspositive radial measure on X with continuous Radon-Nikodym derivative

Ψβ(|z|2) =ν

β(dz)dz

,


where dz is the Lebesgue measure on C. Our basic Hilbert space is the spaceH(X) ∩ L2(ν

β) of square integrable analytic functions, (H(X) is the frechet space

of all analytic functions in X). Clearly the space H(X)∩L2(νβ) is a closed subspace

of L2(νβ).

In the remainder of this Section, we take the following “Bargmann-Segal” con-dition on ν

β,

〈zn, zm〉L2(νβ) = λn,βδn,m, ∀n, m = 0, 1, 2, · · · (4.1)

Later on, we shall delve a little more in detailed study of the existence of suchpositive radial measure ν

βwith condition (4.1).

Under the assumption (4.1), we denoteHL2β(X) the Hilbert spaceH(X)∩L2(ν

β).

HL2β(X) is the (weighted) Bargmann space associated to the measure µβ . It is

easily seen that point evaluations δz : f 7→ f(z), z ∈ X are continuous and(λn,β)−1/2zn∞n=0 is a complete orthonormal basis for HL2

β(X). Therefore, thereproducing kernel is

Kβ(z, w) =∞∑

n=0

znwn

‖zn‖2HL2β(X)

=∞∑

n=0

znwn

λn,β. (4.2)

The convergence radius of the series

Lβ(|z|2) := Kβ(z, z) =∞∑

n=0

|z|2n

λn,β(4.3)

is R =1√2

, for any β > − 12 . Hence, denoting X√2 = z ∈ C ; |z| < 1√

2, the

space HL2β(X√2) is given by

HL2β(X√2) =

F (z) =

∞∑n=0

anzn, holomorphic on X√2,

∞∑n=0

λn,β |an|2 < ∞

.

The mapSβ : Φn ∈ Γβ 7−→ zn ∈ HL2

β(X√2)

can be uniquely extended to a unitary isomorphism. Moreover, the Lie algebragenerated by Sβ A†β S−1

β , Sβ Aβ S−1β and Sβ Nβ S−1

β gives rise to a unitary equiv-alent irreducible representation of the Gegenbauer oscillator algebra Aβ . Thisrepresentation is called Bargmann (or holomorphic) representation of the Gegen-bauer oscillator algebra.

5. Gegenbauer Coherent States

As indicated in the introduction, we consider in this paper, the Klauder-Gazeautype coherent states, Ωβ(z), which are considered to satisfy the following condi-tions:

• for any z, Ωβ(z) is an eigenvector of Aβ , i.e. AβΩβ(z) = zΩβ(z);• normalization: for any z, ‖Ωβ(z)‖Γβ

= 1;• continuity with respect to the complex number z;


• overcompleteness, i.e., there exist a positive radial measure νβ(dz) =

Ψβ(|z|2)dz for which we have the resolution of identity∫

X√2

|Ωβ(z) 〉〈Ωβ(z)|Ψβ(|z|2)dz = Iβ .

5.1. The Gegenbauer coherent vectors Ωβ(z).

Theorem 5.1. The family of Gegenbauer coherent vectors Ωβ(z) is given by

Ωβ(z) =(Lβ(|z|2)

)−1/2∞∑

n=0

zn

λn,βΦn, z ∈ X√2, (5.1)

where Lβ is defined in (4.3).

Proof. For z ∈ X√2, let Ωβ(z) =∞∑

n=0

anΦn. By using (3.3) we have

AβΩβ(z) =∞∑

n=0

anAβΦn

=∞∑

n=1

anλn,β

λn−1,βΦn−1

=∞∑

n=0

an+1λn+1,β

λn,βΦn.

From the equality AβΩβ(z) = zΩβ(z), we get

an+1 =λn+1,β

λn,βzan, ∀n ≥ 0.

Hence, since λ0,β = 1, we have

an = a0zn

λn,β

and therefore

Ωβ(z) = a0

∞∑n=0

zn

λn,βΦn.

By the normalization property of Ωβ(z), we shall choose a0 ∈ C in such a way that

1 = 〈Ωβ(z), Ωβ(z)〉Γβ= |a0|2

∞∑n=0

|z|2n

λn,β= |a0|2Lβ(|z|2),

which gives |a0| =(Lβ(|z|2)

)1/2. In conclusion, the eigenvectors of Aβ are

Ωβ(z) =(Lβ(|z|2)

)−1/2∞∑

n=0

zn

λn,βΦn, z ∈ X√2

as desired. ¤


Through Accardi-Bozejko isomorphism (3.5) one can define the Gegenbauercoherent vectors Ωβ(z; x) in L2(µβ):

Ωβ(z; x) =(Lβ(|z|2)

)−1/2∞∑

n=0

zn

√λn,β

Qn,β(x), z ∈ X√2, |x| < 1. (5.2)

In the following, for future use, we give representation of Ωβ(z; x) in terms ofhypergeometric functions. Recall that the Jacobi Polynomials with parametersa > −1, b > −1 is defined by

P (a,b)n (x) =

(n + a

n

)2F1

( −n, n + a + b + 1a + 1

∣∣∣∣12

(1− x))

(5.3)

where the hypergeometric function 2F1

(u, vw

∣∣∣∣ t

)is given by

2F1

(u, vw

∣∣∣∣ t

)=

∞∑n=0

(u)n(v)n

(w)n

tn

n!

and, for real number γ, the Pochhammer symbol (γ)n is defined by

(γ)0 = 1 and (γ)n = γ(γ + 1) · · · (γ + n− 1) =Γ(γ + n)

Γ(γ), n ≥ 1.

If we replace the integer n by a real number s > −1 in the left hand side of (5.3)we obtain the Jacobi function

P (a,b)s (x) =

Γ(s + a + 1)Γ(s + 1)Γ(a + 1) 2F1

( −s, s + a + b + 1a + 1

∣∣∣∣12

(1− x))

. (5.4)

It is well known that the diagonal case, a = b = β − 12 , gives the Gegenbauer

polynomials Gn,β(x) up to normalization

Gn,β(x) =(2β)n

(β + 12 )n

P(β− 1

2 ,β− 12 )

n (x)

=(2β)n

n! 2F1

( −n, n + 2ββ + 1

2

∣∣∣∣12

(1− x))

, (5.5)

where, we assume β > 0.For β > 0, from Eq. (3.1) we read

λn,β =(2β)n(2−nn!)(β)n(β + 1)n

. (5.6)

It then follows, from (4.3),

Lβ(|z|2) =∞∑

n=0

(β)n(β + 1)n

(2β)n

(2|z|2)n

n!

= 2F1

(β, β + 1

2β

∣∣∣∣ 2|z|2)

. (5.7)


For β = 0, (arcsine case - Tchebychev polynomials of first kind), we easily have

L0(|z|2) =1

1− 2|z|2 , z ∈ X√2. (5.8)

From (5.3) and (5.5) the normalized Gegenbauer polynomials can be rewritten as

Qn,β(x) =

√(2β)n(n + β)

β 2n n! 2F1

( −n, n + 2ββ + 1

2

∣∣∣∣12

(1− x))

. (5.9)

Now, for β > 0, substituting (5.7), (3.1) and (5.9) in (5.2), we get,

Ωβ(z; x) =[2F1

(β, β + 1

2β

∣∣∣∣ 2|z|2)]−1/2

×2F1

(β+1

2 , β+22

β + 12

∣∣∣∣ 2(x2−1)z2

(1−√2 xz)2

)

(1−√2 xz)β+1. (5.10)

For the case β = 0, we have the generating function∞∑

n=0

tnQn,0(x) =1

1− 2tx + t2, |t| < 1.

Put t =√

2 z. Then from (5.8) we deduce

Ω0(z; x) =(1− 2|z|2)1/2

1− 2√

2 xz + 2z2, z ∈ X√2, |x| < 1.

Summing up the above calculus, the following theorem is proved.

Theorem 5.2. In L2(µβ), the Gegenbauer coherent vectors are give by(1) For β = 0, (the arcsine case),

Ω0(z; x) =(1− 2|z|2)1/2

1− 2√

2 xz + 2z2, z ∈ X√2, |x| < 1.

(2) For β > 0,

Ωβ(z; x) =[2F1

(β, β + 1

2β

∣∣∣∣ 2|z|2)]−1/2

×2F1

(β+1

2 , β+22

β + 12

∣∣∣∣ 2(x2−1)z2

(1−√2 xz)2

)

(1−√2 xz)β+1.

Remark 5.3. By using regularity properties of the hypergeometric series [28] andidentity (5.7), we can easily prove the continuity of the map

X√2 3 z 7−→ Ωβ(z) ∈ Γβ

defined by (5.1).


5.2. Overcompleteness of the Gegenbauer coherent vectors. Now we shallinvestigate the Bargmann-Segal condition (4.1) in Section 4. This leads to theproblem of constructing a positive radial measure ν

β(dz) = Ψβ(|z|2) dz in the

partition of unity∫

X√2

|Ωβ(z)〉〈Ωβ(z)| νβ(dz) = Iβ =

∞∑n=0

∣∣∣∣∣Φn√λn,β

⟩⟨Φn√λn,β

∣∣∣∣∣ (5.11)

where, for vectors u, v, x, the one-rank projection |u〉〈v| is defined by |u〉〈v|x =〈v, x〉u.

For z ∈ X√2, write z = reiθ with 0 ≤ r < 1√2, 0 ≤ θ ≤ 2π, then

∞∑n=0


⟩⟨Φn√λn,β

∣∣∣∣∣ =∫

X√2

Ψβ(|z|2) |Ωβ(z)〉〈Ωβ(z)| dz

=∞∑

m=0

∞∑n=0

1

2λn,βλm,β

∫ 1/√

2

0

(Ψβ(r2)Lβ(r2)

)rn+md(r2)

×∫ 2π

0

ei(n−m)θdθ

|Φn〉〈Φm|

=∞∑

n=0

π

λn,β

∫ 1/2

0

xn

(Ψβ(x)Lβ(x)

)dx


⟩⟨Φn√λn,β

∣∣∣∣∣

where we marked the change of variable x = r2. Put Vβ(x) =Ψβ(x)Lβ(x)

, one can

deduce that the overcompleteness of the family of Gegenbauer coherent vectors isequivalent to the classical moment problem:

∫ 1/2

0

xnVβ(x)dx =λn,β

π, n = 0, 1, 2, · · · (5.12)

Now we are in position to prove our two main theorems.

Theorem 5.4. The radial measure ν0 in the partition of unity satisfied by theTchebychev (of the first kind) coherent vectors Ω0(z)z∈X√2

is given by

ν0(dz) =1− |z|21− 2|z|2

[8π

δ0(1− 2|z|2)− 4π

δ0(2|z|2)]

dz, z ∈ X√2. (5.13)

Proof. We recall that

Ω0(z) =[L0(|z|2)

]−1/2∞∑

n=0

zn

λnΦn

with

L0(|z|2) =1− |z|21− 2|z|2 and λn ≡ λn,0 =

1 if n = 0

21−2n otherwise


In this case, after changing t = 2x, (5.12) becomes∫ 1

0

tnV0(t

2)dt =

2π if n = 0

4π if n ≥ 1

The solution of this problem is obviously

V0(x) =8π

δ0(2x− 1)− 4π

δ0(2x), x =t

2.

Finally, from the relation W0(x) = L0(x)V0(x) with x = |z|2, we come to thedesired identity (5.13). ¤Theorem 5.5. Let β > 0. The radial measure ν

βin the partition of unity satisfied

by the Gegenbauer coherent vectors Ωβ(z)z∈X√2is given by

νβ(dz) =

2Γ(β + 1)√π Γ(β + 1

2 ) 2F1

(β, β + 1

2β

∣∣∣∣ 2|z|2)

×[

(1− β)|z|4β−2P(0,2β−1)1−β (4|z|2 − 1)

− 2(β + 1)|z|4βP(1,2β)1−β (4|z|2 − 1)

]+ 2δ0(2|z|2 − 1)

dz. (5.14)

Proof. In view of (5.6) we rewrite (5.12) as∫ 1/2

0

xnVβ(x)dx =1π

n!(2β)n

(β)n(β + 1)n

=1π

Γ(n + 1)Γ(n + 2β)Γ(n + β)Γ(n + β + 1)

Γ(β)Γ(β + 1)2nΓ(2β)

.

Hence ∫ 1/2

0

(2x)n πΓ(2β)2Γ(β)Γ(β + 1)

Vβ(x)d(2x) =Γ(n + 1)Γ(n + 2β)

Γ(n + β)Γ(n + β + 1).

Put t = 2x and Uβ(t) =πΓ(2β)

2Γ(β)Γ(β + 1)Vβ(

t

2). Then we get

∫ 1

0

tnUβ(t)dt =Γ(n + 1)Γ(n + 2β)

Γ(n + β)Γ(n + β + 1), n = 0, 1, 2, · · · (5.15)

From the book [25], formula (7.512.4), we have the identity∫ 1

0

(1− x)n+2β−12F1

(β − 1, β + 1

1

∣∣∣∣ x

)dx =

Γ(n + 2β)Γ(n + 2β + 3)Γ(n + β + 2)Γ(n + β)

.

Changing the variable u = 1− x in this identity∫ 1

0

un+2β−12F1

(β − 1, β + 1

1

∣∣∣∣ 1− u

)du =

Γ(n + 2β)Γ(n + 2β + 3)Γ(n + β + 2)Γ(n + β)

. (5.16)

From (5.4) we know that

2F1

(β − 1, β + 1

1

∣∣∣∣12

(1− x))

= P(0,2β−1)1−β (x),


then (5.16) becomes∫ 1

0

un+2β−1P(0,2β−1)1−β (1− u)du =

Γ(n + 1)Γ(n + 2β)(n + β + 1)Γ(n + β)Γ(n + β + 1)

.

On the other hand, from the identity

P (a,b)s (1) =

Γ(s + a + 1)Γ(s + 1)Γ(a + 1)

,

we have P(0,2β−1)1−β (1) = 1. Thus

limu→1−

[uβ−1P

(0,2β−1)1−β (2u− 1)

]= 1.

The derivative of the hypergeometric function 2F1 is given by (see [32])

d

dz

[2F1

(a, bc

∣∣∣∣ z

)]=

ab

c2F1

(a + 1, b + 1

c + 1

∣∣∣∣ z

),

we then obtaind

du

[P

(0,2β−1)1−β (2u− 1)

]= (β + 1)P (1,2β)

−β (2u− 1). (5.17)

Now defineφβ(u) := − d

du

[uβ−1P

(0,2β−1)1−β (2u− 1)

]

= (1− β)uβ−2P(0,2β−1)1−β (2u− 1)− uβ−1 d

du

[P

(0,2β−1)1−β (2u− 1)

]. (5.18)

Substituting (5.17) in (5.18) we get

φβ(u) = (1− β)uβ−2P(0,2β−1)1−β (2u− 1)− (1 + β)uβ−1P

(1,2β)−β (2u− 1).

Observe that the function φβ verifies∫ 1

u

φβ(y)dy = uβ−1P(0,2β−1)1−β (2u− 1)− 1, (5.19)

then, (5.18) and (5.19) imply∫ 1

0

un+β

(∫ 1

u

φβ(y)dy + 1)

du =Γ(n + 1)Γ(n + 2β)

(n + β + 1)Γ(n + β)Γ(n + β + 1).

A simple integration by parts gives∫ 1

0

un+β+1φβ(u)du + 1 =Γ(n + 1)Γ(n + 2β)

Γ(n + β)Γ(n + β + 1).

Comparing this last equality with (5.15) to get

Uβ(u) = uβ+1φβ(u) + 2δ0(u− 1).

By using the formulaΓ(2β)

πΓ(β)Γ(β + 1)= 22β−1 (see [25]) we obtain

Vβ(t) =Γ(β + 1)√

π 22β−1Γ(β + 12 )

[2β+1tβ+1φβ(2t) + 2δ0(2t− 1)

]

from which, setting t = 2x, we have


Vβ(t) =Γ(β + 1)√

π 22β−1Γ(β + 12 )

[(1− β)22β−1t2β−1P

(0,2β−1)1−β (4t− 1)

−(1 + β)22βt2βP(1,2β)−β (4t− 1) + 2δ0(2t− 1)

].

Taking into account (5.7), the radial measure in the partition of unity (5.11) isgiven by the formula (5.14). ¤

Remark 5.6. In view of Theorem 5.5, for β = 1 we have ν1(dz) = Ψ1(|z|2)dz with

Ψ1(s) =4π

11− 2s

δ0 (2s− 1) .

Clearly ν1 is an absolutely continuous positive measure on X√2, so that the over-completeness property is satisfied, and then, Klauder-Gazeau type coherent statesassociated to tchebychev polynomials of second kind (or for Wigner semi-circledistribution) are well defined.

On the other hand, from (5.13), we should point out that ν0 is not a positivemeasure, and therefore, the overcompleteness relation is not satisfied. So, theKlauder-Gazeau type coherent states associated to tchebychev polynomials of firstkind doesn’t exist. Such singularity property of the arcsine distribution was firstlypointed out by M. Bozejko in [17] and it was our motivation to consider the mainproblem of this paper.

For the general case, by using Maple 8, (package Sumtools [20, 35]), whichcontains summation algorithms for hypergeometric series, one can prove that forβ = 2k−1, k ≥ 0 (k integer) the associated Klauder-Gazeau type coherent statesare well defined. In particular, the answer is positive for Klauder-Gazeau Legendrecoherent states (β = 1/2). Similarly, we prove that Gegenbauer coherent vectorsfails to satisfy the overcompleteness property for β = 2−k+1, k > 2 (k integer).Unfortunately, we have not found a proof at present for the other cases. How-ever, the particular case β = 2k−1, k ≥ 0 is enough for the application we shallinvestigate in the next Section.

6. Complex Hypercontractivity

The quantum mechanical harmonic oscillator is essentially the Weyl represen-tation of the Lie algebra associated to the Euclidean motion group. In Fock-Bargmann model, it can be described by the quadruple [11]

HL2(C, ν), ∂, ∂†,Hwhere

HL2(C, ν) =

f : C→ C, holomorphic, ‖f‖2 :=

(∫

C|f(z)|2ν(dz)

)1/2

< ∞

,

∂f(z) =∂

∂zf(z), ∂†f(z) = zf(z), Hf(z) = z

∂

∂zf(z),

and ν(dz) = 1π e−zzdz is the complex one-dimensional Gaussian measure. They

satisfy the canonical commutation relations (CCR)

[∂, ∂†] = I, [∂, I] = 0, [∂†, I] = 0 (6.1)


and Wigner commutation relations (WCR)

[H, ∂] = −∂, [H, ∂†] = ∂†. (6.2)

The Hamiltonian H = ∂†∂ = z ∂∂z is diagonalized by the orthonormal basis

1√n!

zn∞n=0 and has spectrum 0, 1, 2, · · · . It is remarkable that the semigroupTt := e−tH , t ≥ 0 enjoys the following complex hypercontractivity property: fort satisfying e−2t ≤ p

q , Tt is a contraction from HLp(C) to HLq(C), where, for aninteger p ≥ 1, HLp(C) is the Banach space defined by

HLp(C) =

f : C→ C, holomorphic, ‖f‖p :=

(∫

C|f(z)|pν(dz)

)1/p

< ∞

.

This complex hypercontractivity plays an important role in the study of the Bosonfields theory [27].

In the remainder of this paper we shall study a relativistic analogue for quantummechanical Gegenbauer harmonic oscillator with ultraspherical phase space of onedegree freedom (X√2, νβ

(dz)). According to Theorem 5.5 and Remark 5.6, fromnow on we suppose β = 2k−1, k ≥ 0 (k integer) so that ν

βis a positive measure.

For p ≥ 1, letHLp

β(X√2) =f : X√2 → C, holomorphic, ‖f‖p :=

(∫

X√2

|f(z)|pνβ(dz)

)1/p

< ∞ ,

then HLpβ(X√2) is a Banach space and HL2

β(X√2) is a reproducing kernel Hilbertspace with kernel given by (4.2). Recall that an orthonormal basis for HL2

β(X√2)is

φn,β := (λn,β)−1/2zn, n ≥ 0

.

The geometric symmetry group of X√2 [18] is

G√2 = J SU(1, 1) J−1

where

SU(1, 1) =

A =(

a b

b a

): |a|2 − |b|2 = 1

and J =

( 1√√2

0

0√√

2

).

Consider the following irreducible projective representation of G√2 (see [37])

(Ξβ(X)f) (z) =f(X−1z)

(a− cz)2β+1, X =

(a bc d

)∈ G√2, f ∈ HL2

β(X√2),

where X−1z is defined via the action of G√2 on X√2 given by(

a bc d

)z =

az + b

cz + d.

The matrix Lie algebra [19] of G√2 is generated by

B− = J

(0 11 0

)J−1, B+ = J

(0 i−i 0

)J−1, M = J

(i 00 −i

)J−1


which are essentially Pauli matrices. Then, the associated representation, viadiscrete series [14, 15], is given by

−i Ξ+β :=

d

dt

∣∣∣t=0

(Ξβ(etB+

))

= −i

(√2(2β + 1)z +

√2z2 ∂

∂z+

1√2

∂

∂z

)

−i Ξ−β :=d

dt

∣∣∣t=0

(Ξβ(etB−)

)=√

2(2β + 1)z +√

2z2 ∂

∂z− 1√

2∂

∂z

Ξβ :=d

dt

∣∣t=0

(Ξβ(etM )

)= −i

((2β + 1) + 2z

∂

∂z

).

On HL2β(X√2), Ξ+

β and Ξ−β are understood as position operator and momentumoperator, respectively. Then, the creation and the annihilation operators are de-fined (modulo the factor

√2), respectively, by

∂†β :=1√2

(Ξ+β − iΞ−β ) = 2(2β + 1)z + 2z2 ∂

∂z

∂β :=1√2

(Ξ+β + iΞ−β ) =

∂

∂z.

Set N = z∂

∂zwe get

[∂β , ∂†β ] = 2(2β + 1) + 2(2β + 1)N, [N, ∂β ] = −∂β , [N, ∂†β ] = ∂†β .

Accordingly HL2β(X√2), ∂β , ∂†β , N may be viewed as the Gegenbauer harmonic

oscillator on X√2. It is noteworthy that ∂β , ∂†β does not satisfy the CCR (6.1),while ∂β , ∂†β , N satisfy the WCR (6.2).

In the present context, the Hamiltonian is defined by

Hβ = ∂†β∂β = 2(2β + 1)z∂

∂z+ 2z2 ∂2

∂z2.

By direct computation, we have

Hβφn,β = 2n(n + 2β)φn,β , n = 0, 1, 2, · · ·Then, Hβ is closely related to the ultraspherical operator

Tβ := −2(1− x2)d2

dx2+ (2β + 1)x

d

dx

acting on L2(µβ). It is well-known that Tβ is diagonalized by Qn,β :

TβQn,β = 2n(n + 2β)Qn,β ,

so that, Hβ and Tβ have the same spectrum.The map

Υβ : φn,β ∈ HL2β(X√2) 7−→ Qn,β ∈ L2(µβ)

can be uniquely extended to a unitary isomorphism denoted Υβ again. We seethat, under Υβ , Tβ goes over into Hβ , namely, Hβ and Tβ are intertwined by Υβ :

Hβ = Υ−1β Tβ Υβ .

Let P = P t = e−tHβ , t > 0 be the semigroup generated by Hβ . Motivatedby the applications indicated in [27] we prove the following theorem.


Theorem 6.1. If e−βt ≤ 2−1/8, then P t is a contraction from HL2β(X√2) to

HL4β(X√2).

Proof. Recall thatφn,β = (λn,β)−1/2zn; n ≥ 0

is an orthonormal basis for

HL2β(X√2). Moreover, clearly P tφn,β = e−2n(n+2β)tφn,β . Then, for f(z) =∑∞

n=0 γnφn,β ∈ HL2β(X√2), we have

(P tf

)(z) =

∞∑n=0

γne−2n(n+2β)tφn,β .

Therefore, one can estimate

‖P tf‖44 =∫

X√2

∣∣(P tf)

(z)∣∣4 νβ(dz) =

∫

X√2

((P tf

)(z)(P tf) (z)

)2

νβ(dz)

=∫

X√2

∞∑n1=0

∞∑n2=0

e−2(n21+n2

2)te−4βt(n1+n2)γn1γn2φn1,βφn2,β

2

νβ(dz)

=∫

X√2

∞∑n1,n2,n3,n4=0

e−2t(P4

k=1 n2k)e−4βt(

P4k=1 nk)γn1γn2γn3γn4

× φn1,βφn2,βφn3,βφn4,βνβ(dz)

=∫

X√2

∞∑n1,n2,n3,n4=0

e−2t(P4

k=1 n2k)e−4βt(

P4k=1 nk)γn1γn2γn3γn4

× (λn1,βλn2,βλn3,βλn4,β)−1/2z(n1+n2)(z)(n3+n4) νβ(dz)

Then, by using the obvious inequality e−2t(P4

k=1 n2k) < 1 and the assumption e−βt ≤

2−1/8, we get

‖P tf‖44 ≤∞∑

n=0

e−8βtn∑

n1+n2=n

∑n3+n4=n

γn1γn2γn3γn4

× (λn1,βλn2,βλn3,βλn4,β)−1/2λn,β

=∞∑

n=0

e−8βtn

∣∣∣∣∣∑

n1+n2=n

γn1γn2

(λn,β

λn1,βλn2,β

)1/2∣∣∣∣∣

2

≤∞∑

n=0

(12

)n

∑n1+n2=n

n!n1! n2!

((n1 + β)(n2 + β)Γ(2β)Γ(n + 2β)

(n + β)βΓ(β)Γ(n1 + 2β)Γ(n2 + 2β)

)

×(

Γ(n1 + β)Γ(n2 + β)Γ(β)Γ(n + β)

)2( ∑

n1+n2=n

|γn1γn2 |2)

≤∞∑

n=0

( ∑n1+n2=n

n!n1! n2!

(12

)n1(

12

)n2)( ∑

n1+n2=n

|γn1γn2 |2)

= ‖f‖42.This completes the proof. ¤


7. Concluding Remarks

In the present paper we have defined a new type of an oscillator for which theclass of Gegenbauer polynomials play the same role as the Hermite polynomialsplay for standard boson oscillator. By solving the appropriate classical momentproblem, we defined a distinguished set of coherent states of the Klauder-Gazeautype. Moreover, to illustrate our theory, we developed an example of complexhypercontractivity property for an ultraspherical Hamiltonian naturally associatedto the family of orthogonal polynomials in consideration.

Unfortunately, the class of radial measure given by Eq. (5.14) is rather com-plicated in the general case. A complete description of the Gegenbauer coherentstates of Klauder-Gazeau type in terms of the parameter appearing in the betadistribution is still open problem. A second open problem is indicated in Sec-tion 6 concerning the complex hypercontractivity property for quantum mechani-cal Gegenbauer harmonic oscillator with ultraspherical phase space of one degreefreedom (X√2, νβ

(dz)). The research in this line is in progress.

Acknowledgements. I have profited very much from helpful discussions withProfessor M. Bozejko. I gratefully acknowledges the kind hospitality and the stim-ulating atmosphere of Bozejko group at WrocÃlaw University during my visit of theInstitute of Mathematics Martch/April 2008 where part of this work was com-pleted. I wishes to thank N. Demni and N. Asai for bringing to my attention goodRefs. I also thank I. Dermoul for telling me about the paper [33] which indicatesa physical model for the class of coherent states considered in our present pa-per. This work was supported by the Research Project “Bilateral Polish-TunisianGovernmental Co-operation 07-09”. I am also grateful for the support.

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Abdessatar Barhoumi: Department of Mathematics, Higher School of Sci. and Tech.of Hammam-Sousse, Sousse University, Sousse, Tunisia

E-mail address: [email protected]

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