Generalized coherent states related to the cylindrical Bessel functions and Legendre oscillator

The main objective of this paper is to discuss the Paley-Wiener space PW 1 in the framework of a set of coherent states related to the cylindrical Bessel function and the Legendre oscillator. Thus, we show that the kernel of the Fourier transform of the L2-functions that are supported in [ − 1 , 1 ] form a set of coherent states. This also leads to the construction a coherent states integral transform.


Introduction
Coherent states was originally introduced by Schrödinger in 1926 as a Gaussian wavepacket to describe the evolution of a harmonic oscillator [1]. The notion of coherence associated with these states of physics was first noticed by Glauber [2,3] and then introduced by Klauder [4,5]. The common point here was that all these coherent states were associated to the quantum harmonic oscillator. Because of their important properties these states were then generalized to other systems either from a physical or mathematical point of view. Here, it is necessary to emphasize that quantum coherence of states nowadays pervades many branches of physics such as quantum electrodynamics, solid-state physics, and nuclear and atomic physics from both theoretical and experimental points of view. As the electromagnetic field in free space can be regarded as a superposition of many classical modes, each one governed by the equation of simple harmonic oscillator, the coherent states became significant as the tool for connecting quantum and classical optics. For a review of all of these generalizations see [6][7][8][9].
To construct coherent states, four main different approaches are well used in literature, the so-called Schrödinger, Klauder-Perelomov, Barut-Girardello, and Gazeau-Klauder methods. While the second and the third approaches rely directly on the Lie algebra symmetries with their corresponding generators, the first is established only by means of suitable infinite superposition of wave functions associated with the harmonic oscillator, regardless of the Lie algebraic symmetries. Clearly they introduce coherent states as superpositions of Hamiltons eigenvectors which span complete and (bi-)orthogonal Hilbert spaces [7]. Different types of coherent states for quantum mechanics have been discussed by many authors from various perspectives. Thus, in [10] the authors have constructed coherent states for the Legendre oscillator, in [11][12][13] the authors have introduced new family of coherent states as suitable superposition of the associated Bessel functions and in [14][15][16] the authors also use the generating function approach to construct new kind coherent states associated with the Hermite polynomials and associated Legendre functions, respectively. Here, the interesting fact is that it is not necessary to use the algebraic and group approaches (Barut-Girrardello and Klauder-Perelomov) to construct generalized coherent states.
In the present paper we discuss coherent states associated with a one-dimensional Schrödinger operator found in [10,17] by following the procedure described in ( [18,19]), then we build a family of coherent states through superpositions of the corresponding eigenstates, say y ñ | n which are expressed in terms of the Legendre polynomial [10], where the role of coefficients ! z n n of the canonical coherent states is played by denotes the cylindrical Bessel function [20]. We proceed by determining the wavefunctions of these coherent states in a closed form. The latter gives the kernel of the associated coherent states integral transform which makes correspondence between the quantum states Hilbert space - of the Legendre oscillator and a subspace of a Hilbert space of square integrable functions with respect to a suitable measure on the real line. Then, we discuss the Paley-Wiener space PW a with = a 1, in the framework of a set of coherent states related to the cylindrical Bessel functions for the Legendre oscillator and we show that the kernel of the Fourier transform of the L 2 -functions that are supported in -[ ] 1, 1 form a set of coherent states. Through the Whittaker-Shannon-Kotel'Nikov theorem (in Theorem 3, see below), for the following change of the variable x px  2 , we interpret the function x x px = - defined in (1.1) as the Band-limited signal and the parameter = a 1 as the band limit of x ( ) f and W ≔ 2, the corresponding frequency band [21].
The paper is organized as follows: in section 2 we introduce briefly the Paley-Wiener space PW a . Section 3 is devoted to the coherent states formalism we will be using. In section 4, we recall some notions of the Legendre oscillator. In section 5, we construct a family of coherent states related to the cylindrical Bessel functions for the Legendre oscillator. And finally, section 6 is devoted to some concluding remarks on the paper.

Preliminary
If F is entire function of exponential type, we call a the type of F where Then cleary, is of exponential type at most a.
Hence, by polarization , Theorem 1 ( [22], p.66). Let F be an entire function and > a 0. Then the following are equivalent

Generalized coherent states formalism
In this section we follow the generalization of the canonical coherent states according to the procedure defined in ( [18,19]).
be an orthogonal basis of N 2 satisfying, for arbitrary , is a reproducing kernel, N 2 is the corresponding kernel Hilbert space Definition 4. The coherent states transform associated to the set of coherent states Thereby, we have a resolution of the identity of  which can be expressed in Dirac's bra-ket notation as and where w ( ) x appears as a weight function.
Remark 1. Note that the formula (3.3) can be considered as generalization of the series expansion of the canonical coherent states [23]  k k 0 being an orthonormal basis of eigenstates of the quantum harmonic oscillator. Here the space N 2 is the Fock space   ( )and

The Legendre oscillator
In this section, we summarize the work of Borzov and Demaskinsky ( [10,17]) on the Legendre Hamiltonian where X and P denotes respectively the position and momentum operators, + a anda denotes respectively the creation and annihilation operators (see bellow). The eigenvalues of operators H are equal to The generalized position operator on the Hilbert space  connected with the Legendre polynomials ( ) P x n is an operator of multiplication by argument Taking into account of the relation (4.4), then , the operator X is a self-adjoin operator in the space  (see [24][25][26]). The generalized momentum operator P by the way described in ( [17], p.126) acts on the basis elements in  by the following formula Calculating the usual commutator of operator X and P on the basis elements, we obtain The creation and annihilation operators defined in (4.1) are define by the standard relations On the basis element in  these operators act by the rule   here  x ( ) is a normalization factor, the function In the following results we give some properties verified by coherent states (5.1).
Proposition 1. The normalization factor defined by coherent states (5.1) reads where s x ( ) is an auxiliary density function. Then, according to (5.1) and by writing By comparing (5.15) with (5.13) we obtain finally the desired weight function s x p = ( ) 1 .Therefore, the measure (5.10) has the form (5.9). Indeed (5.12) reduces further to in terms of Dirac's bra-ket notation. This ends the proof. , Thus, according to this construction the states xñ | form an overcomplete basis in the Hilbert space . The coherent states in (5.1) are defined as vectors in the Hilbert space . In the case where these states describe a quantum system, by the usual quantum mechanical convention, the probability of finding the state y ñ | n in some normalized state xñ | of the state Hilbert space  is given by y x á ñ  | | | n n 2 . Thus for coherent states (5.1) it is given by Now, in the following results we will establish a closed form for the construct coherent states in (5.1) and we will discuss the associated coherent states integral transform 1, 1 . Then, the wavefunctions of coherent states defined in (5.1) can be written in a closed form as Proof. We start by writing the expression of the wavefunctions of coherent states (5.1) as follows We now appeal to the Gegenbauer's expansion of the plane wave in Gegenbauer polynomials and Bessel functions ( [28], p.116): Proof. The result follows immediately by using the formula ( [20],p.647): is the spherical Bessel function [20]. This ends the proof. , The carefull reader has certaintly recognized in (5.25) the expression of nonlinear coherent states [23]. Note that, in view of the formula ( [27], p.667): , form an orthonormal basis of PW 1 (see [29]). Next, once we have obtained a closed form of the constructed coherent states, we can look for the associated coherent states transform. In view of (5.1), this transform should map the space  = - 1 2 in the following manner: , the coherent states transform is the unitary map defined by means of (5.19) as Then, as consequence has the integral representation where the function x F ( ) n and the positive sequences r n are given respectively in (5.2) and (5.4), the measure m x ( ) d is given in (5.9), then we find the following expression , 5 . 3 9 n n n ix 1 2 which is recognized as the Fourier transform of the spherical Bessel function (5.27) (see [30], p.267):

5.40
ixt n n n n where P n is the Legendre polynomial [30].
The sampling theorem of band limited functions 1 , which is often named after Whittaker-Shannon-Kotel'Nikov. This is its classical formulation where ( ) J . n denotes the spherical Bessel functions, can be interpreted as the band-limited signal [21]. We see that the spherical Bessel functions expansion may find applications in probability theory, where it can be used to represent classes of characteristic functions and autocorrelation functions [35].
Remark 3. Also note that: a a , , and a is a band limite of f and W ≔ a 2 , the corresponding frequency band.
• The usefulness expansion of coherent states in (5.24) was made very clear in a paper authored by Ismail and Zhang, where it was used to solve the eigenvalue problem for the left inverse of the differential operator, on L 2 -spaces with ultraspherical weights [31,32].
• The exact differential of coherent states  which gives the full information about rates of change of coherent states in the x-direction and in the ξdirection, the field of vector is the gradient of  x ( ) x, i.e. ( ) x e ix , is known as the Gabor's coherent states introduced in signal theory where the property y x y = xˆ( ) T , with  y Î ( ) L 2 , and^x ( ) T the unitary transformation, is obtained by using the standard representation of the Heisenberg group in three dimensions, in  ( ) L 2 , for more information (see [33]). The choosen coefficients (1) and eigenfunctions (4.3) have been used in [35, p 1625] where the authors presented a representation for band-limited functions in terms of spherical Bessel functions. The polynomial approximation of the Fourier transform have been used.

Concluding remarks
In this work, we have built a class of coherent states related to the cylindrical Bessel functions for the Legendre oscillator while discussing the Paley-Wiener space PW 1 in the framework of these coherent states. We found that, the kernel of the Fourier transform of L 2 -function that are supported in -[ ] 1, 1 form a set of coherent states. Furthermore, we were able to define a coherent states transform which maps isometrically - onto a subspace of a Hilbert space of square integrable functions with respect to a suitable measure; here the space PW 1 . Clearly, this subspace is spanned by the coefficients (given in terms of Cylindrical Bessel functions) we have chosen to superpose eigenstates of the Legendre oscillator. In fact, it is well known that, in quantum mechanics, canonical coherent and affine coherent states, and in signal analysis, Gabor and wavelet frames have offered an extremely flexible alternative method for describing functions in  ( ) L 2 and in  + ( ) L 2 . However, few methods exist for the description of L 2 -functions on an interval [ ] a b , , see [34]. Thus, here, we were presenting a contribution in this direction. Note that the constructed coherent states (see definition 5) reproduce the nonlinear coherent states (NLCSs) defined in (5.25). So, it is crucial to analyze some properties arising from these NLCSs. This will be the subject of our future work.