REPRESENTATIONS OF THE GEGENBAUER OSCILLATOR ALGEBRA AND THE OVERCOMPLETENESS OF SEQUENCES OF NONLINEAR COHERENT STATES

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1 Communications on Stochastic Analysis Vol. 3, No. ( Serials Publications REPRESENTATIONS OF THE GEGENBAUER OSCILLATOR ALGEBRA AND THE OVERCOMPLETENESS OF SEQUENCES OF NONLINEAR COHERENT STATES ABDESSATAR BARHOUMI* Abstract. The main purpose of this paper is to investigate a generalized oscillator algebra, naturally associated to the Gegenbauer polynomials and a related class of nonlinear coherent vectors. We derive their overcompleteness relation, in so doing, the partition of unity in terms of the eigenstates of the sequences of coherent vectors is established. It turns out that, in making up such relation, the positivity of the obtained measure is intimately connected, up to a conjecture, to the parameter appearing in the beta distribution. An example of complex hypercontractivity property for an ultraspherical Hamiltonian is developed to illustrate our theory. 1. Introduction Coherent states representation [36] are ubiquitous in the mathematical physics literature. Yet there seems to be a lack of general theory in the context of representations of nonlinear coherent states. This paper is an attempt to partially fill this gap. The original coherent states based on the Heisenberg-Weyl group has been extended for a number of Lie groups with square integrable representations [4], and they have many applications in quantum mechanics [4]. In particular, they are used as bases of coherent states path integrals [4] or dynamical wavepackets for describing the quantum systems in semiclassical approximations [31]. This framework has been given in a general and elegant mathematical form through the work of Perelomov [37]. Many definitions of coherent states exist. The first one, so-called Klauder- Gazeau type [3], defines the coherent states as the eigenstates of the annihilation operator 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 states in 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 Ω(z is smooth, and (iii the system is overcomplete; i.e. Ω(z Ω(z ν(dz = I. (1.1 X Mathematics Subject Classification. Primary 46L53; Secondary 33D45, 44A15. Key words and phrases. Bargmann transform, beta-type distribution, coherent state, Gegenbauer oscillator algebra, hypercontractivity, one-mode interacting Fock space, orthogonal polynomials, overcompleteness relation, Poisson kernel. * Research supported by The Bilateral Polish-Tunisian Governmental Co-operation

2 7 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 z are given by Ω(z = W (zφ, z X. Here the annihilation operator A and the creation operator A are mutually adjoint and Φ is the Fock vacuum. A third definition is based on the Heisenberg uncertainty relation with the position operator Q and momentum operator P given, as usual, by Q = i (A + A and P = i (A A, the coherent states defined above have the minimum-uncertainty value Q P = 1/ and maintain this relation in time (temporal stability of coherent states [5]. Let us stress that coherent states have two important properties. First, they are 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 overcomplete set of states in the interacting Fock space. In fact it is well known that they form a highly overcomplete set in the sense that there are much smaller subsets of coherent states which are also overcomplete. Using them one can express an arbitrary state as a line integral of coherent states [4]. It is therefore clear that knowing that a set of coherent states is overcomplete is not only of theoretical interest; it is also of practical interest in the sense that we are encouraged to search for resolutions of the identity which will make possible the expansion of an arbitrary state in terms of these coherent states. Sometimes it is not easy to find a resolution of the identity and weaker concepts are also sufficient (for example the concept of frames in wavelets. But it is clear that a prerequisite for going down that route is the question of completeness. In the present paper, considering only the first of the above mentioned definitions, we are concerned by a class of nonlinear coherent states: the Gegenbauer coherent states. These coherent states exhibit nonclassical features and can be realized physically as the stationary states of center-of-mass motion of a trapped ion [33]. The questions considered here are interrelated by four basic notions: orthogonal polynomials, oscillator algebra, overcompleteness relation and, finally, hypercontractivity. For the Gegenbauer orthogonal polynomials system on real line, we construct an appropriate oscillator algebra such that these polynomials make up the eigenfunctions system of the associated oscillator hamiltonian. The Gegenbauer oscillator algebras are free algebras generated by four operators: the annihilation operator A and the creation operator A, which are mutually adjoint,

3 ON GEGENBAUER COHERENT STATE REPRESENTATION 71 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. where Θ is a real analytical function. By definition, we qualify nonlinear coherent states those associated with oscillator algebra characterized by (1. with Θ being a nonlinear (non constant real function. Notice that, when Θ(s = σs + τ, we recover the commutation relations of the usual harmonic oscillator algebra. When Θ(s = s(σs + τ, we recover the commutation relations of the generalized harmonic oscillator algebra of Meixnertype. A detailed study of this class is in progress and will appear in a forthcoming paper. In the Gegenbauer class, the function Θ is given by Θ(s = s(s 1 + β 4(s + β(s 1 + β, s 1, β > 1, see Section 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-Bargmann transform in Gaussian analysis yields a unitary map of the L space of the Gaussian measure on R onto the space of L holomorphic functions of the Gaussian measure on C, see [11, 1,, 6]. Later on, based on the work by Accardi-Bożejko [1], Asai [6] has extended the Segal-Bargmann transform to non-gaussian cases. The crucial point is the introduction of a coherent vector as a kernel function in such a way that a transformed function, which is a holomorphic function on a certain domain, becomes a power series expression. Along this line, Asai-Kubo-Kuo [9] have considered the case of the Poisson measure compared with the case of the Gaussian measure. More recently, Asai [7, 8] has constructed a Hilbert space of analytic L functions with respect to a more general family and give examples including Laguerre, Meixner and Meixner-Pollaczek polynomials. However, the Gegenbauer polynomials, and the overcompleteness property are beyond their scope. To our knowledge, these issues have not been systematically (and intrinsically addressed in the literature before. In fact, the representations of coherent states are usually developed in Bosonic Bargmann Fock model. To our opinion this is rather unnatural, and the construction presented in this paper is directly connected to the beta-type distribution. The present paper is organized as follows. In Section, we consider the Hilbert space of square integrable functions L (R, µ β in which the normalized Gegenbauer polynomial system constitutes an orthonormal basis. Using the Poisson kernel, we define the generalized Fourier transform for this system of polynomials, which allows introducing the position and momentum operators. Then, considering the given L -space as a realization of the Fock space, the creation and annihilation operators can be standardly constructed. Together with the standard number operator in this Fock space, they satisfy commutation relations that generalize the Heisenberg relations and generate a Lie algebra that we naturally call the Gegenbauer oscillator algebra A β. In Section 3, following [1] we explicit an equivalent irreducible unitary representation of A β on the basis of an adapted one-mode interacting Fock space. Sections 4, 5 and 6 are devoted to the true subject of this

4 7 ABDESSATAR BARHOUMI paper. We shall first define a class of nonlinear coherent vectors and we study the associated Bargmann representations. Secondly, we shall derive their overcompleteness relation. It turns out that the constructed measure determining a partition of unity for these family of coherent vectors is not always positive. We prove that this is intimately connected, up to a conjecture, to the parameter appearing in the beta distribution. In particular, we found the result proved by Bożejko in a private communication [17] asserting that the answer is negative for the arcsine case. Finally, in Section 6, to illustrate our main results, we give a specific example of complex hypercontractivity property for an ultraspherical Hamiltonian.. The Gegenbauer Oscillator Algebra.1. Gegenbauer polynomials. Let µ β be the beta-type distribution with parameter β > 1 given by µ β (dx = 1 Γ(β + 1 π Γ(β + 1 (1 x β 1 χ] 1,1[ (xdx, (.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 (.1. µ 1 (dx = 1 χ (x dx : the uniform distribution; ] 1,1[ µ (dx = 1 1 π χ (x dx : the arcsine distribution; 1 x ] 1,1[ µ 1 (dx = 1 x χ π ] 1,1[ (x dx : the famous Wigner semi-circle distribution occurring in random matrices and free probability. As easily seen, µ β has finite moments of all orders and the linear span of monomials x n, n =, 1,, is dense in L (µ β := L (R, µ β. The inner product is defined by f, g L (µ β := f(xg(xµ β (dx, f, g L (µ β. R From [1, 8], we recall the following useful background. The Gram-Schmidt procedure applied to {x n, n =, 1,, } gives a complete system {P n,β } of orthogonal polynomials such that P n,β is of degree n with leading coefficient 1 and given explicitly by P n,β (x = Γ(βn! n Γ(n + β G n,β(x, n, where G n,β is the classical ultraspherical Gegenbauer polynomial [3] defined by G n,β (x = ( 1n Γ(β + 1 Γ(n + β (1 x 1 β [ ] n Γ(βΓ(n + β + 1 Dx n (1 x n+β 1. n! From the general theory of orthogonal polynomials, it is well known, (see [1], that there exist two sequences of real numbers α n,β R, ω n,β, so-called the Szegö-Jacobi parameters such that the following relations hold: for n, (x α n,β P n,β (x = P n+1,β (x + ω n,β P n 1,β (x, (.

5 ON GEGENBAUER COHERENT STATE REPRESENTATION 73 P n,β, P m,β L (µ β = (ω 1,β ω,β ω n,β δ n,m, where δ n,m is the Kronecker symbol and by convention Explicitly, we have ω,β = 1, P 1,β =, P,β = 1. ω n,β = and, from the fact µ β is a symmetric measure, n(n 1 + β 4(n + β(n 1 + β, n 1 (.3 α n,β =, n... Gegenbauer oscillator algebra. Denote Q,β (x = 1 and Q n,β (x = (ω 1,β ω,β ω n,β 1 P n,β (x, n 1. Then {Q n,β } is a complete orthonormal system in L (µ β and the recurrence relation (. becomes xq n,β (x = Λ n,β Q n+1,β (x + Λ n 1,β Q n 1,β (x, n 1 (.4 with Λ n,β := (ω n+1,β 1, n. The relation (.4 indicates a manner by which the position operator X β on L (µ β, acts on the basis elements {Q n,β } : (X β Q n,β (x := xq n,β (x = Λ n,β Q n+1,β (x + Λ n 1,β Q n 1,β (x, n. (.5 Now we want to define the momentum operator P β on L (µ β. For this end we use the Poisson kernel κ z β L (µ β L (µ β defined by κ z β(x, y := z n Q n,β (xq n,β (y, z C. We define the integral kernel operator K β,z on L (µ β by (K β,z φ(y := κ z β(x, yφ(xµ β (dx. R It is noteworthy that K β,i is a unitary operator on L (µ β (see [41, 4, 43] and (K β,i 1 = [K β,i ] = K β, i. (.6 The unitary operators K β,i and K β, i are called the generalized Fourier transform and inverse Fourier transform, respectively. We shall denote K β,i simply by K β. We define the momentum operator P β on L (µ β by P β = K 1 β X βk β. The energy operator is then defined by H β = X β + P β. (.7 Proposition.1. The operators X β, P β and H β act on the basis elements of L (µ β by (1 X β Q n,β = Λ n,β Q n+1,β + Λ n 1,β Q n 1,β ;

6 74 ABDESSATAR BARHOUMI ( P β Q n,β = i [ Λ n,β Q n+1,β + Λ n 1,β Q n 1,β ]; (3 H β Q n,β = (ω n,β + ω n+1,β Q n,β. Proof. The statement (1 follows directly from the definition of X β. verify that K β (zq n,β = z n Q n,β, z C, n. Thus, by using (.6, one calculate P β Q n,β = K 1 β [X β (i n Q n,β ] We easily = i n K 1 β [Λ n,βq n+1,β + Λ n 1,β Q n 1,β ] = i [ Λ n,β Q n+1,β + Λ n 1,β Q n 1,β ]. This proves (. The identity (3 follows immediately from (1, ( and (.7. It is noteworthy that relation (3 tells us that the basis vectors are eigenfunctions of the self-adjoint operator H β. Definition.. The creation and annihilation operators are defined as follows a β = 1 (X β + ip β, a β = 1 (X β ip β. Proposition.3. a β = aβ, a β Q n,β = Λ n,β Q n+1,β, a β Q n,β = Λ n 1,β Q n 1,β ; [ ] a β, a β = Θ β (n β + i β Θ β (n β ; (.8 [ ] n β, a β = a β, [n β, a β ] = a β, (.9 where n β is the standard number operator acting on basis vectors by n β Q n,β = nq n,β, n, i β is the identity operator on L (µ β and Θ β is the function given by Θ β (s = Proof. A straightforward verification. s(s 1 + β 4(s + β(s 1 + β. (.1 The right hand side of (.8 is uniquely determined by the spectral theorem. In fact, (.8 can be regarded as a generalization of the usual boson Fock CCR. The label means the conjugation operation and operators are represented by the same notations as their closures. Definition.4. The Lie algebra generated by the operators a β, a β, n β, i β with commutation relations (.8 and (.9 is called the Gegenbauer oscillator algebra and will be denoted A β.

7 ON GEGENBAUER COHERENT STATE REPRESENTATION One-mode Interacting Fock Space Representation In this Section we shall give a unitary equivalent irreducible Fock representation of the algebra A β. Let us consider the Hilbert space K to be the complex numbers which, in physical language, corresponds to a 1-particle space in zero space-time dimension. In this case, for each n N, also K n is 1-dimensional, so we identify it to the multiples of a number vector denoted by Φ +(n. The pre-scalar product on K n can only have the form: z, w n := λ n z w, z, w C, where the λ n s are positive numbers. According to our setting, we define the sequence λ β = { } by := ω,β ω 1,β ω n,β, n. From (.3, can be rewritten explicitly = βγ(n + βγ(β n! n, n. (3.1 (n + βγ(βγ(n + β Definition 3.1. The beta-type one-mode interacting Fock space, denoted Γ β, is the Hilbert space given by taking quotient and completing the orthogonal direct sum ( K n,, n,β where, n,β, n with the choice (3.1. Denote Φ = Φ +( the vacuum vector and, for n 1, Φ n = Φ +(n. For two elements Φ = a n Φ n, Ψ = b n Φ n in Γ β, we have Φ, Ψ Γβ = a n b n. (3. The creation operator is the densely defined operator A β on Γ β satisfying A β : Φ n Φ n+1, n. The annihilation operator A β is given, according to the scalar produces (3. as the adjoint of A β, by A βφ = and Hence, for any n, A β Φ n+1 = λ n+1,β Φ n, n. (3.3 A β A β (Φ n = λ n+1,β Φ n, A β A β(φ n = λ n 1,β Φ n and the following relations arise A β A β = λ N β +I β λ Nβ, A β A β = λ N β λ Nβ I β, (3.4

8 76 ABDESSATAR BARHOUMI where I β is the identity operator on Γ β, N β is the number operator satisfying N β Φ n = nφ n, n, and the right hand side of (3.4 is uniquely determined by the spectral theorem. Proposition 3.. The Lie algebra generated by A β, A β, N β, I β gives rise to a unitary equivalent irreducible representation of the Gegenbauer oscillator algebra A β. Proof. From the relations (.3, (3.1 and (3.4 we read, for any n, ( [A β, A β ]Φ λn+1,β n = λ n,β Φ n λ n 1,β = (ω n+1,β ω n,β Φ n = [Θ β (N β + I β Θ β (N β ] Φ n, where the last equality is in the sense of functional calculus and Θ β is the function defined in (.1. In a similar way we found [N β, A β ]Φ n = A β Φ n, [N β, A β ]Φ n = A β Φ n, n. The irreducibility property is obvious from the completeness of the family {Φ n } in Γ β. As it is expected, we want the Fock space representation of the Gegenbauer oscillator algebra to be unitary equivalent to the one in the L (µ β -space. This question is answered by Accardi-Bożejko isomorphism [1]. Now we state the result of Accardi and Bożejko : there exists a unitary isomorphism U β : Γ β L (µ β satisfying the following relations: (1 U β Φ = 1; ( U β A β U 1 β Q n,β = Λ n,β Q n+1,β ; (3 U β (A β + A β U 1 β Q n,β = x Q n,β. From the above relations (1-(3 we observe that Accardi-Bożejko isomorphism is uniquely determined by the correspondences Φ Q,β ; Φ n (Λ,β Λ 1,β Λ n 1,β Q n,β, n 1. (3.5 Moreover, the unitary equivalence between the two representations of the Gegenbauer oscillator algebra is obtained through the formulas U β A β U 1 β = a β, U β A β U 1 β = a β, U β N β U 1 β = n β. 4. Bargmann Space and Holomorphic Representation Let X be a connected open subset of C and ν β be an absolutely continuous positive radial measure on X with continuous Radon-Nikodym derivative Ψ β ( z = ν (dz β, dz

9 ON GEGENBAUER COHERENT STATE REPRESENTATION 77 where dz is the Lebesgue measure on C. Our basic Hilbert space is the space H(X L (ν β of square integrable analytic functions, (H(X is the fréchet space of all analytic functions in X. Clearly the space H(X L (ν β is a closed subspace of L (ν β. In the remainder of this Section, we take the following Bargmann-Segal condition on ν β, z n, z m L (ν β = δ n,m, n, m =, 1,, (4.1 Later on, we shall delve a little more in detailed study of the existence of such positive radial measure ν β with condition (4.1. Under the assumption (4.1, we denote HL β (X the Hilbert space H(X L (ν β. HL β (X is the (weighted Bargmann space associated to the measure µ β. It is easily seen that point evaluations δ z : f f(z, z X are continuous and { 1/ z n } is a complete orthonormal basis for HL β (X. Therefore, the reproducing kernel is K β (z, w = The convergence radius of the series z n w n z n HL β (X = L β ( z := K β (z, z = is R = 1, for any β > 1. Hence, denoting X space HL β (X is given by HL β(x {F = (z = a n z n, holomorphic on X, The map S β : Φ n Γ β z n HL β(x z n w n. (4. z n (4.3 = {z C ; z < 1 a n < }, the can be uniquely extended to a unitary isomorphism. Moreover, the Lie algebra generated by S β A β S 1 β, S β A β S 1 β and S β N β S 1 β gives rise to a unitary equivalent irreducible representation of the Gegenbauer oscillator algebra A β. This representation is called Bargmann (or holomorphic representation of the Gegenbauer oscillator algebra. 5. Gegenbauer Coherent States As indicated in the introduction, we consider in this paper, the Klauder-Gazeau type coherent states, Ω β (z, which are considered to satisfy the following conditions: for any z, Ω β (z is an eigenvector of A β, i.e. A β Ω β (z = zω β (z; normalization: for any z, Ω β (z Γβ = 1; continuity with respect to the complex number z; }.

10 78 ABDESSATAR BARHOUMI overcompleteness, i.e., there exist a positive radial measure ν β (dz = Ψ β ( z dz for which we have the resolution of identity Ω β (z Ω β (z Ψ β ( z dz = I β. X 5.1. The Gegenbauer coherent vectors Ω β (z. Theorem 5.1. The family of Gegenbauer coherent vectors Ω β (z is given by Ω β (z = ( L β ( z 1/ where L β is defined in (4.3. Proof. For z X, let Ω β(z = z n Φ n, z X, (5.1 a n Φ n. By using (3.3 we have A β Ω β (z = = = a n A β Φ n n=1 a n From the equality A β Ω β (z = zω β (z, we get Hence, since λ,β = 1, we have and therefore λ n 1,β Φ n 1 a n+1 λ n+1,β Φ n. a n+1 = λ n+1,β za n, n. a n = a Ω β (z = a z n z n Φ n. By the normalization property of Ω β (z, we shall choose a C in such a way that 1 = Ω β (z, Ω β (z Γβ = a z n = a L β ( z, which gives a = ( L β ( z 1/. In conclusion, the eigenvectors of Aβ are as desired. Ω β (z = ( L β ( z 1/ z n Φ n, z X

11 ON GEGENBAUER COHERENT STATE REPRESENTATION 79 Through Accardi-Bożejko isomorphism (3.5 one can define the Gegenbauer coherent vectors Ω β (z; x in L (µ β : Ω β (z; x = ( L β ( z 1/ z n Q n,β (x, z X, x < 1. (5. λn,β In the following, for future use, we give representation of Ω β (z; x in terms of hypergeometric functions. Recall that the Jacobi Polynomials with parameters a > 1, b > 1 is defined by P n (a,b n + a n, n + a + b + 1 (x = n F 1 1 a + 1 (1 x (5.3 u, v where the hypergeometric function F 1 w t is given by u, v F 1 w t (u n (v n t n = (w n n! and, for real number γ, the Pochhammer symbol (γ n is defined by (γ = 1 and (γ n = γ(γ + 1 (γ + n 1 = Γ(γ + n, n 1. Γ(γ 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 s (a,b Γ(s + a + 1 s, s + a + b + 1 (x = Γ(s + 1Γ(a + 1 F 1 1 a + 1 (1 x. (5.4 It is well known that the diagonal case, a = b = β 1, gives the Gegenbauer polynomials G n,β (x up to normalization G n,β (x = (β n (β + 1 n = (β n n! where, we assume β >. For β >, from Eq. (3.1 we read P (β 1,β 1 n (x F 1 ( n, n + β β (1 x, (5.5 = (β n( n n! (β n (β + 1 n. (5.6 It then follows, from (4.3, L β ( z (β n (β + 1 n ( z n = (β n n! β, β + 1 = F 1 β z. (5.7

12 8 ABDESSATAR BARHOUMI For β =, (arcsine case - Tchebychev polynomials of first kind, we easily have L ( z = 1 1 z, z X. (5.8 From (5.3 and (5.5 the normalized Gegenbauer polynomials can be rewritten as (β n (n + β n, n + β Q n,β (x = β n F 1 1 n! β + 1 (1 x. (5.9 Now, for β >, substituting (5.7, (3.1 and (5.9 in (5., we get, [ ] β, β + 1 1/ Ω β (z; x = F 1 β z ( β+1 F, β+ 1 β + 1 (x 1z (1 xz (1. (5.1 xz β+1 For the case β =, we have the generating function t n 1 Q n, (x =, t < 1. 1 tx + t Put t = z. Then from (5.8 we deduce Ω (z; x = (1 z 1/ 1 xz + z, z X, x < 1. Summing up the above calculus, the following theorem is proved. Theorem 5.. In L (µ β, the Gegenbauer coherent vectors are give by (1 For β =, (the arcsine case, ( For β >, Ω (z; x = (1 z 1/ 1 xz + z, z X, x < 1. Ω β (z; x = [ F 1 ( β, β + 1 β ] 1/ z ( β+1 F, β+ 1 β + 1 (x 1z (1 xz (1. xz β+1 Remark 5.3. By using regularity properties of the hypergeometric series [8] and identity (5.7, we can easily prove the continuity of the map defined by (5.1. X z Ω β(z Γ β

13 ON GEGENBAUER COHERENT STATE REPRESENTATION Overcompleteness of the Gegenbauer coherent vectors. Now we shall investigate the Bargmann-Segal condition (4.1 in Section 4. This leads to the problem of constructing a positive radial measure ν β (dz = Ψ β ( z dz in the partition of unity Φ n Φ Ω β (z Ω β (z ν β (dz = I β = n (5.11 λn,β λn,β X where, for vectors u, v, x, the one-rank projection u v is defined by u v x = v, x u. For z X, write z = reiθ with r < 1, θ π, then Φ n Φ n = Ψ β ( z Ω β (z Ω β (z dz λn,β λn,β = = { m= π { X 1 λ m,β 1/ } e i(n mθ dθ Φ n Φ m π 1/ x n Ψβ (x dx} L β (x ( Ψβ (r L β (r r n+m d(r Φ n Φ n λn,β λn,β where we marked the change of variable x = r. Put V β (x = Ψ β(x, one can L β (x deduce that the overcompleteness of the family of Gegenbauer coherent vectors is equivalent to the classical moment problem: 1/ x n V β (xdx =, n =, 1,, (5.1 π Now we are in position to prove our two main theorems. Theorem 5.4. The radial measure ν in the partition of unity satisfied by the Tchebychev (of the first kind coherent vectors {Ω (z} z X is given by ν (dz = 1 [ z 8 1 z π δ (1 z 4 ] π δ ( z dz, z X. (5.13 Proof. We recall that with Ω (z = [ L ( z ] 1/ z n λ n Φ n L ( z = 1 1 if n = z 1 z and λ n λ n, = 1 n otherwise

14 8 ABDESSATAR BARHOUMI In this case, after changing t = x, (5.1 becomes 1 t n V ( t dt = The solution of this problem is obviously π if n = 4 π if n 1 V (x = 8 π δ (x 1 4 π δ (x, x = t. Finally, from the relation W (x = L (xv (x with x = z, we come to the desired identity (5.13. Theorem 5.5. Let β >. The radial measure ν β in the partition of unity satisfied by the Gegenbauer coherent vectors {Ω β (z} z X is given by Γ(β + 1 β, β + 1 ν β (dz = π Γ(β + 1 F 1 β z {[ (1 β z 4β P (,β 1 1 β (4 z 1 ] } (β + 1 z 4β P (1,β 1 β (4 z 1 + δ ( z 1 dz. (5.14 Proof. In view of (5.6 we rewrite (5.1 as Hence 1/ 1/ x n V β (xdx = 1 π (x n Put t = x and U β (t = 1 = 1 π n!(β n (β n (β + 1 n Γ(n + 1Γ(n + β Γ(n + βγ(n + β + 1 Γ(βΓ(β + 1 n. Γ(β πγ(β Γ(βΓ(β + 1 V Γ(n + 1Γ(n + β β(xd(x = Γ(n + βγ(n + β + 1. t n U β (tdt = πγ(β Γ(βΓ(β + 1 V β( t. Then we get Γ(n + 1Γ(n + β, n =, 1,, (5.15 Γ(n + βγ(n + β + 1 From the book [5], formula (7.51.4, we have the identity 1 (1 x n+β 1 β 1, β + 1 F 1 1 x Γ(n + βγ(n + β + 3 dx = Γ(n + β + Γ(n + β. Changing the variable u = 1 x in this identity 1 u n+β 1 β 1, β + 1 F u Γ(n + βγ(n + β + 3 du = Γ(n + β + Γ(n + β. (5.16 From (5.4 we know that F 1 ( β 1, β (1 x = P (,β 1 1 β (x,

15 ON GEGENBAUER COHERENT STATE REPRESENTATION 83 then (5.16 becomes 1 u n+β 1 P (,β 1 Γ(n + 1Γ(n + β 1 β (1 udu = (n + β + 1Γ(n + βγ(n + β + 1. On the other hand, from the identity we have P (,β 1 1 β (1 = 1. Thus P s (a,b Γ(s + a + 1 (1 = Γ(s + 1Γ(a + 1, [ lim u β 1 P (,β 1 u 1 1 β (u 1 ] = 1. The derivative of the hypergeometric function F 1 is given by (see [3] [ ] d a, b F 1 dz c z = ab a + 1, b + 1 c F 1 c + 1 z, we then obtain d [ ] P (,β 1 1 β (u 1 = (β + 1P (1,β β (u 1. (5.17 du Now define φ β (u := d [ ] u β 1 P (,β 1 1 β (u 1 du = (1 βu β P (,β 1 1 β (u 1 u β 1 d du Substituting (5.17 in (5.18 we get [ ] P (,β 1 1 β (u 1. (5.18 φ β (u = (1 βu β P (,β 1 1 β (u 1 (1 + βu β 1 P (1,β β (u 1. Observe that the function φ β verifies u 1 u φ β (ydy = u β 1 P (,β 1 1 β (u 1 1, (5.19 then, (5.18 and (5.19 imply 1 ( 1 u n+β Γ(n + 1Γ(n + β φ β (ydy + 1 du = (n + β + 1Γ(n + βγ(n + β + 1. A simple integration by parts gives 1 u n+β+1 φ β (udu + 1 = Comparing this last equality with (5.15 to get By using the formula V β (t = from which, setting t = x, we have Γ(n + 1Γ(n + β Γ(n + βγ(n + β + 1. U β (u = u β+1 φ β (u + δ (u 1. Γ(β πγ(βγ(β + 1 = β 1 (see [5] we obtain Γ(β + 1 [ π β 1 β+1 Γ(β + 1 t β+1 φ β (t + δ (t 1 ]

16 84 ABDESSATAR BARHOUMI V β (t = Γ(β + 1 [ π β 1 Γ(β + 1 (1 β β 1 t β 1 P (,β 1 1 β (4t 1 (1 + β β t β P (1,β β (4t 1 + δ (t 1 Taking into account (5.7, the radial measure in the partition of unity (5.11 is given by the formula (5.14. Remark 5.6. In view of Theorem 5.5, for β = 1 we have ν 1 (dz = Ψ 1 ( z dz with Ψ 1 (s = 4 1 π 1 s δ (s 1. Clearly ν 1 is an absolutely continuous positive measure on X, so that the overcompleteness property is satisfied, and then, Klauder-Gazeau type coherent states associated to tchebychev polynomials of second kind (or for Wigner semi-circle distribution are well defined. On the other hand, from (5.13, we should point out that ν is not a positive measure, and therefore, the overcompleteness relation is not satisfied. So, the Klauder-Gazeau type coherent states associated to tchebychev polynomials of first kind doesn t exist. Such singularity property of the arcsine distribution was firstly pointed out by M. Bożejko in [17] and it was our motivation to consider the main problem of this paper. For the general case, by using Maple 8, (package Sumtools [, 35], which contains summation algorithms for hypergeometric series, one can prove that for β = k 1, k (k integer the associated Klauder-Gazeau type coherent states are well defined. In particular, the answer is positive for Klauder-Gazeau Legendre coherent states (β = 1/. Similarly, we prove that Gegenbauer coherent vectors fails to satisfy the overcompleteness property for β = k+1, k > (k integer. Unfortunately, we have not found a proof at present for the other cases. However, the particular case β = k 1, k is enough for the application we shall investigate in the next Section. 6. Complex Hypercontractivity The quantum mechanical harmonic oscillator is essentially the Weyl representation of the Lie algebra associated to the Euclidean motion group. In Fock- Bargmann model, it can be described by the quadruple [11] {HL (C, ν,,, H} where { ( } 1/ HL (C, ν = f : C C, holomorphic, f := f(z ν(dz <, C ]. f(z = z f(z, f(z = zf(z, Hf(z = z z f(z, and ν(dz = 1 π e zz dz is the complex one-dimensional Gaussian measure. They satisfy the canonical commutation relations (CCR [, ] = I, [, I] =, [, I] = (6.1

17 ON GEGENBAUER COHERENT STATE REPRESENTATION 85 and Wigner commutation relations (WCR [H, ] =, [H, ] =. (6. The Hamiltonian H = = z z is diagonalized by the orthonormal basis { 1 n! z n } and has spectrum {, 1,, }. It is remarkable that the semigroup {T t := e th, t } enjoys the following complex hypercontractivity property: for t satisfying e t p q, T t is a contraction from HL p (C to HL q (C, where, for an integer p 1, HL p (C is the Banach space defined by { ( } 1/p HL p (C = f : C C, holomorphic, f p := f(z ν(dz p <. C This complex hypercontractivity plays an important role in the study of the Boson fields theory [7]. In the remainder of this paper we shall study a relativistic analogue for quantum mechanical Gegenbauer harmonic oscillator with ultraspherical phase space of one degree freedom (X, ν (dz. According to Theorem 5.5 and Remark 5.6, from β now on we suppose β = k 1, k (k integer so that ν β is a positive measure. For p 1, let HL p β (X = ( 1/p f : X C, holomorphic, f p := f(z p ν β (dz <, then HL p β (X is a Banach space and HL β (X is a reproducing kernel Hilbert space with kernel given by (4.. Recall that an orthonormal basis for HL β (X is { } φ n,β := 1/ z n, n. The geometric symmetry group of X [18] is where SU(1, 1 = { ( a b A = b a G = J SU(1, 1 J 1 } : a b = 1 X and J = ( 1 Consider the following irreducible projective representation of G (see [37] (Ξ β (Xf (z = f(x 1 z a b (a cz β+1, X = G c d, f HL β(x, where X 1 z is defined via the action of G on X given by a b z = az + b c d cz + d. The matrix Lie algebra [19] of G is generated by B 1 = J J 1, B + i = J J 1, 1 i M = J ( i i J 1.

18 86 ABDESSATAR BARHOUMI which are essentially Pauli matrices. Then, the associated representation, via discrete series [14, 15], is given by i Ξ + β := d ( (β t= (Ξ β (e tb+ = i + 1z + z dt z + 1 z t= (Ξ β (e tb = (β + 1z + z z 1 i Ξ β := d dt Ξ β := d dt t= ( Ξβ (e tm = i ( (β z z On HL β (X, Ξ+ β and Ξ β are understood as position operator and momentum operator, respectively. Then, the creation and the annihilation operators are defined (modulo the factor, respectively, by. β := 1 (Ξ + β iξ β = (β + 1z + z z β := 1 (Ξ + β + iξ β = z. z Set N = z z we get [ β, β ] = (β (β + 1N, [N, β] = β, [N, β ] = β. Accordingly {HL β (X, β, β, N} may be viewed as the Gegenbauer harmonic oscillator on X. It is noteworthy that { β, β } does not satisfy the CCR (6.1, while { β, β, N} satisfy the WCR (6.. In the present context, the Hamiltonian is defined by By direct computation, we have H β = β β = (β + 1z z + z z. H β φ n,β = n(n + βφ n,β, n =, 1,, Then, H β is closely related to the ultraspherical operator T β := (1 x d d + (β + 1x dx dx acting on L (µ β. It is well-known that T β is diagonalized by {Q n,β } : T β Q n,β = n(n + βq n,β, so that, H β and T β have the same spectrum. The map Υ β : φ n,β HL β(x Q n,β L (µ β can be uniquely extended to a unitary isomorphism denoted Υ β again. We see that, under Υ β, T β goes over into H β, namely, H β and T β are intertwined by Υ β : H β = Υ 1 β T β Υ β. Let P = {P t = e th β, t > } be the semigroup generated by H β. Motivated by the applications indicated in [7] we prove the following theorem.

19 ON GEGENBAUER COHERENT STATE REPRESENTATION 87 Theorem 6.1. If e βt 1/8, then P t is a contraction from HL β (X to HL 4 β (X. Proof. Recall that { φ n,β = 1/ z n ; n } is an orthonormal basis for HL β (X. Moreover, clearly P t φ n,β = e n(n+βt φ n,β. Then, for f(z = γ nφ n,β HL β (X, we have ( P t f (z = γ n e n(n+βt φ n,β. Therefore, one can estimate P t f 4 4 = ( P t f (z 4 ν β (dz = = = = X X { n 1= n = X n 1,n,n 3,n 4 = X ( (P t f (z(p t f (z νβ (dz e (n 1 +n t e 4βt(n 1+n γ n1 γ n φ n1,βφ n,β} ν β (dz e t(p 4 k=1 n k e 4βt(P 4 k=1 n k γ n1 γ n γ n3 γ n4 φ n1,βφ n,βφ n3,βφ n4,βν β (dz e t(p 4 k=1 n k e 4βt(P 4 k=1 nk γ n1 γ n γ n3 γ n4 X n 1,n,n 3,n 4= (λ n1,βλ n,βλ n3,βλ n4,β 1/ z (n1+n (z (n3+n4 ν β (dz Then, by using the obvious inequality e t(p 4 k=1 n k < 1 and the assumption e βt 1/8, we get P t f 4 4 e 8βtn γ n1 γ n γ n3 γ n4 n 1+n =n n 3+n 4=n (λ n1,βλ n,βλ n3,βλ n4,β 1/ ( = e 8βtn γ n1 γ n n 1+n =n { ( 1 n n 1+n =n n 1+n =n n! n 1! n! λn,β λ n1,βλ n,β 1/ (n1 + β(n + βγ(βγ(n + β (n + ββγ(βγ(n 1 + βγ(n + β } γ n1 γ n ( Γ(n1 + βγ(n + β Γ(βΓ(n + β n 1+n =n ( n1 ( n n! 1 1 n 1! n! = f 4. This completes the proof. n 1+n =n γ n1 γ n

20 88 ABDESSATAR BARHOUMI 7. Concluding Remarks In the present paper we have defined a new type of an oscillator for which the class of Gegenbauer polynomials play the same role as the Hermite polynomials play for standard boson oscillator. By solving the appropriate classical moment problem, we defined a distinguished set of coherent states of the Klauder-Gazeau type. Moreover, to illustrate our theory, we developed an example of complex hypercontractivity property for an ultraspherical Hamiltonian naturally associated to the family of orthogonal polynomials in consideration. Unfortunately, the class of radial measure given by Eq. (5.14 is rather complicated in the general case. A complete description of the Gegenbauer coherent states of Klauder-Gazeau type in terms of the parameter appearing in the beta distribution is still open problem. A second open problem is indicated in Section 6 concerning the complex hypercontractivity property for quantum mechanical Gegenbauer harmonic oscillator with ultraspherical phase space of one degree freedom (X, ν (dz. The research in this line is in progress. β Acknowledgements. I have profited very much from helpful discussions with Professor M. Bożejko. I gratefully acknowledges the kind hospitality and the stimulating atmosphere of Bożejko group at Wroc law University during my visit of the Institute of Mathematics Martch/April 8 where part of this work was completed. I wishes to thank N. Demni and N. Asai for bringing to my attention good Refs. I also thank I. Dermoul for telling me about the paper [33] which indicates a physical model for the class of coherent states considered in our present paper. This work was supported by the Research Project Bilateral Polish-Tunisian Governmental Co-operation 7-9. I am also grateful for the support. References 1. Accardi, L. and Bożejko, M.: Interacting Fock space and Gaussianization of probability measures, Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 ( Accardi, L. and Das, P. K.: Phase measurement in interacting Fock space, Int. J. Th. Physics 4 no. 11 ( Accardi, L. and Lu, Y. G.: The Wigner Semi-circle law in Quantum Electro-dynamics, Comm. Math. Phys. 18 ( Ali, S. T., Antoine, J.-P., Gazeau, J.-P., and Müller, U.-A.: Coherent states and their generalizations: A mathematical overview, Rev. Math. Phys. 7 ( Antoine, J.-P., Gazeau, J.-P., Klauder, J. R., and Monceau, P.: Temporally stable coherent states for infinite well and Pöschl-Teller potentials, J. Math. Phys. 4 ( Asai, N.: Analytic characterization of one-mode interacting Fock space, Infinite Dimensional Analysis, Quantum Probability and Related Topics 4 ( Asai, N.: Hilbert space of analytic functions associated to the modified Bessel function and related orthogonal polynomials, Infinite Dimensional Analysis, Quantum Probability and Related Topics 8 no. 3 ( Asai, N.: Riesz potentials derived by one-mode interacting Fock space approach, Colloquium Mathematicum 19 ( Asai, N., Kubo, I., and Kuo, H.-H.: Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures, Proc. Amer. Math. Soc. 131 no. 3 (

21 ON GEGENBAUER COHERENT STATE REPRESENTATION Asai, N., Kubo I., and Kuo, H.-H.: Multiplicative renormalization and generating functions II, Taiwanese J. Math. 8 ( Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform I, Commun. Pure Appl. Math. 14 ( Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform II, Commun. Pure Appl. Math. ( Ben Saïd, S. and Orsted, B.: Segal-Bargmann transforms associated with finite Coxeter groups, Math. Ann. 334 ( Berndt, R. and Böcherer, S.: Jacobi forms and discrete series representations of the Jacobi group, Math. Z. 4 ( Berndt, R. and Schmidt, R.: Elements of the representation theory of the Jacobi group, Progress in Mathematics 163, Birkhäuser Verlag, Basel, Biane, P.: Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems, J. Funct. Anal. 144 no. 1 ( Bożejko, M.: Private communication, Tunis ( Burger, M. and de la Harpe, P.: Constructing irreducible representations of discrete groups, Proc. Indian Acad. Sci. (Math. Sci. 17 ( Carey, A. I.: Group representations in reproducing kernel Hilbert spaces, Report on Math. Phys. 14 ( Char, B. W.: Maple 8 Learning Guide, Waterloo Maple, Waterloo,. 1. Chihara, T. S.: An Introduction to Orthogonal Polynomials, Gordon and Breach, Dwyer, T.: Partial differential equations in Fischer-Fock spaces for Hilbert-Schmidt holomorphy type, Bull. Amer. Math. Soc. 77 ( Gazeau, J.-P. and Klauder, J. R.: Coherent states for systems with discrete and continuous spectrum, J. Phys. A: Math. Gen. 3 ( Glauber, R. J.: Coherent and incoherent states of the radiation field, em Phys. Rev. 131 ( Gradshtein, I. S. and Ryzhik, I. M.: Tables of Integrals, Series and Products, Acad. Press, San Diego, Calif.. 6. Gross, L. and Malliavin, P.: Hall s transform and the Segal-Bargmann map, in: Itô Stochastic Calculus and Probability Theory, N. Ikeda et al. (eds. Springer, ( Hoghn-Krohn, R. and Simon, B.: Hypercontractivitive semigroup and two dimensional self coupled Bose fields, J. Funct. Anal. 9 ( Ismail, M. E. H.: Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press Janson, S.: On complex hypercontractivity, J. Funct. Anal. 151 ( Kemp, T.: Hypercontractivity in non-commutative holomorphic spaces, Commun. Math. Phys. 59 ( Klauder, J. R. and Skagerstam, B. S.: Coherent States - Applications in Physics and Mathematical Physics, World Scientific, Singapore, Koekoek, R. and Swarttouw, R. F.: The Askey-scheme of Hypergeometric Orthogonal Polynomials and its q-analogue, Delft Univ. of Tech., Report No ( Manko, V. I., Marmo, G., Porzio, A., Solimeno, S., and Zaccaria, F.: Trapped ions in laser fields: a benchmark for deformed quantum oscillators, Phys. Rev. A 6 ( Mueller, C. E. and Wissler, E. B.: Hypercontractivity for the heat semigroup for ultraspherical polynomials on the n-sphere, J. Funct. Anal. 48 ( Le, H. Q., Abramov, S. A., and Geddes, K. O.: A Maple Package for Finding Closed Forms of Indefinite and Definite Sums of Hypergeometric Type. fttp://cs-archive.uwaterloo.ca/csarchive/cs-1-4.,. 36. Lisiecki, L.: Coherent state representations. A survey, Reports on Mathematical Physics 35 no. 13 ( Perelomov, A.: Generalized Coherent States and Their Applications, Texts and Monographs in Physics, Springer, Berlin, 1985.

22 9 ABDESSATAR BARHOUMI 38. Sontz, S. B.: The µ-deformed Segal-Bargmann transform as a Hall type transform, Inf. Dim. Anal. Q. Prob. Rel. Top., to appear, Vitonofrio, C.: Interacting Fock Spaces: Central Limit Theorems and Quantum Stochastic Calculus, Tesi di Dottorato, Università Degli Studi Di Bari, Vourdas, A.: The growth of Bargmann functions and the completeness of sequences of coherent states, J. Phys. A: Math. Gen. 3 ( Watson, G. N.: Notes on generating functions of polynomials: (1 Laguerre polynomials, Journ. London Math. Soc. 8 ( Watson, G. N.: Notes on generating functions of polynomials: (3 Polynomials of Legendre and Gegenbauer, Journ. London Math. Soc. 8 ( Watson, G. N.: Notes on generating functions of polynomials: (4 Jacobi polynomials, Journ. London Math. Soc. 9 ( Abdessatar Barhoumi: Department of Mathematics, Higher School of Sci. and Tech. of Hammam-Sousse, Sousse University, Sousse, Tunisia address: abdessatar.barhoumi@ipein.rnu.tn