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

Abstract

Consider the space 𝐿₂(ℂ,𝑑𝜇(𝑧)), where 𝑑𝜇(𝑧)=𝑒^−|𝑧|2𝑑𝑧⋀𝑑𝑧 is the Gaussian measure, and its generalized Bargmann subspaces 𝐸_𝑚 which are the null kernels of the operator Δ=−𝜕²/𝜕𝑧𝜕𝑧+𝑧(𝜕/𝜕𝑧)−𝑚𝐼;   𝑚=0,1,…. In this work, we present an other construction of 𝐸_𝑚 following the Hermite functions which allows us to define a family of generalized Bargmann transform 𝐵_𝑚 which maps isometrically 𝐸_𝑚 into 𝐿₂(ℝ). The generalized coherent states ∣𝑧⟩_𝑚 associated to 𝐸_𝑚 are constructed and some properties of them are given.

1. Introduction

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

[𝑎,𝑎^∗]=1. (1.1)

Of special interest is a representation of these operators 𝑎 and 𝑎^∗ as linear operators in a separable Hilbert 𝐻 spanned by the eigenvectors |𝑛⟩; 𝑛=0,1,… of the positive semidefinite number operator 𝑁=𝑎^∗𝑎.

One has the well-known relations

𝑎|𝑛⟩=√𝑛|𝑛−1⟩,𝑎^∗|𝑛⟩=√𝑛+1|𝑛+1⟩. (1.2)

The coherent states analysis is a very well-known tool in physics, in particular in quantum optics and in quantum mechanics. The name “coherent states” 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 𝐈₀ of coherent states spanned by the set {|𝑧⟩₀}, where

|𝑧⟩₀=𝑒^{−1/2|𝑧|2∞}_𝑛=0𝑧^𝑛√𝑛!|𝑛⟩ (1.3) for each complex 𝑧=𝑥+𝑖𝑦.

(i)The vectors |𝑧⟩₀ are the eigenvectors of the annihilation operator; one has 𝑎|𝑧⟩₀=𝑧|𝑧⟩₀.(ii)The space 𝐈₀ of coherent states is a separable Hilbert space which is isomorphic to the Hilbert space 𝐻 spanned by {|𝑛⟩}.(iii)|𝑛⟩=𝑎^∗𝑛/√𝑛!|0⟩.(iv)|𝑧⟩₀=𝑒^−|𝑧|2/2𝑒^𝑧𝑎∗|0⟩.(v)₀⟨z|=⟨0|𝑒^𝑧𝑎𝑒^−|𝑧|2/2 is the adjoint vector of coherent state |𝑧⟩₀, where 𝑧=𝑥−𝑖𝑦.

From the above properties, we obtain

(vi)𝑎^∗|𝑧⟩₀=(𝑧/2+𝜕/𝜕𝑧)|𝑧⟩₀,(vii)𝑎|𝑧⟩₀=(𝑧/2−𝜕/𝜕𝑧)|𝑧⟩₀,

where 𝜕/𝜕𝑧 and 𝜕/𝜕𝑧 are the linear partial differential operators on ℝ² given by 𝜕=𝜕𝜕𝑧𝜕𝜕𝑥−𝑖𝜕𝜕𝑦,𝑧=𝑥+𝑖𝑦,𝜕𝑧=𝜕𝜕𝜕𝑥+𝑖,𝜕𝑦𝑧=𝑥−𝑖𝑦. (1.4) Let 𝑍=𝜕/𝜕𝑧+𝑧/2 and 𝑍=𝜕/𝜕𝑧−𝑧/2

Then we define the linear partial differential operator 𝐿 on ℝ² by

1𝐿=−2(𝑍𝑍+𝑍𝑍). (1.5)

The vector fields Z and 𝑍 and the identity operator form a basis for a Lie algebra in which the Lie bracket of two elements is their commutator. In fact, −𝑍 is the formal adjoint of Z and 𝐿 is an elliptic partial differential operator on ℝ² given by

𝜕𝐿=−(²𝜕𝑥²+𝜕²𝜕𝑦²1)+4(𝑥²+𝑦²𝜕)−𝑖(𝑥𝜕𝜕𝑦−𝑦𝜕𝑥). (1.6)

Thus, 𝐿 is the usual Hermite operator

𝜕−(²𝜕𝑥²+𝜕²𝜕𝑦²1)+4(𝑥²+𝑦²) (1.7) perturbed by the rotation operator

𝜕−𝑖(𝑥𝜕𝜕𝑦−𝑦𝜕𝑥). (1.8) The strongly continuous one parameter semigroup 𝑒^−𝑡𝐿,𝑡>0 is ultracontractive in the sense that for any 𝑡>0, 𝑒^−𝑡𝐿 is a bounded linear operator from 𝐿₂(ℝ²) to 𝐿_∞(ℝ²), and it is hypercontractive in the sense that for all 𝑡>0, 𝑒^−𝑡𝐿 is a bounded linear operator from 𝐿₂(ℝ²) to 𝐿₄(ℝ²).

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 𝐈₀ 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 𝑎 and 𝑎^∗.

For any |𝜙₀⟩∈𝐈₀, we can define an entire analytic function by

𝜙₀(𝑧)=𝑒^1/2|𝑧|2⟨𝜙₀∣𝑧⟩₀=^∞_𝑛=0𝑧^𝑛√𝑛!⟨𝜙₀∣𝑛⟩. (1.9) As ∫_ℂ|⟨𝜙₀∣z⟩₀|²𝑑𝑥𝑑𝑦⟨∞^⋅ then

_ℂ|𝜙₀(𝑧)|²𝑒^−|𝑧|2𝑑𝑥𝑑𝑦<∞. (1.10) We denote the Bargmann space by:

𝐸₀={𝜙₀∶ℂ→ℂentire;_ℂ|𝜙₀(𝑧)|²𝑒^−|𝑧|2𝑑𝑥𝑑𝑦<∞}. (1.11)𝐸₀ is closed in 𝐿₂(ℂ,𝑑𝜇(𝑧)), where the measure 𝑑𝜇(𝑧)=𝑒^−|𝑧|2𝑑𝑥𝑑𝑦 and is closed related to 𝐿₂(ℝ) by an unitary transform of 𝐿₂(ℝ) onto 𝐸₀ given in [6] by the following integral transform (for some appropriate constant 𝑐):

𝜙₀(𝑧)=[𝐵₀𝑓](𝑧)=𝑐_ℝ𝑒^{−(1/2)𝑧2+√2𝑢𝑧−(1/2)𝑢2}𝑓(𝑢)𝑑𝑢. (1.12)

If 𝑓∈𝐿₂(ℝ), the integral converges absolutely.

In 𝐿₂(ℂ,𝑑𝜇(𝑧)), Askour et al. [9, 10] have introduced the so-called“generalized” Bargmann spaces 𝐸_𝑚, 𝑚=0,1,…, as null kernels of the operator

𝜕Δ=−²𝜕𝑧𝜕𝑧+𝑧𝜕𝜕𝑧−𝑚𝐼,𝑚=0,1,… (1.13) and have proved that 𝐿₂(ℂ,𝑑𝜇(𝑧)) is the direct sum of the Bargmann space 𝐸₀ and all “generalized” Bargmann spaces 𝐸_𝑚.

Otherwise, Vasilevski in [11] concerning the structure of polyanalytic Fock spaces has introduced the so-called “true-poly-Fock” spaces and proved that 𝐿₂(ℂ,𝑑𝜇(𝑧)) is the direct sum of the Bargmann space (Fock) and all “true-poly-Fock” spaces.

According to chapter 2 of Daubechies book [12], we give in this work a family of embedding operators 𝐵_𝑚 from 𝐿₂(ℝ) into 𝐿₂(ℂ,𝑑𝜇(𝑧)) such that the range 𝐵₀(𝐿₂(ℝ)) is a Bargmann space and 𝐵_𝑚(𝐿₂(ℝ)) are reproducing subspaces.

The construction of 𝐵_𝑚 is based on the Fourier-Bargmann transform which is an isometry from 𝐿₂(ℝ) into 𝐿₂(ℝ²).

To do this, we consider ℎ∈𝐿₂(ℝ) as a windowed function and denote ℎ^𝑢,𝑣(𝑥)=𝑒^iuxℎ(𝑥−𝑣). Then we can consider the windowed transform (the Fourier-Bargmann transform or the Gabor Fourier transform):

𝐵_ℎ∶𝐿₂(ℝ,𝑑𝑥)→𝐿₂(ℝ²),𝑓→𝐵_ℎ𝑓(𝑢,𝑣)=⟨𝑓,ℎ^𝑢,𝑣⟩_𝐿2(ℝ). (1.14)

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 ℎ(𝑥)=(1/𝜋⁴)𝑒^−𝑥2/2, Daubechies has pointed out that 𝐵_ℎ(𝐿₂(ℝ)) is isomorphic to the Bargmann space.

In this work, taking ℎ₀(𝑥)=(1/𝜋⁴)𝑒^−𝑥2/2 and the Hermite functions ℎ_𝑚(𝑥)=(1/√2^𝑚𝑚!)(𝑥−𝑑/𝑑𝑥)^𝑚ℎ₀(𝑥) as windowed functions, we point out that 𝐵_ℎ𝑚(𝐿₂(ℝ)) (Gabor space with the 𝑚^𝑡ℎ Hermite window ℎ_𝑚) is isomorphic to a “generalized” Bargmann space 𝐸_𝑚.

We adopt this last construction to give a “generalized” Bargmann transform 𝐵_𝑚 associated to 𝐸_𝑚 and we construct the phase spaces 𝐼_𝑚 of “generalized coherent states” |𝑧⟩_𝑚 associated to 𝐸_𝑚.

To do this, let 𝑧=𝑢+𝑖𝑣 and 𝑚=0,1,…, we use the following integral transform:

𝐵_ℎ𝑚∶𝐿₂(ℝ,𝑑𝑥)→𝐿₂(ℝ²,𝑑𝑢𝑑𝑣),𝑓→𝐵_ℎ𝑚𝑓 (1.15) such that

𝐵_ℎ𝑚𝑓∶ℝ²→ℂ,(𝑢,𝑣)→𝐵_ℎ𝑚𝑓(𝑢,𝑣)=⟨𝑓,ℎ^𝑧_𝑚⟩_𝐿2(ℝ)=_ℝ𝑒^−iuxℎ_𝑚(𝑥−𝑣)𝑓(𝑥)𝑑𝑥, (I) where

ℎ^𝑧_𝑚(𝑥)=𝑒^iuxℎ_𝑚(𝑥−𝑣). (1.16) 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 “generalized” Bargmann spaces following the Hermite functions by using the integral transform (I) for all 𝑚, and we define the “generalized” Bargmann transform associated to 𝐸_𝑚. In Section 3, we construct the “generalized coherent states” |𝑧⟩_𝑚, 𝑚=0,1,…, associated to “generalized” Bargmann spaces 𝐸_𝑚 and give some properties of them.

2. Construction of Generalized Bargmann Spaces Following the Hermite Functions

Let us consider the Hamiltonian oscillator 𝐻_osc=1/2(𝑃²+𝑄²) on 𝐿₂(ℝ), where 𝑄, 𝑃 are the usual position and momentum operators ([𝑄,𝑃]=𝑖𝐼). It is well-known that the eigenvalues and the eigenvectors of 𝐻_osc are given by

𝐻_oscℎ_𝑚1=(𝑚+2)ℎ_𝑚, (2.1) where ℎ_𝑚 are the following Hermite functions:

ℎ_𝑚1(𝑥)=√2^𝑚𝑑𝑚!(𝑥−)𝑑𝑥^𝑚1𝜋⁴1√2^𝑚𝑒𝑚!^−𝑥2/2=1𝜋⁴1√2^𝑚𝑒𝑚!^−𝑥2/2𝐻_𝑚(𝑥) (2.2) and 𝐻_𝑚(𝑥)=∑^[𝑚/2]_𝑗=0(−1)^𝑗(𝑚!/𝑗!(𝑚−2𝑗)!)(2𝑥)^𝑚−2𝑗 are the Hermite polynomials satisfying the recursion relations 𝑥𝐻_𝑚(𝑥)=𝑚𝐻_𝑚−1(𝑥)+(1/2)𝐻_𝑚+1(𝑥).

Remark 2.1. (i) It is well-known that 𝜆_𝑚=𝑚+1/2 may also be seen as the eigenvectors of the usual 𝑆𝑐ℎ𝑟𝑜𝑑̈𝑖𝑛𝑔𝑒𝑟 operator 𝐻_𝑆 on ℝ² in presence of a constant magnetic field. Namely 𝐻_𝑆 acting in 𝐿₂(ℝ²) is given by
𝐻_𝑆1=−4𝜕[(𝜕𝑥+𝑖𝑦)²𝜕+(𝜕𝑦−𝑖𝑥)²]. (2.3)
 (ii) 𝐻_𝑆 has an infinite degeneracy.
(iii) It is well-known that the complex Hermite polynomials
𝐻_𝑛,𝑘(𝑧,𝑧)=(−1)^𝑚+𝑘𝑒^|𝑧|2𝜕^𝑚+𝑘𝜕𝑧^𝑚𝜕𝑧^𝑘𝑒^−|𝑧|2 (2.4) are an orthogonal basis of the Hilbert space 𝐿₂(ℂ,𝑑𝜇(𝑧)) and we can see in [15] how it was used to give some spectral properties of Cauchy transform on 𝐿₂(ℂ,𝑑𝜇(𝑧)).

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

Now, let 𝑧=𝑢+𝑖𝑣 and 𝑚=0,1,…, we define the following integral transforms:

𝐵_ℎ𝑚∶𝐿₂(ℝ,𝑑𝑥)→𝐿₂(ℝ²,𝑑𝑢𝑑𝑣),𝑓→𝐵_ℎ𝑚𝑓, (2.5) by

𝐵_ℎ𝑚𝑓(𝑢,𝑣)=⟨𝑓,ℎ^𝑧_𝑚⟩_𝐿2(ℝ)=_ℝ𝑒^−iuxℎ_𝑚(𝑥−𝑣)𝑓(𝑥)𝑑𝑥, (2.6) where

ℎ^𝑧_𝑚(𝑥)=𝑒^iuxℎ_𝑚(𝑥−𝑣). (2.7)

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

_ℝℎ_𝑚+𝑘𝑣(𝑥−2)𝑒^−iuxℎ_𝑚𝑣(𝑥+2)𝑑𝑥=(−𝑖√2)^𝑘𝑚!𝑒(𝑚+𝑘)!^{−1/4[𝑢2+𝑣2]}(𝑢−𝑖𝑣)^𝑘𝐿^𝑘_𝑚(|𝑧|²2), (2.8) where 𝐿^𝑘_𝑚(𝑡) is the Laguerre polynomial defined by the Rodriguez formula as

𝐿^𝑘_𝑚1(𝑡)=𝑡𝑚!^−𝑘𝑒^𝑡(𝑑)𝑑𝑡^𝑚(𝑡^𝑘+𝑚𝑒^−𝑡),𝑡∈ℝ. (2.9)

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

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

_ℝℎ_𝑚+𝑘(𝑥)𝑒^−iuxℎ_𝑚(𝑥+𝑣)𝑑𝑥=(−𝑖√2)^𝑘𝑚!𝑒(𝑚+𝑘)!^{−1/4[𝑢2+𝑣2−2𝑖𝑢𝑣]}(𝑢−𝑖𝑣)^𝑘𝐿^𝑘_𝑚(|𝑧|²2). (2.10)

From (2.10), we deduce for 𝑘=0,1,2,… that

𝐵_ℎ𝑚ℎ_𝑚+𝑘(𝑢,𝑣)=(−𝑖√2)^𝑘𝑚!𝑒(𝑚+𝑘)!^{−1/4[𝑢2+𝑣2+2𝑖𝑢𝑣]}(𝑢+𝑖𝑣)^𝑘𝐿^𝑘_𝑚(|𝑧|²2) (2.11) or

𝐵_ℎ𝑚ℎ_𝑚+𝑘(𝑢,𝑣)=𝜔(𝑢,𝑣)(−𝑖√2)^𝑘𝑚!(𝑚+𝑘)!(𝑢+𝑖𝑣)^𝑘𝐿^𝑘_𝑚(|𝑧|²2) (2.12) where

𝜔(𝑢,𝑣)=𝑒^{−1/4[𝑢2+𝑣2+2𝑖𝑢𝑣]}=𝜔(𝑧). (2.13)

 (2) If 0≤k≤m, we have the following formula:

_ℝℎ_𝑚−𝑘(𝑥)𝑒^−iuxℎ_𝑚(𝑥−𝑣)𝑑𝑥=(−𝑖√2)^𝑘(𝑚−𝑘)!𝑒𝑚!^{−1/4[𝑢2+𝑣2+2𝑖𝑢𝑣]}(𝑢−𝑖𝑣)^𝑘𝐿^𝑘_𝑚−𝑘(|𝑧|²2). (2.14)

That permits us to write for 0≤𝑘≤𝑚

𝐵_ℎ𝑚ℎ_𝑚−𝑘(𝑢,𝑣)=𝜔(𝑢,𝑣)(−𝑖√2)^𝑘(𝑚−𝑘)!𝑒𝑚!^{−1/4[𝑢2+𝑣2+2𝑖𝑢𝑣]}(𝑢−𝑖𝑣)^𝑘𝐿^𝑘_𝑚−𝑘(|𝑧|²2). (2.15)

Denoting 𝑧^′=𝑢^′+𝑖𝑣^′, then the reproducing kernel 𝑁_ℎ𝑚 of 𝐵_ℎ𝑚(𝐿₂(ℝ)) can be computed as

𝑁_ℎ𝑚(𝑢,𝑣,𝑢^′,𝑣^′)=⟨ℎ^𝑧_𝑚,ℎ^𝑧′_𝑚⟩_𝐿2(ℝ2). (2.16)

Hence we have

𝑁_ℎ𝑚(𝑢,𝑣,𝑢^′,𝑣^′)=_ℝ𝑒^−iuxℎ_𝑚(𝑥−𝑣)𝑒^{𝑖𝑢′𝑥}ℎ_𝑚(𝑥−𝑣^′)𝑑𝑥. (2.17) After a simple variable change, for example 𝑥=𝑦+𝑣^′, then 𝑁_ℎ𝑚(𝑢,𝑣,𝑢^′,𝑣^′) can be alsoexpressed as 𝑁_ℎ𝑚(𝑢,𝑣,𝑢^′,𝑣^′)=_ℝℎ_𝑚(𝑦)ℎ_𝑚(𝑦+𝑣^′−𝑣)𝑒^{−𝑖(𝑢−𝑢′)(𝑦+𝑣′)}𝑑𝑦. (2.18)

Hence

𝑁_ℎ𝑚(𝑢,𝑣,𝑢^′,𝑣^′)=𝑒^{−1/4[𝑢2+𝑣2+2𝑖𝑢𝑣]}𝑒^{−1/4[𝑢′2+𝑣′2−2𝑖𝑢′𝑣′]}𝑒^{(𝑢+𝑖𝑣)(𝑢′−𝑖𝑣′)/2}𝐿_𝑚((𝑢−𝑢^′)²+(𝑣−𝑣^′)²2) (2.19) or

𝑁_ℎ𝑚(𝑢,𝑣,𝑢^′,𝑣^′)=𝜔(𝑢,𝑣)𝜔(𝑢,𝑣)𝑒^{(𝑢+𝑖𝑣)(𝑢′−𝑖𝑣′)/2}𝐿_𝑚((𝑢−𝑢^′)²+(𝑣−𝑣^′)²2). (2.20)

The reproducing relation in 𝐵_ℎ𝑚(𝐿₂(ℝ)) can be written as

𝐵_ℎ𝑚𝑓(𝑢,𝑣)=_ℝ2𝑁_ℎ𝑚(𝑢,𝑣,𝑢^′,𝑣^′)𝐵_ℎ𝑚𝑓(𝑢^′,𝑣^′)𝑑𝑢^′𝑑𝑣^′, (2.21) then 𝐵_ℎ𝑚(𝐿₂(ℝ)) is a reproducing space with the reproducing kernel 𝑁_ℎ𝑚.

Introducing the complex structures 𝑧 and 𝑧^′, theabove formulas can be rewriten as

𝐵_ℎ𝑚ℎ_𝑚+𝑘(𝑧)=𝜔(𝑧)(−𝑖𝑧√2)^𝑘𝑚!𝐿(𝑚+𝑘)!^𝑘𝑚(|𝑧|²2𝐵),_ℎ𝑚ℎ_𝑚−𝑘(𝑧)=𝜔(𝑧)(−𝑖𝑧√2)^𝑘(𝑚−𝑘)!𝐿𝑚!^{𝑘𝑚−𝑘}(|𝑧|²2𝑁),_ℎ𝑚(𝑧,𝑧^′)=𝜔(𝑧)𝜔(𝑧)𝑁_𝑚(𝑧,𝑧^′), (2.22)

with

𝑁_𝑚(𝑧,𝑧^′)=𝑒^{𝑧𝑧′/2}𝐿_𝑚(|𝑧−𝑧^′|²2). (2.23)

We also can see that by changing the variable 𝑧 into √2𝜉 and 𝑧^′ into √2𝜉^′ that 𝑁_𝑚(𝑧,𝑧^′) can be transformed to kernels given in [9, Theorem 2]

𝐾_𝑚(𝜉,𝜉^′)=𝑒^𝜉𝜉′𝐿_𝑚(|𝜉−𝜉^′|²). (2.24) Now, we can define the “generalized” Bargmann transforms by

𝐵_𝑚1𝑓(𝑧)=𝐵𝜔(𝑧)^ℎ𝑚𝑓(𝑧) (2.25) and the “generalized” Bargmann spaces by

𝐸_𝑚=𝐵_𝑚[𝐿₂(ℝ)]. (2.26) Now, the consequences of all the above computations are summarized in the following form.

Theorem 2.2. (i)𝐵_𝑚[𝐿₂(ℝ)] is a subspace of 𝐿₂(ℂ,𝑒^−|𝑧|2𝑑𝑢𝑑𝑣).
(ii)𝐵_𝑚ℎ_𝑚+𝑘(𝑧)=(−𝑖𝑧√2)^𝑘𝑚!𝐿(𝑚+𝑘)!^𝑘𝑚(|𝑧|²2),𝑘=0,1,…. (2.27)
(iii) For 𝑚=0, 1/√𝑘!(−𝑖𝑧/√2)^𝑘 is a basis of classical Bargmann space 𝐸₀.
(iv)𝐵_𝑚ℎ_𝑚−𝑘(𝑧)=(−𝑖𝑧√2)^𝑘(𝑚−𝑘)!𝐿𝑚!^{𝑘𝑚−𝑘}(|𝑧|²2),𝑘=0,…,𝑚. (2.28)
(v)𝐵_𝑚1𝑓(𝑧)=𝜔(𝑧)_ℝ𝑒^iux𝑑𝑓(𝑥)(𝑥−𝑣−)𝑑𝑥^𝑚𝑒^{−(𝑥−𝑣)2}=√𝑑𝑥2^𝑚𝑚!𝜔(𝑧)_ℝ𝑒^{−1/2(𝑥2+𝑣2)+𝑖𝑧𝑥}𝐻_𝑚(𝑥−𝑣)𝑓(𝑥)𝑑𝑥. (2.29)
(vi)𝐵_𝑚𝑓(𝑖𝑧)=√2^𝑚𝑚!_ℝ𝑒^{−(𝑧/2−𝑥)2}𝐻_𝑚(𝑥−𝑧+𝑧2)𝑓(𝑥)𝑑𝑥. (2.30)

Remark 2.3. (i) The operator 𝐴^∗=−𝜕/𝜕𝑧+𝑧 is the adjoint of the operator 𝐴=𝜕/𝜕𝑧 in 𝐿₂(ℂ,𝑒^−|𝑧|2𝑑𝑢𝑑𝑣) and Δ=𝐴^∗𝐴; see for example [19] or [20].
(ii) In [9, 10] the “generalized” Bargmann spaces 𝐸_𝑚 are constructed as null spaces of the second-order differential operator Δ=−𝜕²/𝜕𝑧𝜕𝑧+𝑧(𝜕/𝜕𝑧)−𝑚𝐼 in 𝐿₂(ℂ,𝑒^−|𝑧|2𝑑𝑢𝑑𝑣) but in this section we have given another construction of these spaces which allows us to construct the “generalized” Bargmann transforms (2.25).

3. Construction of Generalized Coherent States |𝑧⟩_𝑚 Associated to Generalized Bargmann Spaces 𝐸_𝑚

In the separable Hilbert space 𝐻 spanned by the eigenvectors |𝑛⟩𝑛=0,1,2,… of the positive semidefinite number operator 𝑁=𝑎^∗𝑎, one can define for each complex number 𝑧=𝑥+𝑖𝑦

|𝑧⟩_𝑚=𝑒^{−1/2|𝑧|2∞}_𝑛=0𝐴^∗𝑚𝑧^𝑛√𝑛!|𝑛⟩, (3.1) where

𝐴^∗𝜕=−+𝜕𝑧𝑧. (3.2)

The space 𝐈_𝑚 spanned by the set {|𝑧⟩_𝑚} is a separable Hilbert space which is in fact isomorphic to 𝐻. This space is called the “generalized” space of coherent states. It is closely related to the “generalized” Bargmann space 𝐸_𝑚.

For any |𝜙_𝑚⟩∈𝐼_𝑚, we can define a function by

𝜙_𝑚(𝑧)=𝑒^1/2|𝑧|2⟨𝜙_𝑚∣𝑧⟩_𝑚=^∞_𝑛=0𝐴^∗𝑚𝑧^𝑛√𝑛!⟨𝜙_𝑚∣𝑛⟩. (3.3)

As ∫_ℂ|⟨𝜙_𝑚∣𝑧⟩_𝑚|²𝑑𝑥𝑑𝑦<∞, then

_ℂ|𝜙_𝑚(𝑧)|²𝑒^−|𝑧|2𝑑𝑥𝑑𝑦<∞. (3.4)

Theorem 3.1. Let 𝐴=𝜕/𝜕𝑧 and 𝐴^∗=−𝜕/𝜕𝑧+𝑧, then the following statements hold.(i)𝐴𝐴^∗𝑚𝑓(𝑧)=𝑚𝐴^∗𝑚−1𝑓(𝑧); 𝑚=1,2,… for any holomorphic function 𝑓(𝑧). (ii)Let 𝜙_𝑚(𝑧)=𝑒^1/2|𝑧|2⟨𝜙_𝑚∣𝑧⟩_𝑚, then 𝜙_𝑚(𝑧) is in the “generalized” Bargmann space 𝐸_𝑚; 𝑚=0,1,…(iii)|𝑧⟩_𝑚=^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀, (3.5) where 𝑎^∗|𝑛⟩=√𝑛+1|𝑛+1⟩ and 𝐶^𝑗_𝑚=𝑚!/𝑗!(𝑚−𝑗)!

Proof. (i) Let 𝑓(𝑧) be holomorphic function; then by applying the commutation relation [𝐴,𝐴^∗]=𝐼 and the Cauchy-Riemann equation, we deduce that (i) is satisfied for 𝑚=1 and by induction that 𝐴𝐴^∗𝑚+1𝑓(𝑧)=𝐴^∗𝐴𝐴^∗𝑚𝑓(𝑧)+𝐴^∗𝑚𝑓(𝑧)=(𝑚+1)𝐴^∗𝑚𝑓(𝑧). (3.6)
  (ii) For 𝑚=0, we have |𝑧⟩₀=𝑒^{−1/2|𝑧|2}∑^∞_𝑛=0(𝑧^𝑛/√𝑛!)|𝑛⟩, and 𝜙₀(𝑧) is in the classical Bargmann space 𝐸₀.
For 𝑚≠0 and 𝜙_𝑚(𝑧)=∑^∞_𝑛=0𝐴^∗𝑚(𝑧^𝑛/√𝑛!)⟨𝜙_𝑚∣𝑛⟩ we have
𝐴^∗𝐴𝜙_𝑚(𝑧)=^∞_𝑛=0𝐴^∗𝐴𝐴^∗𝑚𝑧^𝑛√𝑛!⟨𝜙_𝑚∣𝑛⟩. (3.7) As the function 𝑓(𝑧)=𝑧^𝑛 is holomorphic, we can apply property (i). Premultiplying by 𝐴^∗ the equation 𝐴𝐴^∗𝑚𝑓(𝑧)=𝑚𝐴^∗𝑚−1𝑓(𝑧), we obtain 𝐴^∗𝐴𝜙_𝑚(𝑧)=^∞_𝑛=0𝐴^∗𝐴𝐴^∗𝑚𝑧^𝑛√𝑛!⟨𝜙_𝑚∣𝑛⟩=^∞_𝑛=0𝑚𝐴^∗𝑚𝑧^𝑛√𝑛!⟨𝜙_𝑚∣𝑛⟩=𝑚𝜙_𝑚(𝑧). (3.8)
  (iii) Let 𝑎^∗𝑗|𝑧⟩₀=𝑒^{−1/2|𝑧|2∞}_𝑛=0𝑧^𝑛√𝑎𝑛!^∗𝑗|𝑛⟩=𝑒^{−1/2|𝑧|2∞}_𝑛=0(𝜕)𝜕𝑧^𝑗𝑧^𝑛√𝑛!|𝑛⟩,|𝑧⟩_𝑚=𝑒^{−1/2|𝑧|2∞}_𝑛=0𝐴^∗𝑚𝑧^𝑛√𝑛!|𝑛⟩. (3.9)
As 𝜕/𝜕𝑧(𝑧𝑓(𝑧))=𝑧(𝜕/𝜕𝑧)𝑓(𝑧), we have
𝐴^∗𝑚=(𝜕𝑧−)𝜕𝑧^𝑚=^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗(𝜕)𝜕𝑧^𝑗, (3.10) where 𝐶^𝑗_𝑚=𝑚!/𝑗!(𝑚−𝑗)!Then |𝑧⟩_𝑚=^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀. (3.11) The action of the annihilation operator 𝑎 on |𝑧⟩_𝑚 for 𝑚≠0 is presented below, in the form of a theorem.

Theorem 3.2. (i)𝑎|𝑧⟩_𝑚=𝑧|𝑧⟩_𝑚−𝑚|𝑧⟩_𝑚−1.
(ii) There exists an operator 𝑇_𝑚 such that 𝑇_𝑚|𝑧⟩_𝑚=𝑧|𝑧⟩_𝑚.

Proof. (i) Let |𝑧⟩_𝑚=𝑒^{−1/2|𝑧|2}∑^∞_𝑛=0𝐴^∗𝑚𝑧^𝑛/√𝑛!|𝑛⟩, where 𝐴^∗=−𝜕/𝜕𝑧+𝑧.
By applying the operator 𝑎=𝑧/2−𝜕/𝜕𝑧 on |𝑧⟩_𝑚, we get
𝑎|𝑧⟩_𝑚=𝑧2|𝑧⟩_𝑚−𝜕𝜕𝑧(𝑒^{−1/2|𝑧|2∞}_𝑛=0𝐴^∗𝑚𝑧^𝑛√=𝑧𝑛!|𝑛⟩)2|𝑧⟩_𝑚+𝑧2|𝑧⟩_𝑚−𝑒^{−1/2|𝑧|2}𝜕𝜕𝑧(^∞_𝑛=0𝐴^∗𝑚𝑧^𝑛√𝑛!|𝑛⟩)=𝑧|𝑧⟩_𝑚−𝑒^{−1/2|𝑧|2∞}_𝑛=0𝜕𝜕𝑧𝐴^∗𝑚𝑧^𝑛√𝑛!|𝑛⟩ (3.12)
Now as 𝑓(𝑧)=𝑧^𝑛 is a holomorphic function and 𝐴=𝜕/𝜕𝑧, we have 𝜕/𝜕𝑧𝐴^∗𝑚𝑓(𝑧)=𝑚𝐴^∗𝑚−1𝑓(𝑧) and we get
𝑎|𝑧⟩_𝑚=𝑧|𝑧⟩_𝑚−𝑚𝑒^{−1/2|𝑧|2∞}_𝑛=0𝐴^∗𝑚−1𝑧^𝑛√𝑛!|𝑛⟩; (3.13) that is, 𝑎|𝑧⟩_𝑚=𝑧|𝑧⟩_𝑚−𝑚|𝑧⟩_𝑚−1. (3.14)
(ii)In the above property (i) of the present theorem, it is shown that the “generalized” coherent states are not eigenstates of the annihilation operator. Nevertheless, we will show that these “generalized” 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 |𝑧⟩_𝑚 obeys to the equation
𝑇_𝑚|𝑧⟩_𝑚=𝑧|𝑧⟩_𝑚 (3.15) with a suitable choice for the operator 𝑇_𝑚. Let us now construct the explicit form of 𝑇_𝑚.
From (3.11), we have 𝑧⟩_𝑚=∑^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀, then 𝑧⟩_𝑚 is in the space 𝐹_𝑚 spanned by {𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀}_{𝑗=0,1,…,𝑚}.
By definition, the coherent states |𝑧⟩₀ satisfy
𝑎|𝑧⟩₀=𝑧|𝑧⟩₀. (3.16)
Premultiplying the both sides of this equation by 𝑎^∗𝑗 leads
𝑎^∗𝑗𝑎|𝑧⟩₀=𝑧𝑎^∗𝑗|𝑧⟩₀. (3.17)
Using the commutation relation [𝑎,𝑎^∗]=𝐼, the above equation is written as
𝑎𝑎^∗𝑗−𝑗𝑎^∗𝑗−1|𝑧⟩₀=𝑧𝑎^∗𝑗|𝑧⟩₀. (3.18) which, making use of the identity (𝑎^∗𝑎+𝐼)⁻¹𝑎𝑎^∗=𝐼, leads to [𝑎−𝑗(𝑎^∗𝑎+𝐼)⁻¹𝑎]𝑎^∗𝑗|𝑧⟩₀=𝑧𝑎^∗𝑗|𝑧⟩₀ (3.19) In the following, we introduce 𝑅_𝑗=(𝐼−𝑗(𝑎^∗𝑎+𝐼)⁻¹)𝑎 (3.20) and 𝑃_𝑗 the projection operator on the space spanned by the system {𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀}, 𝑗=0,1,…,𝑚𝑃_𝑘𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀=𝛿_𝑗𝑘𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀, (3.21) where 𝛿_𝑗𝑘=1if𝑘=𝑗 and 𝛿_𝑗𝑘=0if𝑘≠𝑗.
Let
𝑇_𝑚=^𝑚_𝑗=0𝑅_𝑗𝑃_𝑗. (3.22) Acting 𝑇_𝑚 on |𝑧⟩_𝑚 leads to 𝑇_𝑚|𝑧⟩_𝑚=^𝑚_𝑗=0𝑅_𝑗𝑃_𝑗^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀=^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑅_𝑗𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀=^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗𝑅_𝑗𝑎^∗𝑗|𝑧⟩₀=^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗𝑧𝑎^∗𝑗|𝑧⟩₀=𝑧^𝑚_𝑗=0(−1)^𝑗𝐶^𝑗_𝑚𝑧^𝑚−𝑗𝑎^∗𝑗|𝑧⟩₀, (3.23) that is, 𝑇_𝑚|𝑧⟩_𝑚=𝑧|𝑧⟩_𝑚. (3.24) To sum up, we have used the Gabor spaces 𝐵_ℎ𝑚(𝐿₂(ℝ)) with the 𝑚^𝑡ℎ Hermite window ℎ_𝑚 to give another construction of generalized Bargmann spaces 𝐸_𝑚 which is different from that given in [9, 10] or [11] and have constructed a family of “generalized” Bargmann transforms attached to these spaces. The states |𝑧⟩_𝑚 constructed in this work with 𝑚≠0 are nonclassical and can be interpreted as nonlinear coherent states in nonanalytic representation. The spaces ⋀_𝑚∶=𝐸_𝑚−1,𝑚=1,2,…, are constituted by the polyanalytic functions 𝜙(𝑧) of order 𝑚, that is, (𝜕𝜕𝑧)^𝑚𝜙(𝑧)=0,𝑚=1,2,… (3.25) which are not polyanalytic of any lower order and satisfy _ℂ|𝜙(𝑧)|²𝑒^−|𝑧|2𝑑𝑥𝑑𝑦⟨∞. (3.26) The functions 𝜙(𝑧) in ⋀_𝑚,𝑚=1,2,…, can be uniquely expressed in the form 𝜙(𝑧)=_{0≤𝑘≤𝑚−1}𝑧^𝑘∞_𝑗=0𝑎_𝑗,𝑘𝑧^𝑗. (3.27) Finally, the spaces 𝐸_𝑚 defined in this work are the “true polyanalytic Fock spaces” of [11] or the “generalized Bargmann spaces” of [9, 10].
When 𝑚 is not zero, we give (in a further work) a complete spectral analysis of the nonselfadjoint operator 𝐻_𝐼=𝐵^∗(𝐵+𝐵^∗)𝐵 on ⋀_𝑚 or on ⊕^𝑚_𝑗=1⋀_𝑚 (the natural generalization of the Bargmann space, since (3.25) generalizes the Cauchy-Riemann equation (𝜕/𝜕𝑧)𝜙(𝑧)=0), where 𝐵=𝜕/𝜕𝑧 and 𝐵^∗=−𝜕/𝜕𝑧+𝑧.
The restriction of 𝐻_𝐼 on the Bargmann space 𝐸₀ 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.