International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 357050 | 15 pages | https://doi.org/10.1155/2010/357050

An Elementary Construction on Nonlinear Coherent States Associated to Generalized Bargmann Spaces

Academic Editor: Ricardo Estrada
Received27 Jul 2009
Revised28 Dec 2009
Accepted11 Feb 2010
Published07 Mar 2010

Abstract

Consider the space , where is the Gaussian measure, and its generalized Bargmann subspaces which are the null kernels of the operator ;โ€‰โ€‰ In this work, we present an other construction of following the Hermite functions which allows us to define a family of generalized Bargmann transform which maps isometrically into . The generalized coherent states associated to are constructed and some properties of them are given.

1. Introduction

The annihilation operator and the creator operator are well-known from the quantum theory of harmonic oscillator.They are defined by the commutation relation

Of special interest is a representation of these operators and as linear operators in a separable Hilbert spanned by the eigenvectors ; of the positive semidefinite number operator .

One has the well-known relations

The coherent states analysis is a very well-known tool in physics, in particular in quantum optics and in quantum mechanics. The name โ€œcoherent statesโ€ was first used by Glauber, Nobel prize in physics (2005), for his works in quantum optics and electrodynamics. In the book of Klauder and Skagerstam [1], the reader can get an idea of the coherent states application fields in physics and mathematical-physics.

A general mathematical theory of coherent states is detailed in [2].

Let us begin by reviewing the most important properties of the space of coherent states spanned by the set , where

for each complex

(i)The vectors are the eigenvectors of the annihilation operator; one has (ii)The space of coherent states is a separable Hilbert space which is isomorphic to the Hilbert space spanned by .(iii).(iv)(v) is the adjoint vector of coherent state , where

From the above properties, we obtain

(vi)(vii)

where and are the linear partial differential operators on given by Let and

Then we define the linear partial differential operator on by

The vector fields Z and and the identity operator form a basis for a Lie algebra in which the Lie bracket of two elements is their commutator. In fact, is the formal adjoint of Z and is an elliptic partial differential operator on given by

Thus, is the usual Hermite operator

perturbed by the rotation operator

The strongly continuous one parameter semigroup is ultracontractive in the sense that for any , is a bounded linear operator from to and it is hypercontractive in the sense that for all , is a bounded linear operator from to .

The theory of ultracontractive semigroups can be found in Davie's book [3] and the connections of hypercontractivity with constructive quantum field theory are attributed to Nelson [4] and are explained in the Simon's book [5].

Coming back to the space of coherent states, it is closely related to Bargmann's space which was used in [6] for the canonical commutation rules as representation space of quantum mechanics. Since then, it had occurred in many different contents, that is, in representation theory of nilpotent Lie groups and in Reggeon field theory. This last theory is governed by a nonselfadjoint Gribov operator [7, 8] constructed as a polynomial in and .

For any , we can define an entire analytic function by

As then

We denote the Bargmann space by:

is closed in where the measure and is closed related to by an unitary transform of onto given in [6] by the following integral transform (for some appropriate constant ):

If the integral converges absolutely.

In , Askour et al. [9, 10] have introduced the so-calledโ€œgeneralizedโ€ Bargmann spaces , as null kernels of the operator

and have proved that is the direct sum of the Bargmann space and all โ€œgeneralizedโ€ Bargmann spaces .

Otherwise, Vasilevski in [11] concerning the structure of polyanalytic Fock spaces has introduced the so-called โ€œtrue-poly-Fockโ€ spaces and proved that is the direct sum of the Bargmann space (Fock) and all โ€œtrue-poly-Fockโ€ spaces.

According to chapter 2 of Daubechies book [12], we give in this work a family of embedding operators from into such that the range is a Bargmann space and are reproducing subspaces.

The construction of is based on the Fourier-Bargmann transform which is an isometry from into .

To do this, we consider as a windowed function and denote . Then we can consider the windowed transform (the Fourier-Bargmann transform or the Gabor Fourier transform):

In the nice Folland's book [13], the Fourier-Bargmann transform is called wave packet transform and is very close to the Bargmann transform and FBI transform; see also [14].

By taking , Daubechies has pointed out that is isomorphic to the Bargmann space.

In this work, taking and the Hermite functions as windowed functions, we point out that (Gabor space with the Hermite window ) is isomorphic to a โ€œgeneralizedโ€ Bargmann space .

We adopt this last construction to give a โ€œgeneralizedโ€ Bargmann transform associated to and we construct the phase spaces of โ€œgeneralized coherent statesโ€ associated to .

To do this, let and we use the following integral transform:

such that

where

We try to be self-contained and elementary as far as possible in this paper which is organized as follows: in Section 2, we present a construction of โ€œgeneralizedโ€ Bargmann spaces following the Hermite functions by using the integral transform (I) for all and we define the โ€œgeneralizedโ€ Bargmann transform associated to . In Section 3, we construct the โ€œgeneralized coherent statesโ€ , associated to โ€œgeneralizedโ€ Bargmann spaces and give some properties of them.

2. Construction of Generalized Bargmann Spaces Following the Hermite Functions

Let us consider the Hamiltonian oscillator on where , are the usual position and momentum operators (). It is well-known that the eigenvalues and the eigenvectors of are given by

where are the following Hermite functions:

and are the Hermite polynomials satisfying the recursion relations .

Remark 2.1. (i) It is well-known that may also be seen as the eigenvectors of the usual operator on in presence of a constant magnetic field. Namely acting in is given by

โ€‰(ii) has an infinite degeneracy.
(iii) It is well-known that the complex Hermite polynomials
are an orthogonal basis of the Hilbert space and we can see in [15] how it was used to give some spectral properties of Cauchy transform on .

If we formally take to be real in (2.4), the complex Hermite polynomials become the well-known real Hermite polynomials .

Now, let and we define the following integral transforms:

by

where

โ€‰(1) If we take in (2.6) f(x)=hm+k(x) for k=0, 1,โ€ฆ, then we can use the formulas on the Hermite functions given in [16, 17], in particular the famous formula:

where is the Laguerre polynomial defined by the Rodriguez formula as

The same formula appeared in [18, Chapter 23].

After a simple variable change, the formula (2.8) can be also expressed as

From (2.10), we deduce for that

or

where

โ€‰(2) If 0โ‰คkโ‰คm, we have the following formula:

That permits us to write for

Denoting , then the reproducing kernel of can be computed as

Hence we have

After a simple variable change, for example then can be alsoexpressed as

Hence

or

The reproducing relation in can be written as

then is a reproducing space with the reproducing kernel .

Introducing the complex structures and theabove formulas can be rewriten as

with

We also can see that by changing the variable into and into that can be transformed to kernels given in [9, Theorem ]

Now, we can define the โ€œgeneralizedโ€ Bargmann transforms by

and the โ€œgeneralizedโ€ Bargmann spaces by

Now, the consequences of all the above computations are summarized in the following form.

Theorem 2.2. (i) is a subspace of .
(ii)
(iii) For , is a basis of classical Bargmann space
(iv)
(v)
(vi)

Remark 2.3. (i) The operator is the adjoint of the operator in and ; see for example [19] or [20].
(ii) In [9, 10] the โ€œgeneralizedโ€ Bargmann spaces are constructed as null spaces of the second-order differential operator in but in this section we have given another construction of these spaces which allows us to construct the โ€œgeneralizedโ€ Bargmann transforms (2.25).

3. Construction of Generalized Coherent States Associated to Generalized Bargmann Spaces

In the separable Hilbert space spanned by the eigenvectors of the positive semidefinite number operator , one can define for each complex number

where

The space spanned by the set is a separable Hilbert space which is in fact isomorphic to . This space is called the โ€œgeneralizedโ€ space of coherent states. It is closely related to the โ€œgeneralizedโ€ Bargmann space .

For any , we can define a function by

As then

Theorem 3.1. Let and then the following statements hold.(i); for any holomorphic function . (ii)Let then is in the โ€œgeneralizedโ€ Bargmann space ; (iii) where and

Proof. (i) Let be holomorphic function; then by applying the commutation relation and the Cauchy-Riemann equation, we deduce that (i) is satisfied for and by induction that
โ€‰ (ii) For , we have and is in the classical Bargmann space .
For and we have
As the function is holomorphic, we can apply property (i). Premultiplying by the equation , we obtain
โ€‰ (iii) Let
As , we have
where Then The action of the annihilation operator on for is presented below, in the form of a theorem.

Theorem 3.2. (i)
(ii) There exists an operator such that

Proof. (i) Let where
By applying the operator on we get

Now as is a holomorphic function and , we have and we get
that is,
In the above property (i) of the present theorem, it is shown that the โ€œgeneralizedโ€ coherent states are not eigenstates of the annihilation operator. Nevertheless, we will show that these โ€œgeneralizedโ€ coherent states can be interpreted as nonlinear coherent states, see for example the theory of Photon-added coherent states used in [21] or the approach of nonlinear coherent states given in [22].
To do this, we have to show that obeys to the equation
with a suitable choice for the operator . Let us now construct the explicit form of .
From (3.11), we have then is in the space spanned by .
By definition, the coherent states satisfy

Premultiplying the both sides of this equation by leads

Using the commutation relation , the above equation is written as
which, making use of the identity , leads to In the following, we introduce and the projection operator on the space spanned by the system , where and if.
Let
Acting on leads to that is, To sum up, we have used the Gabor spaces with the Hermite window to give another construction of generalized Bargmann spaces which is different from that given in [9, 10] or [11] and have constructed a family of โ€œgeneralizedโ€ Bargmann transforms attached to these spaces. The states constructed in this work with are nonclassical and can be interpreted as nonlinear coherent states in nonanalytic representation. The spaces are constituted by the polyanalytic functions of order , that is, which are not polyanalytic of any lower order and satisfy The functions in , can be uniquely expressed in the form Finally, the spaces defined in this work are the โ€œtrue polyanalytic Fock spacesโ€ of [11] or the โ€œgeneralized Bargmann spacesโ€ of [9, 10].
When is not zero, we give (in a further work) a complete spectral analysis of the nonselfadjoint operator on or on (the natural generalization of the Bargmann space, since (3.25) generalizes the Cauchy-Riemann equation , where and
The restriction of on the Bargmann space is the Heun's operator and it has been studied in detail in [23].

Acknowledgment

I congratulate my son Jean-Karim for his dazzling academic performance in college and for those that get in football.

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Copyright © 2010 Abdelkader Intissar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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