Gelfand Triplets, Ladder Operators and Coherent States

Abstract

Inspired by a similar construction on Hermite functions, we construct two series of Gelfand triplets, each one spanned by Laguerre–Gauss functions with a fixed positive value of one parameter, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre–Gauss functions with negative values of the fundamental parameter are proven to be continuous functionals on one of these triplets. Different sorts of coherent states are considered and proven to be in some spaces of test functions corresponding to Gelfand triplets.

1. Introduction and Motivation

As is well known, observables in the usual formalism of Quantum Mechanics are given by self-adjoint operators on a Hilbert space. Although the modern characterization of observables has shifted from the self-adjoint property to the PT-invariance property, we should keep Hamiltonians self-adjoint for two reasons: (i) The evolution law for a given Hamiltonian H, $e^{-itH}$, is well defined and is unitary; therefore, it preserves the probabilities defined by wave functions. (ii) The current density is preserved if the Hamiltonian is self-adjoint.

However, self-adjoint and the vast majority of quantum observables are given by unbounded self-adjoint operators on a separable infinite-dimensional Hilbert space. This destroys the intuition derived from matrix calculus to deal with operators. The use of unbounded operators involves many subtleties that do not appear in the matrix calculus or in its direct generalization, the calculus with bounded operators. The simple notion of unbounded operator implies the existence of its domain, a subspace of the Hilbert space on which the operator is well defined, which cannot be the whole Hilbert space. The situation is quite different in some other aspects. Let A be an operator with dense (the domain of an operator should be dense; otherwise, its adjoint is not well defined) domain $\mathcal{D}\left(A\right)\subset \mathcal{H}$, where $\mathcal{H}$ is a separable infinite-dimensional Hilbert space. Contrary to what happens for bounded operators (i.e., matrices on a finite-dimensional Hilbert space), the relation

$$⟨\psi |A\phi ⟩=⟨A\psi |\phi ⟩,\forall \varphi ,\phi \in \mathcal{D}\left(A\right),$$

(1)

does not imply the self-adjointness of A, just that its adjoint $A^{†}$ extends (This means that $\mathcal{D}\left(A\right)\subset \mathcal{D}\left(A^{†}\right)$ and that for all $\psi \in \mathcal{D}\left(A\right)$, $A\psi =A^{†}\psi$). A, $A\prec A^{†}$. Self-adjointness of A implies not only $A\prec A^{†}$ but also that $\mathcal{D}\left(A\right)\equiv \mathcal{D}\left(A^{†}\right)$, so that $A\equiv A^{†}$.

Bounded operators are continuous linear actions on the Hilbert space, while unbounded operators, although linear, are not continuous on Hilbert space. However, for a densely defined unbounded self-adjoint operator A, there is always a dense subdomain thereof, $\Phi$, endowed with a topology finer (A topology is finer than another on the same set if it has more open sets.) than the Hilbert space topology, on which A is continuous. This result is the well-known Gelfand–Maurin theorem [1,2]. Note that this discussion can take place on infinite-dimensional Hilbert spaces only.

The study of quantum one-dimensional systems is quite important for two particular reasons. One is that one-dimensional systems provide simple models with full quantum properties. In addition, these models are very often solvable or quasi-solvable, which makes their study particularly interesting. Among these one-dimensional systems, there are some special ones in which the Hamiltonian admits a factorization in terms of some operators called ladder operators (plus some additional term) [3,4,5,6,7]. This factorization may have an important consequence as it allows the construction of a sequence of Hamiltonians, each one with a similar spectrum to the precedent. Here, Hamiltonians and the ladder operators that factorize them are unbounded.

Then, our question is as follows: Can we find a domain of the Hilbert space and a topology on this domain for which the Hamiltonian and ladder operators that factorize it are continuous? This may go beyond a mathematical curiosity, as continuous linear operators have properties that may serve to cancel some of the complicated subtleties of non-continuous operators. The answer passes through a mathematical concept widely used in quantum physics, such that the notion of Gelfand triplet is also known as rigged Hilbert space (RHS). We define this notion and give some basic results in the next section.

The present mathematical discussion has its origins in a study on Quantum Optics, which analyzes classes of solutions of a paraxial wave equation for parabolic media in order to study the observable effects of orbital angular momentum carried by laser beams [8]. The research in [8] was motivated by some previous studies on the production and observation of light modes with a given orbital angular momentum [9,10,11]. Solutions studied in [8] refer to these solutions of the mentioned paraxial wave equation, known as stationary Laguerre–Gauss modes or simply Laguerre–Gauss modes. Functions, ladder operators and all the ingredients, whose properties are discussed in the present article, were introduced after these studies on Quantum Optics.

At the same time, the present study is a sort of continuation of a series of works showing that Gelfand triplets represent the appropriate framework where discrete and continuous bases, the basis of special functions and Lie algebras of symmetries of a given system coexist, represented by continuous operators [12,13,14]. This is not possible on Hilbert spaces, where continuous bases are not defined and Lie algebras are represented by unbounded operators. A similar analysis is conducted here, where the Laguerre–Gauss modes give a discrete basis, which helps in the construction of the topology of a test space and where the ladder operators are continuous.

The organization of the present article goes as follows: In Section 2, we define the notion of Gelfand triplet, also known as rigged Hilbert space, and give the Schwartz space as the most important example. In Section 3, we discuss the application to the harmonic oscillator, which is a representative example and helps to understand the latter analysis. On Section 4, we introduce the Laguerre–Gauss functions and define the relevant ladder operators. Section 5 contains the main part of this presentation. We construct test spaces with their topologies and show the continuity of the relevant operators under these topologies. Some comments on coherent states are given in Section 6. Concluding remarks are left for Section 7. Finally, we have added Appendix A and Appendix B, with some technical comments and results that we consider relevant.

2. Gelfand Triplets: Definition and Properties

A Gelfand triplet or rigged Hilbert space is a term of linear spaces over the complex field [15,16,17,18]

$$\Phi \subset \mathcal{H}\subset {\Phi }^{\times },$$

(2)

where (i) $\mathcal{H}$ is an infinite-dimensional separable Hilbert space; (ii) $\Phi$ is a dense subspace of $\mathcal{H}$ endowed with its own topology, which is finer than the topology that $\Phi$ has inherited from (In particular, this implies that for the canonical injection, $i\left(\phi \right)=\phi$, $\forall \phi \in \Phi$ is continuous.) $\mathcal{H}$; (iii) ${\Phi }^{\times }$ is the space of continuous antilinear (Or linear. The advantage of antilinear functionals is that their action may be represented with a notation compatible with the Dirac notation of Quantum Mechanics.) functionals (A mapping $F:\Phi ⟼\mathbb{C}$ is an antilinear functional on $\Phi$ if, for any pair $\varphi ,\phi \in \Phi$ and any pair of complex numbers $\alpha ,\beta \in \mathbb{C}$, one has that

$$F(\alpha \varphi +\beta \phi )={\alpha }^{*}F\left(\varphi \right)+{\beta }^{*}F\left(\phi \right),$$

where the asterisk denotes complex conjugation. In addition, if F were continuous with respect to the topologies on $\Phi$ and on $\mathbb{C}$, then $F\in {\Phi }^{\times }$. Note that ${\Phi }^{\times }$ has a natural structure as a linear space.) on $\Phi$ endowed with a topology compatible (This notion of compatibility is rather technical and, usually, we shall choose on ${\Phi }^{\times }$ its weak topology to be defined later.) with the dual pair $\{\Phi ,{\Phi }^{\times }\}$. Often, ${\Phi }^{\times }$ is called the antidual space of $\Phi$. The space $\Phi$ is sometimes called the test vector space.

The space $\Phi$ is a locally convex space [19], with a topology defined by a set of seminorms, (A seminorm is a mapping $p:\Phi ⟼\mathbb{C}$, such that, for all $\varphi \in \Phi$, (i) $p\left(\varphi \right)\ge 0$; (ii) for all $\alpha \in \mathbb{C}$, $p\left(\alpha \varphi \right)=\left|\alpha \right|p\left(\varphi \right)$; (iii) for all $\varphi ,\phi \in \Phi$, $p(\varphi +\phi )\le p\left(\varphi \right)+p\left(\phi \right)$. Thus, a seminorm is like a norm in which we admit that $p\left(\varphi \right)=0$ with $\varphi \ne \mathbf{0}$. In particular, any norm is a seminorm) one of which has to be the Hilbert space norm defined on $\Phi$ as a subspace of $\mathcal{H}$. In our particular context, the topology on $\Phi$ will be defined by a countably infinite set of seminorms so that the space $\Phi$ is metrizable (We always may choose $\Phi$ to be complete under its topology. A complete metrizable locally convex space is a Frèchet space).

A sequence $\left\{{\varphi }_n\right\}\in \Phi$ converges to $\varphi \in \Phi$ if and only if, for any seminorm p giving the topology on $\Phi$, it happens that $p\left({\varphi }_n\right)⟼p\left(\varphi \right)$. If $\Phi$ is metrizable, a linear (Linearity is irrelevant here, but we shall use linear mappings only.) mapping $F:\Phi ⟼\mathbb{C}$ is continuous if and only if, for any convergent sequence ${\varphi }_n⟼\varphi$ in $\Phi$, we have that $F\left({\varphi }_n\right)⟼F\left(\varphi \right)$. Similarly, an operator A on $\Phi$ is continuous if $A{\varphi }_n⟼A\varphi$.

Nevertheless, if $\Phi$ were a locally convex space, another more operative method to check the continuity of linear or antilinear mappings is in order. If $F:\Phi ⟼\mathbb{C}$ is linear, then it is continuous if and only if there exists a positive constant $C>0$ and a finite number of seminorms $\{p_1,p_2,\cdots ,p_n\}$ from those that define the topology on $\Phi$ such that for all $\varphi \in \Phi$, we have

$$\left|F\left(\varphi \right)\right|\le C\{p_1\left(\varphi \right)+p_2\left(\varphi \right)+\cdots +p_n\left(\varphi \right)\}.$$

(3)

Analogously, if $A:\Phi ⟼\Phi$ is linear, it is continuous if and only if, for any seminorm $p_i$ defining the topology on $\Phi$, there is a positive constant $C>0$ and k seminorms out of those that define the topology on $\Phi$ such that for all $\varphi \in \Phi$ [20],

$$p_i\left(A\varphi \right)\le C\{p_{i_1}\left(\varphi \right)+p_{i_2}\left(\varphi \right)+\cdots +p_{i_k}\left(\varphi \right)\}.$$

(4)

In principle, the constant C, the seminorms $p_{i_j}$ and its number k, depend on the particular $p_i$ chosen. Obviously, C and the choice of the seminorms is not unique.

This result may be extended to any linear operator $A\Phi ⟼\Psi$, where $\Phi$ and $\Psi$ are different locally convex spaces. If $\left\{p_i\right\}$ and $\left\{q_j\right\}$ are the families of seminorms that produce the topologies on $\Phi$ and $\Psi$, respectively, (4) is now written as

$$q_j\left(A\varphi \right)\le C_j\{p_{i_{j_1}}\left(\varphi \right)+p_{i_{j_2}}\left(\varphi \right)+\cdots +p_{i_{j_k}}\left(\varphi \right)\},$$

(5)

for all seminorms $q_j$ on $\Psi$, where $C_j$ and the seminorms $p_{i_{j_s}}$ depend on $q_j$.

At this point, let us define the notion of weak topology on the antidual space ${\Phi }^{\times }$. This is a locally convex topology, for which the seminorms are defined like this: for each $\varphi \in \Phi$, we define a seminorm $p_{\varphi }$ on ${\Phi }^{\times }$, such that, for any $F\in {\Phi }^{\times }$, one has

$$p_{\varphi }\left(F\right):=\left|F\left(\varphi \right)\right|.$$

(6)

Finally, it is convenient to use $⟨F|\varphi ⟩$ instead of $F\left(\varphi \right)$ to denote the action of $F\in {\Phi }^{\times }$ on $\varphi \in \Phi$. As a matter of fact, the weak topology (As any other compatible with duality such as the strong or the McKey topologies.) induces on $\mathcal{H}$ a topology weaker (it has less open sets) than the Hilbert space topology. Note that any $\psi \in \mathcal{H}$ gives an $F_{\psi }\in {\Phi }^{\times }$ defined as

$$⟨F_{\psi }|\varphi ⟩=F_{\psi }\left(\varphi \right):=⟨\psi |\varphi ⟩,\forall \varphi \in \Phi ,$$

(7)

where $⟨-|-⟩$ denotes the scalar product on $\mathcal{H}$. The functional $F_{\psi }$ is obviously linear on $\Phi$. Furthermore, it is also continuous since

$$|⟨F_{\psi }|\varphi ⟩|=|⟨\psi |\varphi ⟩|\le |\left|\psi \right|\left|\right|\left|\varphi \right||=C{p}_0\left(\varphi \right),\forall \varphi \in \Phi ,$$

(8)

with $C=\left|\right|\psi \left|\right|$. Being given $\psi \in \mathcal{H}$, the $F_{\psi }\in {\Phi }^{\times }$ is unique. This mapping $i\left(\psi \right)=F_{\psi }$, for all $\psi \in \mathcal{H}$, is antilinear and continuous (The antilinearity is obvious. To show continuity, let us choose an arbitrary $\phi \in \Phi$. Then,

$$p_{\phi }\left(i\left(\psi \right)\right)=p_{\phi }\left(F_{\psi }\right)=⟨F_{\psi }|\phi ⟩=⟨\psi |\phi ⟩\le \left|\right|\phi \left|\right|\left|\right|\psi \left|\right|\le C\left|\right|\psi \left|\right|,$$

which proves the continuity of $i:\mathcal{H}⟼{\Phi }^{\times }$). Note that the choice of antilinear functionals, instead of linear functionals, in order to comply with the Dirac notation, becomes obvious after (6).

Now, one understands the structure in (2). These spaces are related through continuous canonical injections (always $i\left(\psi \right)=\psi$, whatever $\psi$ is), $\Phi ⟼\mathcal{H}⟼{\Phi }^{\times }$.

The Schwartz Space

Let us give an example of Gelfand triplet, which is more than a simple example, since it will help us to construct other triplets for our purposes. The Schwartz space $\mathcal{S}$ is the linear space of all functions $f\left(x\right):\mathbb{R}⟼\mathbb{C}$, where $\mathbb{R}$ is the real line, such that the following apply:

(i)

Any $f\left(x\right)\in \mathcal{S}$ is indefinitely differentiable at all points of the real line $\mathbb{R}$.

(ii)

Any $f\left(x\right)\in \mathcal{S}$ converges to zero at the infinite faster than the inverse of any polynomial. This means that, for any (complex, x real) polynomial $P\left(x\right)$ and any $f\left(x\right)\in \mathcal{S}$, one has that

$$\underset{x\to \pm \infty }{lim}{\displaystyle \frac{f\left(x\right)}{P\left(x\right)}}=0.$$

(9)

All functions $f\left(x\right)\in \mathcal{S}$ are in $L^2\left(\mathbb{R}\right)$. Furthermore, $\mathcal{S}$ is dense in $L^2\left(\mathbb{R}\right)$ with the topology of the latter.

We may endow $\mathcal{S}$ with a locally convex metrizable topology into three equivalent forms, although we choose one here that is particularly interesting for our purposes. This comes after an interesting characterization of the Schwartz space $\mathcal{S}$ in terms of the Hermite functions, $\left\{{\psi }_n\left(x\right)\right\}$, to be defined in the next section. Hermite functions form an orthonormal basis (complete orthonormal set) for the Hilbert space $L^2\left(\mathbb{R}\right)$ so that any $f\left(x\right)\in L^2\left(\mathbb{R}\right)$ takes the form

$$f\left(x\right)=\sum _{n=0}^{\infty }{a}_n{\psi }_n{\left(x\right),\left|\right|f\left(x\right)\left|\right|}^2=\sum _{n=0}^{\infty }{\left|a_n\right|}^2<\infty .$$

(10)

Then, $f\left(x\right)\in L^2\left(\mathbb{R}\right)$ is in $\mathcal{S}$ if and only if, for the sequence $\left\{a_n\right\}$ in (10), one has that, for all $k=0,1,2,\dots$, [20]

$$\sum _{n=0}^{\infty }{(n+1)}^{2k}{\left|a_n\right|}^2<\infty .$$

(11)

Then, let us define the following set of norms $p_k(-)=\left|\right|-{\left|\right|}_k$ for all $f\left(x\right)\in \mathcal{S}$:

$$p_k^2\left(f\right):={\left|\right|f\left(x\right)\left|\right|}_k^2:=\sum _{n=0}^{\infty }{(n+1)}^{2k}{\left|a_n\right|}^2,k=0,1,2,\dots .$$

(12)

These norms (which are also seminorms) define the topology on $\mathcal{S}$. Note the following: (i) Since $k=0$ is a possible value, the list includes the norm on $L^2\left(\mathbb{R}\right)$. This implies that the topology on $\mathcal{S}$ is finer than the Hilbert space topology on $L^2\left(\mathbb{R}\right)$. (ii) Since the number of norms (seminorms) is countably infinite, the space $\mathcal{S}$ is metrizable. We may add that $\mathcal{S}$ is a Frèchet space having the property of nuclearity (This is a very technical property, with interesting implications that we shall not use here. For instance, the unit ball is compact, contrary to what happens in infinite-dimensional Hilbert spaces, or that the canonical injection $i:\mathcal{S}⟼L^2\left(\mathbb{R}\right)$ admits a spectral decomposition similar to those of compact operators [21]). Thus,

$$\mathcal{S}\subset L^2\left(\mathbb{R}\right)\subset {\mathcal{S}}^{\times }$$

(13)

is a Gelfand triplet or RHS. The antidual is algebraically and topologically isomorphic to the space of tempered distributions, usually defined as continuous linear functionals on $\mathcal{S}$.

3. The Harmonic Oscillator

Although the contents of this section are well known, we consider that a pedagogical account of them will help reach a much better understanding of the motivation, purpose and proofs for the results contained in the main body of the present article. We begin with the ubiquitous Harmonic oscillator.

As is well known, the Hamiltonian of one-dimensional quantum harmonic oscillator is given by

$$H={\displaystyle \frac{p^2}{2m}}+{\displaystyle \frac{1}{2}}m{\omega }^2{q}^2,\omega :=\sqrt{k/m}.$$

(14)

This Hamiltonian has a pure non-degenerate discrete spectrum with infinite values given by $E_n=\hslash \omega (n+1/2)$, $n=0,1,2,\dots$. The respective eigenfunctions are the normalized Hermite functions:

$${\psi }_n\left(x\right):={\displaystyle \frac{1}{2^{n}n!}}\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)e^{-{\displaystyle \frac{m\omega x^2}{2\hslash }}}{H}_n\left(\sqrt{{\displaystyle \frac{m\omega }{\hslash }}}x\right),n=0,1,2,\dots ,$$

(15)

where $H_n\left(y\right)$ are the Hermite polynomials. The annihilation and creation operators are, respectively, given by

$$a:=\sqrt{{\displaystyle \frac{m\omega }{2\hslash }}}\left(q+{\displaystyle \frac{i}{m\omega }}p\right),a^{†}:=\sqrt{{\displaystyle \frac{m\omega }{2\hslash }}}\left(q-{\displaystyle \frac{i}{m\omega }}p\right).$$

(16)

These operators are usually called the ladder operators. Note that q and p are the multiplication and differentiation operators, respectively, i.e., $q\psi \left(x\right)=x\psi \left(x\right)$ and $p\psi \left(x\right)=-i\hslash {\psi }^{\prime }\left(x\right)$, where the prime denotes derivation with respect to x. In terms of the ladder operators, the Hamiltonian (14) is written as

$$H=\hslash \omega \left(N+{\displaystyle \frac{1}{2}}\right),N:=a^{†}a.$$

(17)

The operator N is the number operator. When using the ladder operators, a change in the notation is somehow convenient for the sake of simplicity. Henceforth, we shall use $|n⟩:={\psi }_n\left(x\right)$, where all $n=0,1,2,\dots$. The action of the ladder operators on the normalized Hermite function is given in this latter notation as

$$a|n⟩=\sqrt{n}|n-1⟩,a|0⟩=0,a^{†}|n⟩=\sqrt{n+1}|n+1⟩.$$

(18)

Some properties are as follows:

• The Hilbert space on which Hamiltonian and ladder operators act is $L^2\left(\mathbb{R}\right)$.
• The Hermite functions form an orthonormal basis (also called complete orthonormal set) in $L^2\left(\mathbb{R}\right)$. Then, the subspace of $L^2\left(\mathbb{R}\right)$ of (finite) linear combinations of Hermite functions is dense in $L^2\left(\mathbb{R}\right)$.
• Hermite functions are Schwartz functions.
• The Hamiltonian and the ladder operators are unbounded operators. Hence, they do not act on the whole $L^2\left(\mathbb{R}\right)$ but just on subspaces thereof, called the domains of the operators. In any case, Hermite functions lie in the domains of all these three operators, so that these domains are always dense in $L^2\left(\mathbb{R}\right)$. All these domains contain the Schwartz space as a subspace.

As shown in the Introduction, (13) is a Gelfand triplet. Let us show that H, a and $a^{†}$ are continuous operators on the Schwartz space $\mathcal{S}$. Let $\psi \left(x\right)\in \mathcal{S}$ and write

$$\psi \left(x\right)=\sum _{n=0}^{\infty }{a}_n{\psi }_n\left(x\right)=\sum _{n=0}^{\infty }{a}_n|n⟩.$$

(19)

Since $\psi \left(x\right)\in \mathcal{S}$, the coefficients $\left\{a_n\right\}$ must satisfy (12). Let us define $a\psi$, the action of the operator a on $\psi \left(x\right)$, as

$$a\psi :=\sum _{n=1}^{\infty }{a}_n\sqrt{n}|n-1⟩.$$

(20)

In a unique operation, we show that a as in (19) is well defined and is continuous on on $\mathcal{S}$. For any norm $p_k(-)$ as in (12), $k=0,1,2,\dots$, we have for any $\psi \in \mathcal{S}$ that ($m=n+1$)

$$\begin{array}{c}\hfill p_k\left(a\psi \right)=\sum _{n=0}^{\infty }{b}_n^2{(n+1)}^{2k}=\sum _{n=0}^{\infty }|a_{n+1}{|}^2{(2n+1)}^{2k+1}=\sum _{m=1}^{\infty }{\left|a_m\right|}^2{m}^{2k+1}\\ \hfill \le \sum _{m=0}^{\infty }{\left|a_m\right|}^2{(m+1)}^{2k+2}=p_{k+1}\left(\psi \right),\end{array}$$

(21)

where the convergence of these sums for all k shows that (20) is well defined. Now, let us define the action of $a^{†}$ on $\psi \left(x\right)\in \mathcal{S}$ as

$$a^{†}\psi :=\sum _{n=0}^{\infty }{a}_n\sqrt{n+1}|n+1⟩.$$

(22)

Thus, for $k=0,1,2,\dots$,

$$p_k\left(a^{†}\psi \right)=\sum _{n=0}^{\infty }{b}_n^2{(n+1)}^{2k}=\sum _{n=0}^{\infty }|a_n{|}^2{(n+1)}^{2k+1}\le \sum _{n=0}^{\infty }{\left|a_n\right|}^2{(n+1)}^{2k+2}=p_{k+1}\left(\psi \right),$$

(23)

which shows both that (22) is well defined and that $a^{†}$ is continuous on $\mathcal{S}$. Once we have shown the continuity of a and $a^{†}$, the continuity of H is obvious after (16). Nevertheless, this continuity may be proven directly just by noting that for any $k=0,1,2,\dots$,

$$p_k\left(H\psi \right)={\hslash }^2{\omega }^2\sum _{n=0}^{\infty }{(n+1/2)}^2|a_n{|}^2{(n+1)}^{2k}\le {\hslash }^2{\omega }^2\sum _{n=0}^{\infty }{\left|a_n\right|}^2{(n+1)}^{2k+2}={\hslash }^2{\omega }^2{p}_{k+1}\left(\psi \right).$$

(24)

Here, we present a note about the coherent states. Let $\alpha \in \mathbb{C}$ be arbitrary but fixed. Its coherent state is given by

$$|\alpha ⟩:=e^{-{\displaystyle \frac{1}{2}}{\left|\alpha \right|}^2}\sum _{n=0}^{\infty }{\displaystyle \frac{{\alpha }^n}{\sqrt{n!}}}|n⟩.$$

(25)

Coherent states are eigenvectors of the annihilation operator so that $a|\alpha ⟩=\alpha |\alpha ⟩$. Coherent states evolve classically. In the present case, $|\alpha ⟩\in \mathcal{S}$ for all $\alpha \in \mathbb{C}$. To prove it, we just have to show that for all $k=0,1,2,\dots$,

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\left|\alpha \right|}^{2n}}{n!}}{(n+1)}^{2k}<\infty .$$

(26)

Then, we just need to show that the general term in the series (26) goes to zero at infinity faster than $1/n$. To see that this is true, just note that

$$\underset{n\to \infty }{lim}{n}^2{\displaystyle \frac{{\left|\alpha \right|}^{2n}}{n!}}{(n+1)}^{2k}=0,$$

(27)

which is a simple exercise. Thus, coherent states for the Harmonic oscillator are Schwartz functions. In the sequel, we shall consider more general types of coherent states.

4. On Laguerre–Gaussian Ladder Operators

The Laguerre–Gauss functions have recently been considered by some authors in the study of two-dimensional systems. As an example, they appear as the radial part of common eigenfunctions of the angular momentum and number operators for the two-dimensional harmonic oscillator written in cylindric coordinates [22]. In the present paper, we are considering another type of model: the paraxial wave equation for parabolic media, given by the following three-dimensional partial differential equation:

$$\left[-\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right)+{\displaystyle \frac{{\Omega }^2}{2}}{\rho }^2\right]U=i{\displaystyle \frac{\partial }{\partial z}}U,{\Omega }^2{\rho }^2<<1,$$

(28)

where $\Omega$ is a constant and $\rho =\sqrt{x^2+y^2}$. This model describes the z-propagation of electromagnetic waves through media with square refractive index $1-{\Omega }^2{\rho }^2$. This model has been studied in [8,23,24,25,26] and its physical motivation is not particularly relevant to our purposes.

What really concerns us is the form of (28). It is like a two-dimensional time-dependent Schrödinger equation for the harmonic oscillator, where the variable z plays the role of time. In this sense, it may be looked at as a two-dimensional model in spatial coordinates. The plane $(x,y)$ may also be described by its polar coordinates $(\rho ,\theta )$. Then, (28) admits as solutions functions of the following type (see [8]):

$$U_{l,p}=C_{l,p}(z,\theta ){\Phi }_{\left|l\right|,p}\left(\rho \right)l\in \mathbb{Z},p=0,1,2,\dots .$$

(29)

Under some conditions [8], these solutions are square integrable, a fact that we are assuming from now on. We are not interested in the explicit form of the functions $C_{l,p}(z,\theta )$, our concern goes to ${\Phi }_{\left|l\right|,p}\left(\rho \right)$, which are of the form [8]:

$${\Phi }_{\left|l\right|,p}\left(\rho \right)={\displaystyle \frac{2{(-1)}^p}{w_0}}\sqrt{{\displaystyle \frac{\Gamma (p+1)}{\Gamma \left(\right|l|+p+1)}}}{\left[{\displaystyle \frac{\sqrt{2}\rho }{w_0}}\right]}^{\left|l\right|}exp\left(-{\displaystyle \frac{{\rho }^2}{w_0^2}}\right)L_p^{\left(\right|l\left|\right)}\left({\displaystyle \frac{2{\rho }^2}{w_0^2}}\right),$$

(30)

where $w_0$ is a constant and $L_p^{\left(\right|l\left|\right)}\left(x\right)$ are the associated Laguerre polynomials of degree p and order $\left|l\right|$. These special functions are the mentioned Laguerre–Gauss functions. The most important properties of these functions that concerns us are as follows [8,22]:

• For each fixed value of l, ${\Phi }_{\left|l\right|,p}\left(\rho \right)$ form an orthonormal basis for $L^2({\mathbb{R}}^{+},\rho d\rho )$. Note that ordinary Laguerre functions

  $$M_n^{\alpha }\left(y\right)=\sqrt{{\displaystyle \frac{\Gamma (n+1)}{\Gamma (n+\alpha +1)}}}{y}^{\alpha /2}{e}^{-y/2}{L}_n^{\alpha }\left(y\right),$$

  (31)

  form an orthonormal basis for $L^2\left({\mathbb{R}}^{+}\right)$ for any fixed $\alpha \in (-1,\infty )$.
• The Laguerre–Gauss functions satisfy the following differential equation:

  $$\left[-{\displaystyle \frac{w_0^2}{2}}\left({\displaystyle \frac{{\partial }^2}{\partial {\rho }^2}}+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial \rho }}-{\displaystyle \frac{l^2}{{\rho }^2}}\right)+2{\displaystyle \frac{{\rho }^2}{w_0^2}}-2{\beta }_{\left|l\right|,p}\right]{\Phi }_{\left|l\right|,p}\left(\rho \right)=0,$$

  (32)

  with

  $${\beta }_{\left|l\right|,p}=\left|l\right|+2p+1.$$

  (33)

Next, we define the following functions:

$${\psi }_{\left|l\right|,p}\left(\rho \right):={\rho }^{1/2}{\Phi }_{\left|l\right|,p}\left(\rho \right).$$

(34)

After the aforementioned properties of ${\Phi }_{\left|l\right|,p}\left(\rho \right)$, it is straightforward to show that, for each fixed value of l, the sequence $\left\{{\psi }_{\left|l\right|,p}\left(\rho \right)\right\}$, $p=0,1,2,\dots$, is an orthonormal basis in $L^2\left({\mathbb{R}}^{+}\right)$. Functions ${\psi }_{\left|l\right|,p}\left(\rho \right)$ are also called Laguerre–Gauss.

Now, for a fixed value of the integer, $l\in \mathbb{Z}$ and $\rho \ge 0$, let us consider the following operator on $L^2\left({\mathbb{R}}^{+}\right)$ [27]:

$$H_l:={\displaystyle \frac{w_0^2}{2}}\left[-{\displaystyle \frac{d^2}{d{\rho }^2}}+{\displaystyle \frac{l^2-\frac{1}{4}}{{\rho }^2}}\right]+2{\displaystyle \frac{{\rho }^2}{w_0^2}}.$$

(35)

After (32) and (34), we obviously have for $l\in \mathbb{Z}$ and $p=0,1,2,\dots$,

$$H_l{\psi }_{\left|l\right|,p}\left(\rho \right)=2{\beta }_{\left|l\right|,p}{\psi }_{\left|l\right|,p}\left(\rho \right).$$

(36)

In the sequel, it is convenient to choose $w_0^2=2$ in (35) for simplicity in the notation. This will not change any of our relevant results. Since for fixed l, the sequence ${\psi }_{\left|l\right|,p}\left(\rho \right)$, $p=0,1,2,\dots$, forms an orthonormal basis in $L^2\left({\mathbb{R}}^{+}\right)$, and these basis functions are eigenfunctions of $H_l$, being obviously $H_l$ symmetric, then $H_l$ is essentially self-adjoint on any subspace of $L^2\left({\mathbb{R}}^{+}\right)$ containing this basis for l. In particular, each of the $H_l$, $l\in \mathbb{Z}$, is essentially self-adjoint on the subspaces ${\Phi }_l$ to be defined later.

Then, (35) defines a family of Hamiltonians on $L^2\left({\mathbb{R}}^{+}\right)$ indexed by $\mathbb{Z}$ (or rather by $\left|l\right|=0,1,2,\dots$, since $H_l$ depends on $l^2$). In any case, Hamiltonians of this family, $l\in \mathbb{Z}$, can be factorized in terms of the following operators [27]:

$$A_l^{-}:=\left({\displaystyle \frac{d}{d\rho }}-{\displaystyle \frac{l+\frac{1}{2}}{\rho }}+\rho \right),A_l^{+}:=\left(-{\displaystyle \frac{d}{d\rho }}-{\displaystyle \frac{l+\frac{1}{2}}{\rho }}+\rho \right),$$

(37)

and

$$B_l^{-}:=\left({\displaystyle \frac{d}{d\rho }}+{\displaystyle \frac{l-\frac{1}{2}}{\rho }}+\rho \right),B_l^{+}:=\left(-{\displaystyle \frac{d}{d\rho }}+{\displaystyle \frac{l-\frac{1}{2}}{\rho }}+\rho \right).$$

(38)

These objects, as defined in (37) and (38), are the Laguerre–Gauss ladder operators. This factorization of $H_l$, $l\in \mathbb{Z}$, can be performed in four different modes, and this goes as follows:

$$H_l=B_l^{+}{B}_l^{-}-{\epsilon }_{l-2}=B_{l+1}^{-}{B}_{l+1}^{+}-{\epsilon }_l=A_l^{+}{A}_l^{-}+{\epsilon }_l=A_{l-1}^{-}{A}_{l-1}^{+}+{\epsilon }_{l-2},$$

(39)

where $\left\{{\epsilon }_l\right\}$ is a sequence of real numbers. Some other relations are the following:

$$\begin{array}{c}\hfill A_l^{-}{H}_l=(H_{l+1}+2)A_l^{-},H_l{A}_l^{+}=A_l^{+}(H_{l+1}+2),\\ \hfill B_l^{-}{H}_l=(H_{l-1}+2)B_l^{-},H_l{B}_l^{+}=B_l^{+}(H_{l-1}+2).\end{array}$$

(40)

5. Gelfand Triplets and Continuous Operators

Let $\phi \left(\rho \right)\in L^2\left({\mathbb{R}}^{+}\right)$ be arbitrarily chosen. Then, for fixed l, we know that

$$\phi \left(\rho \right)=\sum _{p=0}^{\infty }{\alpha }_{l,p}{\psi }_{\left|l\right|,p}\left(\rho \right),$$

(41)

where $\left\{{\alpha }_{l,p}\right\}$ is a sequence of complex numbers such that

$$\sum _{p=0}^{\infty }{\left|{\alpha }_{l,p}\right|}^2<\infty .$$

(42)

The convergence of the series in (41) is in the norm of $L^2\left({\mathbb{R}}^{+}\right)$, as is well known.

For fixed $l\ge 0$, let us now construct a Gelfand triplet in the following form: Let us choose the subspace ${\Phi }_l$ of $L^2\left({\mathbb{R}}^{+}\right)$ of all vectors $\phi \left(p\right)$ as in (41) such that

$$p_{s,\left|l\right|}^2\left(\phi \right):=\sum _{p=0}^{\infty }|{\alpha }_{l,p}{|}^2{\left(\right|l|+2p+1)}^{2s}<\infty ,s=0,1,2,3,\dots .$$

(43)

Then, $p_{s,\left|l\right|}$, $s=0,1,2\dots$ is a family of seminorms (These are indeed norms. Recall that norms are also seminorms.) that produce the topology on ${\Phi }_l$. In Appendix B, we shall prove some topological properties of this space for $l>0$.

Note that ${\Phi }_l$ contains all the functions on the orthonormal basis ${\left\{{\psi }_{\left|l\right|,p}\left(\rho \right)\right\}}_{p=0}^{\infty }$, so that it is dense in $L^2\left({\mathbb{R}}^{+}\right)$. Since the choice $s=0$ in (43) gives the norm on $L^2\left({\mathbb{R}}^{+}\right)$, the topology induced on ${\Phi }_l$ by the Hilbert space norm is coarser than the topology on ${\Phi }_l$. Hence, the canonical injection $j:{\Phi }_l⟼L^2\left({\mathbb{R}}^{+}\right)$, $j\left(\phi \right)=\phi$, $\forall \phi \in {\Phi }_l$ is continuous. Therefore, if ${\Phi }_l^{\times }$ is the antidual space of ${\Phi }_l$, we conclude that

$${\Phi }_l\subset L^2\left({\mathbb{R}}^{+}\right)\subset {\Phi }_l^{\times },$$

(44)

is a Gelfand triplet or rigged Hilbert space (RHS in the sequel).

Let us prove that the Hamiltonian $H_l$ leaves invariant ${\Phi }_l$, which means that $H_l\psi \in {\Phi }_l$ for any $\phi \in {\Phi }_l$, and that it is a continuous operator on ${\Phi }_l$. First of all, we need to define the action of $H_l$ on each of the $\phi \in {\Phi }_l$, with the requirement that it extends (36). This definition should go as

$$H_l\sum _{p=0}^{\infty }{\alpha }_{l,p}{\psi }_{\left|l\right|,p}\left(\rho \right)=\sum _{p=0}^{\infty }{\alpha }_{l,p}\left(\right|l|+2p+1){\psi }_{\left|l\right|,p}\left(\rho \right).$$

(45)

To see that (45) is well defined, note that

$$p_{s,\left|l\right|}^2\left(H_l\phi \right)=\sum _{p=0}^{\infty }|{\alpha }_{l,p}{|}^2{\left(\right|l|+2p+1)}^{2(s+1)}=p_{s+1}^2\left(\phi \right).$$

(46)

This shows that the coefficients ${\alpha }_{l,p}\left(\right|l|+2p+1)$ of $H_l\phi$ verify relations (43) if $\phi \in {\Phi }_l$, so that $H_l\phi \in {\Phi }_l$ and $H_l$ leaves ${\Phi }_l$ invariant for all $l\in \mathbb{Z}$. In addition, after (8), one concludes that $H_l$ is continuous on ${\Phi }_l$. Note that none of the $H_l$ is a bounded (continuous) operator on $L^2\left({\mathbb{R}}^{+}\right)$.

5.1. The Ladder Operators

Although the index l could, in principle, run out the set of entire numbers, the Laguerre–Gauss basis functions (30) and (34) are just defined for $l\ge 0$. Laguerre–Gauss functions for negative values of l can be defined, although they are highly singular at the origin. This means that such functions cannot be square integrable for the Lebesgue measure on the positive semiaxis, although some other measures of the type ${\rho }^{\left|l\right|}d\rho$ could be used.

The action of the ladder operators (37) and (38) on the basis functions ${\psi }_{\left|l\right|,p}\left(\rho \right)$ has been well studied [8,28]. For $l\ge 0$, we have that

$$\begin{array}{c}\hfill A_l^{-}{\psi }_{l,p}\left(\rho \right)=2\sqrt{p}{\psi }_{l+1,p-1}\left(\rho \right),A_{l-1}^{+}{\psi }_{l,p}\left(\rho \right)=2\sqrt{p+1}{\psi }_{l-1,p+1}\left(\rho \right)(l\ne 0)\\ \hfill B_l^{-}{\psi }_{l,p}\left(\rho \right)=2\sqrt{p+l}{\psi }_{l-1,p-1}\left(\rho \right)(l\ne 0),B_{l+1}^{+}{\psi }_{l,p}\left(\rho \right)=2\sqrt{p+l+1}{\psi }_{l+1,p}\left(\rho \right).\end{array}$$

(47)

Functions with subindex $p-1$ are identically equal to zero if $p=0$.

Next, we have the following result: Ladder operators are linear continuous mappings between the following respective spaces:

$$\begin{array}{ccc}\hfill A_l^{-}& :& {\Phi }_l⟼{\Phi }_{l+1},\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill A_{l-1}^{+}& :& {\Phi }_l⟼{\Phi }_{l-1},(l\ne 0),\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill B_l^{-}& :& {\Phi }_l⟼{\Phi }_{l-1},(l\ne 0),\hfill \end{array}$$

(50)

$$\begin{array}{ccc}\hfill B_{l+1}^{+}& :& {\Phi }_l⟼{\Phi }_{l+1}.\hfill \end{array}$$

(51)

The proof is simple. First, take $\phi \in {\Phi }_l$ such that taking into account (41), we have

$$\left(A_l^{-}\psi \right)\left(\rho \right)=A_l^{-}\sum _{p=0}^{\infty }{\alpha }_{l,p}{\psi }_{l,p}\left(\rho \right)=\sum _{p=0}^{\infty }{\alpha }_{l,p}2\sqrt{p}{\psi }_{l+1,p-1}\left(\rho \right).$$

(52)

Then,

$$p_{s,|l+1|}^2\left(A_l^{-}\phi \right)=4\sum _{p=0}^{\infty }p{\left|{\alpha }_{l,p}\right|}^2{\left(\right|l+1|+2p+1)}^{2s},$$

(53)

with $s=0,1,2,\dots$. Clearly, (henceforth, we omit the modulus for l since we are assuming that $l\ge 0$)

$$p\le (l+1+2p+1),$$

(54)

and

$$\begin{array}{c}\hfill l+1+2p+1=l+2p+1+1\le 2(l+2p+1),\end{array}$$

(55)

so that (53) becomes

$$p_{s,l+1}^2\left(A_l^{-}\phi \right)\le 8\sum _{p=0}^{\infty }{\left|{\alpha }_{l,p}\right|}^2{(l+2p+1)}^{2(s+1)}=8p_{s+1,l}^2\left(\phi \right),\forall \phi \in {\Phi }_l.$$

(56)

Equation (56) shows the following:

• The series in (53) converges for all values of $s=0,1,2,\dots$ for $l+1$, which implies that $A_l^{-}\phi$ belongs to ${\Phi }_{l+1}$ for all $\phi \in {\Phi }_l$.
• In accordance with (5), the mapping $A_l^{-}:{\Phi }_l⟼{\Phi }_{l+1}$ is continuous with the topologies defined on both spaces.

The definition for the remainder ladder operators on ${\Phi }_l$ is similar. In fact,

$$A_{l-1}^{+}\phi :=2\sum _{p=1}^{\infty }\sqrt{p+1}{\alpha }_{l,p}{\psi }_{l-1,p+1}\left(\rho \right),$$

(57)

$$B_l^{-}\phi :=2\sum _{p=1}^{\infty }\sqrt{p+l}{\alpha }_{l,p}{\psi }_{l-1,p-1}\left(\rho \right)$$

(58)

and

$$B_{l+1}^{+}\phi :=2\sum _{p=0}^{\infty }\sqrt{p+l+1}{\alpha }_{l,p}{\psi }_{l+1,p}\left(\rho \right).$$

(59)

The proof of the continuity of these operators is made analogously with the proof we have given for $A_l^{-}$. Take, for instance,

$$\begin{array}{c}\hfill p_{s,l+1}^2\left(B_{l+1}^{+}\phi \right)=4\sum _{p=0}^{\infty }(p+l+1){\left|{\alpha }_{l,p}\right|}^2{(l+1+2p+1)}^{2s}\\ \hfill \le 8\sum _{p=0}^{\infty }|{\alpha }_{l,p}{|}^2{(l+2p+1)}^{2s+3}\le 8\sum _{p=0}^{\infty }{\left|{\alpha }_{l,p}\right|}^2{(l+2p+1)}^{2(s+2)}=8p_{s,l}^2\left(\phi \right),\forall \phi \in {\Phi }_l.\end{array}$$

(60)

This first proves that $B_{l+1}^{+}$ maps ${\Phi }_l$ into ${\Phi }_{l+1}$ and, then, that this map is continuous with respect to the topologies of the initial and final spaces. The same kind of arguments goes for $A_{l-1}^{+}$ and $B_l^{-}$.

5.2. The Laguerre–Gauss Functions with Negative Value of L

Let us go back to the family of functions in (30), where for simplicity we chose $w_0^2=2$. This choice simplifies the notation and does not affect the mathematical discussion under process. Then, the functions in (30) for fixed $l\ge 0$ and $p=0,1,2,\dots$ form an orthonormal basis for $L^2({\mathbb{R}}^{+},\rho d\rho )$. Then, for any $\psi \left(\rho \right)\in L^2({\mathbb{R}}^{+},\rho d\rho )$, we have the following span:

$$\psi \left(\rho \right)=\sum _{p=0}^{\infty }{a}_{l,p}{\Phi }_{l,p}\left(\rho \right),\sum _{p=0}^{\infty }{\left|a_{l,p}\right|}^2<\infty ,$$

(61)

where the first sum in (61) converges in the norm of $L^2({\mathbb{R}}^{+},\rho d\rho )$.

Now, let ${\Psi }_l$, $l\ge 0$, the space of all $\psi \left(\rho \right)\in L^2({\mathbb{R}}^{+},\rho d\rho )$, such that for fixed $l\ge 0$ we have that

$$q_{s,l}^2\left(\psi \right):=\sum _{p=0}^{\infty }{\left|a_{l,p}\right|}^2{[(l+2p+1)!]}^{2s}<\infty ,s=0,1,2,\dots .$$

(62)

No matter how small the spaces ${\Psi }_l$ could be as subspaces of $L^2({\mathbb{R}}^{+},\rho d\rho )$, they are dense in $L^2({\mathbb{R}}^{+},\rho d\rho )$ with the Hilbert space topology, since all ${\Psi }_l$ contain an orthonormal basis of $L^2({\mathbb{R}}^{+},\rho d\rho )$. In addition, the seminorm in (62) $q_{l,0}$ is nothing else than the Hilbert space norm, so that the canonical injection $i:{\Psi }_l⟼L^2({\mathbb{R}}^{+},\rho d\rho )$, $i\left(\psi \right)=\psi$ is continuous for all values of $l=0,1,2,\dots$. We have a new series of Gelfand triplets given by

$${\Psi }_l\subset L^2({\mathbb{R}}^{+},\rho d\rho )\subset {\Psi }_l^{\times },l=0,1,2,\dots .$$

(63)

Let us go back to the beginning of Section 4. Also, for $l\ge 0$ and a subspace ${\tilde{\Phi }}_l$ of vectors $\psi \left(\rho \right)\in L^2({\mathbb{R}}^{+},\rho d\rho )$, we may define the set of norms (43), which for an “abus de langage” shall be denoted as $p_{s,l}\left(\psi \right)$, exactly as in the case described there (Obviously, the space ${\tilde{\Phi }}_l$ is algebraic and topologically isomorphic to ${\Phi }_l$ for $l\ge 0$). This gives a new Gelfand triplet:

$${\tilde{\Phi }}_l\subset L^2({\mathbb{R}}^{+},\rho d\rho )\subset {\tilde{\Phi }}_l^{\times }.$$

(64)

Furthermore, it is obvious that for $l=0,1,2,\dots$, ${\Psi }_l\subset {\tilde{\Phi }}_l$. Let us call $j_l:{\Psi }_l⟼{\tilde{\Phi }}_l$ to the corresponding canonical injection. Then, for any $\psi \in {\Psi }_l$ as in (61), we have that

$$p_{s,l}\left(j_l\left(\psi \right)\right)=\sum _{p=0}^{\infty }|a_{l,p}{|}^2{(l+2p+1)}^{2s}\le \sum _{p=0}^{\infty }{\left|a_{l,p}\right|}^2{[(l+2p+1)!]}^{2s}=q_{s,l}\left(\psi \right),$$

(65)

so that $j_l$ is continuous. As a consequence, we have the following sequence of locally convex spaces for which all the canonical mappings are continuous:

$${\Psi }_l\subset {\tilde{\Phi }}_l\subset L^2({\mathbb{R}}^{+},\rho d\rho )\subset {\tilde{\Phi }}_l^{\times }\subset {\Psi }_l^{\times },l=0,1,2,\dots .$$

(66)

The reason to adopt such a strong topology will be evident later.

Next, let us take the “l negative version of the functions (30)”, which is ($l\ge 0$)

$${\Phi }_{-l,p}\left(\rho \right)={(-1)}^p\sqrt{{\displaystyle \frac{\Gamma (p+1)}{\Gamma (l+p+1)}}}{\rho }^{-l}exp(-{\rho }^2/2)L_p^l\left({\rho }^2\right).$$

(67)

Note that the associated Laguerre polynomial $L_m^{\alpha }\left(x\right)$ makes sense only if $\alpha \in (-1,\infty )$, so that we cannot use $\alpha =-l$. The objective is to show that ${\Phi }_{-l,p}\left(\rho \right)\in {\Psi }_l^{\times }$, $l=0,1,2,\dots$.

First of all, let us prove that if $\psi \left(\rho \right)\in {\Psi }_l$, then the first series in (61) also converges uniformly. To this end, let us consider the following relation concerning Laguerre functions:

$$e^{-x}{x}^{k/2}{L}_n^k\left(x\right)={\displaystyle \frac{1}{n!}}{\int }_0^{\infty }{e}^{-t}{t}^{n+k/2}{J}_k\left(2\sqrt{tx}\right)dt,$$

(68)

where $J_k\left(t\right)$ is the Bessel function with index k. If k is an integer number, which is the case in our discussion, the Bessel function $J_k\left(t\right)$ is given by

$$J_k\left(y\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }cos(k\tau -ysin\tau )d\tau ,$$

(69)

so that $|J_k\left(y\right)|\le 1$, with k being an integer number. For us, $k=l$ and $n=p$. Then,

$$|e^{-x}{x}^{l/2}{L}_p^l\left(x\right)|\le {\displaystyle \frac{1}{p!}}{\int }_0^{\infty }{e}^{-t}{t}^{p+l/2}dt={\displaystyle \frac{\Gamma (p+l/2+1)}{p!}}.$$

(70)

Taking $x={\rho }^2$, we have that

$$|e^{-{\rho }^2}{\rho }^l{L}_p^l\left({\rho }^2\right)|\le {\displaystyle \frac{\Gamma (p+l/2+1)}{p!}}.$$

(71)

If $\psi \left(\rho \right)\in {\Psi }_l$ as in (61), one has the following series of inequalities for each $l=0,1,2,\dots$:

$$\begin{array}{c}\hfill \left|\right|\psi \left(\rho \right)\left|\right|=\left|\sum _{p=0}^{\infty }{a}_{l,p}{\Phi }_{l,p}\left(\rho \right)\right|\le \sum _{p=0}^{\infty }|a_{l,p}\left|\right|{\Phi }_{l,p}\left(\rho \right)|\le \sum _{p=0}^{\infty }\left|a_{l,p}\right|\sqrt{{\displaystyle \frac{\Gamma (p+1)}{\Gamma (l+p+1)}}}{\displaystyle \frac{\Gamma (p+l/2+1)}{p!}}\\ \hfill \le \sum _{p=0}^{\infty }|a_{l,p}|(l+2p+1)!\le \sum _{p=0}^{\infty }\left|a_{l,p}\right|{[(l+2p+1)!]}^2{\displaystyle \frac{1}{(l+2p+1)!}}\\ \hfill \le \sqrt{\sum _{p=0}^{\infty }{\left|a_{l,p}\right|}^2{[(l+2p+1)!]}^4}\times \sqrt{\sum _{p=0}^{\infty }{\displaystyle \frac{1}{{[(l+2p+1)!]}^2}}}=C_l\left[q_{2,l}\left(\psi \right)\right],\end{array}$$

(72)

where $C_l:=\sqrt{{\sum }_{p=0}^{\infty }{\displaystyle \frac{1}{{[(l+p+1)!]}^2}}}$. The last inequality in (72) is the Schwartz inequality. Due to the Weierstrass M-test, the series ${\sum }_{p=0}^{\infty }|a_p\left|\right|{\Phi }_{l,p}\left(\rho \right)|$ converges uniformly for $\rho \in [0,\infty )$ (Similarly, we may prove exactly the same result if, instead of the norms $q_{s,l}$, we use the norms $p_{s,l}$).

Now, let $\psi \left(\rho \right)$ be an arbitrary function in ${\Psi }_l$. The action of ${\Phi }_{-l,q}\left(\rho \right)$ as in (66), $q=0,1,2,\dots$, as a functional on ${\Psi }_l$ should be, for all $\psi \left(\rho \right)\in {\Psi }_l$,

$$\begin{array}{c}\hfill {\int }_0^{\infty }\psi \left(\rho \right){\Phi }_{-l,q}\left(\rho \right)\rho d\rho ={\int }_0^{\infty }\left(\sum _{p=0}^{\infty }{a}_{l,p}{\Phi }_{l,p}\left(\rho \right)\right){\Phi }_{-l,q}\left(\rho \right)\rho d\rho \\ \hfill =\sum _{p=0}^{\infty }{a}_{l,p}{\int }_0^{\infty }{\Phi }_{l,p}\left(\rho \right){\Phi }_{-l,q}\left(\rho \right)\rho d\rho ,\end{array}$$

(73)

where the last identity in (73) is due to the uniform convergence of the series. The last integral in (73) gives

$${\int }_0^{\infty }{\Phi }_{l,p}\left(\rho \right){\Phi }_{-l,q}\left(\rho \right)\rho d\rho =\sqrt{{\displaystyle \frac{\Gamma (p+1)\Gamma (q+1)}{\Gamma (l+p+1)\Gamma (l+q+1)}}}{\int }_0^{\infty }{e}^{-{\rho }^2}{L}_p^l\left({\rho }^2\right)L_q^l\left({\rho }^2\right)\rho d\rho .$$

(74)

Next, let us use the explicit form for the associated Laguerre polynomials

$$L_p^l\left({\rho }^2\right)={\displaystyle \frac{1}{p!}}\sum _{s=0}^p{\displaystyle \frac{p!}{s!}}\left(\begin{array}{c}p+l\\ p-s\end{array}\right){(-{\rho }^2)}^s,$$

(75)

so that

$$\begin{array}{c}\hfill \left|{\int }_0^{\infty }{e}^{-{\rho }^2}{L}_p^l\left({\rho }^2\right)L_q^l\left({\rho }^2\right)\rho d\rho \right|\\ \hfill \le \sum _{s=0}^p{\displaystyle \frac{1}{s!}}{\displaystyle \frac{(p+l)!}{(p-s)!(l+s)!}}\sum _{r=0}^{p^{\prime }}{\displaystyle \frac{1}{r!}}{\displaystyle \frac{(p^{\prime }+l)!}{(p^{\prime }-r)!(l+r)!}}{\int }_0^{\infty }{e}^{-{\rho }^2}{\rho }^{2(r+s)}\rho d\rho \end{array}$$

(76)

Let us make the change of variables ${\rho }^2=t$ in the integral. It gives

$${\displaystyle \frac{1}{2}}{\int }_0^{\infty }{e}^{-t}{t}^{r+s}dt={\displaystyle \frac{1}{2}}\Gamma (r+s+1)={\displaystyle \frac{1}{2}}(r+s)!,$$

(77)

so that (75) is smaller or equal to

$$\sum _{s=0}^p\sum _{r=0}^{p^{\prime }}(p+l)!(p^{\prime }+l)!(s+r)!.$$

(78)

Now, we have two possibilities. Either $p^{\prime }>p$, so that $(s+r)!<\left(2p^{\prime }\right)!$ or $p^{\prime }\le p$, in which case $(s+r)!\le \left(2p\right)!$. Also, note that $rs\le (r+s)!$, so if $p^{\prime }>p$, then (78) is smaller or equal to

$$(p+l)!(p^{\prime }+l)!\left(2p^{\prime }\right)!\left(rs\right)\le (p+l)!\{(p^{\prime }+l)!{[\left(2p^{\prime }\right)!]}^2\}\le (l+2p+1)!\{(p^{\prime }+l)!{[\left(2p^{\prime }\right)!]}^2\}.$$

(79)

Note that the term between the brackets does not depend on p and only on l and $p^{\prime }$, which are fixed here. Then, if $F_{p^{\prime }}$ represents the functional on ${\Psi }_l$ defined by ${\Phi }_{-l,p^{\prime }}\left(\rho \right)$ with (73), we have, for all $\psi \in {\Psi }_l$,

$$\begin{array}{c}\hfill \left|F_q\left(\psi \right)\right|=\left|{\int }_0^{\infty }\psi \left(\rho \right){\Phi }_{-l,q}\left(\rho \right)\rho d\rho \right|\le \sum _{p=0}^{\infty }\left|a_{l,p}\right|q!\left|{\int }_0^{\infty }{e}^{-{\rho }^2}{L}_p^l\left({\rho }^2\right)L_q^l\left({\rho }^2\right)\rho d\rho \right|\\ \hfill \le C_{q,l}\sum _{p=0}^{q-1}\left|a_{l,p}\right|(l+2p+1)!+\sum _{p=q}^{\infty }\left|a_{l,p}\right|q!\left|{\int }_0^{\infty }{e}^{-{\rho }^2}{L}_p^l\left({\rho }^2\right)L_q^l\left({\rho }^2\right)\rho d\rho \right|,\end{array}$$

(80)

with $C_{q,l}:=q!(q+l)!{[\left(2q\right)!]}^2$. We analyze both summands in the last row of (80) separately. For the first one, we have

$$\begin{array}{c}\hfill C_{q,l}\sum _{p=0}^{q-1}\left|a_{l,p}\right|(l+2p+1)!=C_{p^{\prime },l}\sum _{p=0}^{q-1}\left|a_{l,p}\right|{((l+2p+1)!)}^2{\displaystyle \frac{1}{(l+2p+1)!}}\\ \hfill \le C_{q,l}\sqrt{{\sum }_{p=0}^{q-1}|a_{l,p}{{|}^2(l+2p+1)!)}^4}\sqrt{{\sum }_{p=0}^{q-1}{\displaystyle \frac{1}{{((l+2p+1)!)}^2}}}\\ \hfill \le C_{q,l}\sqrt{{\sum }_{p=0}^{\infty }{\left|a_{l,p}\right|}^2{(l+2p+1)!)}^4}\sqrt{{\sum }_{p=0}^{\infty }{\displaystyle \frac{1}{{((l+2p+1)!)}^2}}}=C_{q,l}{C}_l\left[q_{2,l}\left(\psi \right)\right],\end{array}$$

(81)

where the second inequality is none other than the Cauchy–Schwarz inequality and the meaning of $C_l$ is obvious. For the second term in the last row of (80) ($p\ge p^{\prime }$), we just have to replace ${[\left(2p^{\prime }\right)!]}^2$ in (80) by ${[\left(2p\right)!]}^2$, which is smaller than or equal to ${[(l+2p+1)!]}^2$ and $\left(p^{\prime }\right)!$ by $p!\le (l+2p+1)!$. Then, if $K_{p^{\prime },l}:=(p^{\prime }+l)!$ and following similar arguments as just before, we have that

$$\begin{array}{c}\hfill \sum _{p=q}^{\infty }|a_{l,p}|q!\left|{\int }_0^{\infty }{e}^{-{\rho }^2}{L}_p^l\left({\rho }^2\right)L_q^l\left({\rho }^2\right)\rho d\rho \right|\le K_{q,l}\sum _{p=q}^{\infty }\left|a_{l,p}\right|{[(l+2p+1)!]}^4\\ \hfill \le K_{q,l}\sum _{p=0}^{\infty }\left|a_{l,p}\right|{[(l+2p+1)!]}^4\le \sqrt{\sum _{p=0}^{\infty }{\left|a_{l,p}\right|}^2{[(l+2p+1)!]}^{10}}\sqrt{{\displaystyle \frac{1}{{[(l+2p+1)!]}^2}}}\\ \hfill \le {\tilde{K}}_{q,l}\left[q_{4,l}\left(\psi \right)\right],\end{array}$$

(82)

where ${\tilde{K}}_{q,l}$ is a constant. Let $K:=max\{C_{q,l}{C}_l,K_{q,l}\}$ and take into account that for all $\psi \in {\Psi }_l$, $q_{2,l}\left(\psi \right)\le q_{4,l}\left(\psi \right)$, we finally arrive to

$$|F_q\left(\psi \right)|\le 2K\left[q_{4,l}\left(\psi \right)\right],\forall \psi \in {\Psi }_l,$$

(83)

which proves the continuity of $F_q$ on ${\Psi }_l$. This goes for all $q=0,1,2,\dots$, which ends the proof of our statement.

The need for a strong topology is due to some difficulties produced by the terms in the first sum of (76), which cannot be simplified, so that the use of the topology (43) can be feasible. Since this new topology given by the norms (62) is stronger than (43), it would be interesting to check whether the properties of continuity of creation and annihilation operators are preserved under the new topology.

5.3. On the Continuity of Ladder Operators

In this subsection, we would make a brief analysis on the continuity of the ladder operators when they act between the spaces of the type ${\Psi }_l$. It is noteworthy that Equations (48)–(51) also hold with the spaces ${\Psi }_l$ with continuity. We just need to check this result with the new version of (48), since the other cases follow similarly. The equivalent formula to (48) is

$$A_l^{-}:{\Psi }_l⟼{\Psi }_{l+1}$$

(84)

Now, take $\psi \left(\rho \right)$ as in (61) and assume that $\psi \left(\rho \right)\in {\Psi }_l$. Then, ($l\ge 0$) after (52), we have for $s=0,1,2,\dots$

$$q_{s,l+1}^2\left(A_l^{-}\psi \right)=4\sum _{p=0}^{\infty }p{\left|{\alpha }_{l,p}\right|}^2{[(l+1+2p+1)!]}^{2s}.$$

(85)

Clearly,

$$(l+2p+2)!=(l+2p+2)\left[\right(l+2p+1)!]=2(l/2+p)\left[\right(l+2p+1)!].$$

(86)

Since

$${\displaystyle \frac{l}{2}}+p<l+2p+1<(l+2p+1)!\mathrm{and}p<(l+2p+1)!,$$

(87)

we have that

$$p{[(l+1+2p+1)!]}^{2s}\le 2^{2s}{[(l+2p+1)!]}^{4s+2},$$

(88)

so that

$$q_{s,l+1}^2\left(A_l^{-}\psi \right)\le 2^{2s+2}\sum _{p=0}^{\infty }{\left|{\alpha }_{l,p}\right|}^2{[(l+2p+1)!]}^{4s+2}=2^{2s+2}{q}_{2s+1,l}^2\left(\psi \right)\forall \psi \in {\Phi }_l.$$

(89)

This shows, at the same time, that $A_l^{-}\psi \in {\Psi }_l$ for all $\psi \in {\Psi }_l$ and that the mapping $A_l^{-}$, as in (83), is continuous. Similar proofs will follow for the equivalent of mappings (49)–(51).

Note that the interest of the topology introduced here, lately by (62), is due to a proper characterization of the functions (66) as continuous functionals. For the purpose of the continuity of the ladder operators, it is not only enough but also looks more appropriate to use the topology introduced by the norms (43). The topology given by (62) makes the spaces ${\Psi }_l$ too small, and the topology itself is too strong. This is not a serious inconvenience for the characterization of the ladder operators as continuous operators, although it seems an excessive use of the resources. We have just pointed out that the topology given by norms (43) seems not to be appropriate to prove the continuity of the functionals ${\Phi }_{-l,q}\left(\rho \right)$.

5.4. The Laguerre–Gauss Modes

Along the present subsection, we intend to recall a discussion already presented in [8], which deals fundamentally with the physical aspects of the problem. Here, we wish to make a series of comments from the mathematical point of view. Let us go back to (29), which is explicitly given in [8] as

$$U_{l,p}(\theta ,z,\rho )={\displaystyle \frac{e^{il\theta }}{2\pi }}exp\left[-iz{\beta }_{\left|l\right|,p}\right]{\Phi }_{\left|l\right|,p}\left(\rho \right),$$

(90)

where ${\beta }_{\left|l\right|,p}$ is given by (33). For each fixed value of $z\in \mathbb{R}$, the functions $U_{l,p}(\theta ,z,\rho )$, $l\in \mathbb{Z}$ and $p=0,1,2,\dots$ form an orthonormal basis on $L^2\left({\mathbb{R}}^2\right)$. These functions are called the Laguerre–Gauss (LG) modes. Thus, for each fixed z, the orthogonality (we omit the dependence on the variables for simplicity)

$${\int }_{-\pi }^{\pi }{\int }_0^{\infty }{U}_{l,p}{U}_{l^{\prime },p}\rho d\rho d\theta ={\delta }_{l,l^{\prime }}{\delta }_{p,p^{\prime }},$$

(91)

and completeness

$$\sum _{l\in \mathbb{Z}}\sum _{p=0}^{\infty }{U}_{l,p}(\theta ,\rho )U_{l,p}({\theta }^{\prime },{\rho }^{\prime })=\delta (\theta -{\theta }^{\prime })\delta (\rho -{\rho }^{\prime })$$

(92)

relations hold.

In [8], the following operator for each fixed value of $l\in \mathbb{Z}$ was defined (we have chosen $w_0^2=2$ in the formulas given in [8]):

$${\mathcal{L}}_l:={\rho }^{-1/2}{L}_l{\rho }^{1/2},\mathrm{with}{L}_l={\displaystyle \frac{1}{2}}\left[-{\displaystyle \frac{{\partial }^2}{\partial {\rho }^2}}+{\displaystyle \frac{(l+1/2)(l-1/2)}{{\rho }^2}}\right]+{\rho }^2$$

(93)

It is proven in [8] that the LG modes $U_{l,p}(\theta ,\rho )$, $p=0,1,2,\dots$ are eigenfunctions of ${\mathcal{L}}_l$ for each fixed value of $l=0,1,2,\dots$ with eigenvalue ${\displaystyle \frac{1}{2}}{\beta }_{\left|l\right|,p}$, i.e.,

$${\mathcal{L}}_l{U}_{l,p}={\displaystyle \frac{1}{2}}{\beta }_{\left|l\right|,p}{U}_{l,p},p=0,1,2,\dots ,l\in \mathbb{Z}.$$

(94)

Let us call ${\mathcal{H}}_l$ the subspace of $L^2\left({\mathbb{R}}^2\right)$ by the functions $U_{l,p}(\theta ,\rho )$, $p=0,1,2,\dots$ and $l\in \mathbb{Z}$ fixed, which form an orthonormal basis for ${\mathcal{H}}_l$. Obviously, $L^2\left({\mathbb{R}}^2\right)={\oplus }_{l\in \mathbb{Z}}{\mathcal{H}}_l$. The functions $U_{l,p}(\theta ,\rho )$ differ from the functions ${\Phi }_{\left|l\right|,p}\left(\rho \right)$ in (30) just for a phase and a multiplicative constant. Thus, they could be identified by an “abuse of language”; hence, ${\mathcal{H}}_l$ and $L^2({\mathbb{R}}^{+},\rho d\rho )$ for each value of l. Any ${\psi }_l(\theta ,\rho )\in {\mathcal{H}}_l$ with fixed l has the form

$${\psi }_l(\theta ,\rho )=\sum _{p=0}^{\infty }{a}_{l,p}{U}_{l,p}(\theta ,\rho )\mathrm{with}\sum _{p=0}^{\infty }{\left|a_{l,p}\right|}^2<\infty .$$

(95)

Then, let us consider the subspace of functions ${\psi }_l$ as in (95) such that

$$\sum _{p=0}^{\infty }|a_{l,p}{|}^2{\left(\right|l|+2p+1)}^{2s}<\infty ,s=0,1,2,\dots .$$

(96)

This is a subspace of ${\mathcal{H}}_l$ that may be identified with ${\tilde{\Phi }}_l$ after the aforementioned “abus de langage”, so that ${\tilde{\Phi }}_l\subset {\mathcal{H}}_l\subset {\tilde{\Phi }}_l^{\times }$ is an RHS for all values of l. If we define the action of ${\mathcal{L}}_l$ on each ${\psi }_l\in {\tilde{\Phi }}_l$, for fixed l, as

$${\mathcal{L}}_l{\psi }_l(\theta ,\rho ):={\displaystyle \frac{1}{2}}\sum _{p=0}^{\infty }{a}_{l,p}\left(\right|l|+2p+1)U_{l,p}(\theta ,\rho ),$$

(97)

we immediately see—using the arguments at the beginning of the present section—that (90) is well defined and that ${\mathcal{L}}_l$ is continuous on ${\tilde{\Phi }}_l$.

Ladder operators, ${\mathcal{L}}_l^{\pm }$, have also been defined in [8], and this definition can be extended to ${\Phi }_l$ as follows:

$${\mathcal{L}}_l^{\pm }{\psi }_l(\theta ,\rho )=\sum _{p=0}^{\infty }{a}_{l,p}{U}_{l,p\pm 1},\mathrm{with}{a}_{l,-1}=0.$$

(98)

The same arguments prove that ${\mathcal{L}}_l^{\pm }$ are continuous on ${\tilde{\Phi }}_l$.

This construction may be extended trivially to $L^2\left({\mathbb{R}}^2\right)$. In fact, if $\psi (\theta ,\rho )\in L^2\left({\mathbb{R}}^2\right)$, we have

$$\psi (\theta ,\rho )=\sum _{l=-\infty }^{\infty }\sum _{p=0}^{\infty }{a}_{l,p}{U}_{l,p}(\theta ,\rho ),\mathrm{with}\sum _{l=-\infty }^{\infty }\sum _{p=0}^{\infty }{\left|a_{l,p}\right|}^2<\infty .$$

(99)

Then, select the subspace $\mathbf{\Phi }$ of all vectors, $\psi (\theta ,\rho )$, in $L^2\left({\mathbb{R}}^2\right)$ such that, for all $l\in \mathbb{Z}$ and $p=0,1,2,\dots$,

$$\sum _{l=-\infty }^{\infty }\sum _{p=0}^{\infty }|a_{l,p}{|}^2{\left(\right|l|+2p+1)}^{2s}<\infty ,s=0,1,2,\dots .$$

(100)

Since $\mathbf{\Phi }$ contains all the LG modes $\left\{U_{l,p}\right\}$, $l\in \mathbb{Z}$, $p=0,1,2,\dots$, it is dense in $L^2\left({\mathbb{R}}^2\right)$. Since $s=0$ gives the Hilbert space topology, then the topology on $\mathbf{\Phi }$ given by the seminorms for which their squares are given in (100), the topology on $\mathbf{\Phi }$ is finer than the Hilbert space topology. Consequently,

$$\mathbf{\Phi }\subset L^2\left({\mathbb{R}}^2\right)\subset {\mathbf{\Phi }}^{\times }$$

(101)

is an RHS. On $\mathbf{\Phi }$, the following operators are continuous:

$$\left[\underset{l=-\infty }{\overset{\infty }{⨁}}{\mathcal{L}}_l\right]\psi (\theta ,\rho ):={\displaystyle \frac{1}{2}}\sum _{l=-\infty }^{\infty }\sum _{p=0}^{\infty }{a}_{l,p}\left(\right|l|+2p+1)U_{l,p}(\theta ,\rho ),$$

(102)

and

$$\left[\underset{l=-\infty }{\overset{\infty }{⨁}}{\mathcal{L}}_l^{\pm }\right]\psi (\theta ,\rho ):=\sum _{l=-\infty }^{\infty }\sum _{p=0}^{\infty }{a}_{l,p}{U}_{l,p\pm 1}(\theta ,\rho ),$$

(103)

with $a_{l,-1}=0$ for all values of l. Proofs are as in previous examples.

6. Coherent States and Continuous Generators for the Algebra $\mathbf{Su}(\mathbf{1},\mathbf{1})$

Let us give a brief analysis of different types of coherent states [29,30] from the point of view of the above-defined Gelfand triplets. To begin with, let us define the so-called index-free operators, which have been introduced in [8]. These are ladder operators defined on ${\Phi }_l$ for fixed values of $l>0$. On the basis vectors of ${\Phi }_l$, these operators act as follows [27]:

$$A^{+}{\psi }_{l,p}:=\sqrt{p+1}{\psi }_{l,p+1}\left(\rho \right),A^{-}{\psi }_{l,p}:=\sqrt{p}{\psi }_{l,p-1}\left(\rho \right),$$

(104)

so that their action on an arbitrary vector in ${\Phi }_l$, $l>0$, of the form (41) with (43) takes the form

$$A^{+}\phi \left(\rho \right):=\sum _{p=0}^{\infty }{a}_{l,p}\sqrt{p+1}{\psi }_{l,p+1}\left(\rho \right),A^{-}\phi \left(\rho \right):=\sum _{p=1}^{\infty }{a}_{l,p}\sqrt{p}{\psi }_{l,p-1}\left(\rho \right).$$

(105)

Following similar techniques to those given in Section 5.1, we easily show that both ladder operators $A^{\pm }$ are well defined and continuous on each of the ${\Phi }_l$, $l=1,2,\dots$.

Since we are focusing our attention on coherent states, we are mainly interested in the operator $A^{-}$, since coherent states are eigenvectors of the annihilation operator with an arbitrary eigenvalue. Recall that coherent states are the eigenstates of the annihilation operator, so, if ${\eta }^{\alpha }\left(\rho \right)$ is a coherent state, $A^{-}{\eta }^{\alpha }\left(\rho \right)=\alpha {\eta }^{\alpha }\left(\rho \right)$ with $\alpha \in \mathbb{C}$. Clearly, coherent states must satisfy the following relation:

$$A^{-}{\eta }^{\alpha }\left(\rho \right)=\sum _{p=0}^{\infty }{b}_{l,p}\sqrt{p}{\psi }_{l,p-1}\left(\rho \right)=\alpha \sum _{p=0}^{\infty }{b}_{l,p}{\psi }_{l,p}\left(\rho \right).$$

(106)

A straightforward calculation gives

$${\eta }^{\alpha }\left(\rho \right)\equiv {\eta }_l^{\alpha }\left(\rho \right)=\sum _{p=0}^{\infty }{\displaystyle \frac{{\alpha }^p}{\sqrt{p!}}}{e}^{-{\left|\alpha \right|}^2/2}{\psi }_{l,p}\left(\rho \right).$$

(107)

Thus, coherent states are defined for each value of $l=1,2,\dots$.

Proposition 1. 

For fixed $\alpha \in \mathbb{C}$ and $l=1,2,\dots$, ${\eta }_l^{\alpha }\left(\rho \right)\in {\Phi }_l$.

Proof. 

We just need to show that

$$\sum _{p=0}^{\infty }{\displaystyle \frac{{\left|\alpha \right|}^{2p}}{p!}}{e}^{-{\left|\alpha \right|}^2}{(l+2p+1)}^{2s}<\infty ,s=0,1,2,\dots .$$

(108)

Note that the term $e^{-{\left|\alpha \right|}^2}$ in (108) is a constant and, hence, irrelevant. To show this, it is sufficient to prove that, for large values of p, we have (Since ${\sum }_{p=1}^{\infty }{\displaystyle \frac{1}{p^2}}<\infty$)

$${\displaystyle \frac{{\left|\alpha \right|}^{2p}}{p!}}{(l+2p+1)}^{2s}<{\displaystyle \frac{1}{p^2}}s=0,1,2,\dots .$$

(109)

Note that, according to the Stirling formula,

$$p!\sim \sqrt{2\pi p}{\left({\displaystyle \frac{p}{e}}\right)}^p,$$

(110)

so that it is enough to show that for large values of p,

$${\displaystyle \frac{a^p}{\sqrt{2\pi p}{p}^p}}{(l+2p+1)}^{2s}{p}^2<1,a={\left|\alpha \right|}^{2}e.$$

(111)

Then, it is enough to show two inequalities valid for large values of p. The former is

$$a^p<p^{p/2}⟺p>a^2,$$

(112)

for any fixed value of the positive constant a and large p. The second one is

$${(l+2p+1)}^{2s}{p}^2<p^{p/2},$$

(113)

which is straightforward for large p. Then, (111) comes and, hence, (108). □

These coherent states are often referred to as Glauber–Klauder–Sudarshan coherent states. Note that the coherent state ${\eta }_l^{\alpha }\left(\rho \right)\notin {\Psi }_l$, since for $s=3$,

$$\sum _{p=0}^{\infty }{\displaystyle \frac{{\left|\alpha \right|}^{2p}}{{(p!)}^2}}{[(l+2p+1)!]}^6=\infty .$$

(114)

Next, let us consider a new set of ladder operators, ${\mathcal{A}}_l^{\pm }$, formally defined as

$${\mathcal{A}}_l^{\pm }:=\left({\rho }^2\mp \rho {\displaystyle \frac{d}{d\rho }}-\left(l+2p\mp {\displaystyle \frac{1}{2}}\right)\right).$$

(115)

Their action on the basis of functions ${\psi }_{l,p}\left(\rho \right)$ is given by

$${\mathcal{A}}_l^{+}{\psi }_{l,p-1}\left(\rho \right)=\sqrt{p(l+p)}{\psi }_{l,p}\left(\rho \right),{\mathcal{A}}_l^{-}{\psi }_{l,p}\left(\rho \right)=\sqrt{p(l+p)}{\psi }_{l,p-1}\left(\rho \right).$$

(116)

Their extension to any vector $\phi \left(\rho \right)\in {\Phi }_l$ is straightforward. This extension is well defined and continuous on ${\Phi }_l$.

As in the previous case, we may here define coherent states for each positive value of l, ${\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)$, as ${\mathcal{A}}_l^{-}{\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)=\alpha {\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)$, $\forall \alpha \in \mathbb{C}$. They are called Barut–Girardello coherent states. They have the following form:

$${\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)=\sum _{p=0}^{\infty }{\displaystyle \frac{{\alpha }^p}{\sqrt{p!(l+p)!}}}{\displaystyle \frac{{\alpha }^{l/2}}{\sqrt{I_l\left(2\alpha \right)}}}{\psi }_{l,p}\left(\rho \right),$$

(117)

where $I_l\left(x\right)$ is the modified Bessel function. Again, ${\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)\in {\Phi }_l$ and ${\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)\notin {\Psi }_l$.

Finally, we have the Perelomov coherent states. Their construction requires some analysis. First of all, let us define a new operator ${\mathcal{A}}_0$ as

$$2{\mathcal{A}}_l^0{\psi }_{l,p}\left(\rho \right):=[{\mathcal{A}}_l^{-},{\mathcal{A}}_l^{+}]{\psi }_{l,p}\left(\rho \right)=(l+2p+1){\psi }_{l,p}\left(\rho \right).$$

(118)

Clearly, ${\mathcal{A}}_l^0$ may be extended to ${\Phi }_l$ with continuity. Note that the operators ${\mathcal{A}}_l^{\pm }$ and ${\mathcal{A}}_l^0$ give a system of generators on each of the ${\Phi }_l$ of the Lie algebra $su(1,1)$, since

$$[{\mathcal{A}}_l^0,{\mathcal{A}}_l^{\pm }]=\pm {\mathcal{A}}_l^{\pm },[{\mathcal{A}}_l^{-},{\mathcal{A}}_l^{+}]=2{\mathcal{A}}_l^0.$$

(119)

In consequence, each of the ${\Phi }_l$ supports an irreducible representation of the algebra $su(1,1)$ given by continuous operators with the topology on ${\Phi }_l$. Note that the operators on the corresponding enveloping algebras are also continuous on each of the ${\Phi }_l$. Their canonical extension to the duals ${\Phi }_l^{\times }$ are also continuous with any topology on ${\Phi }_l^{\times }$ compatible with duality.

In order to construct the Perolomov coherent states, let us consider for each $l>0$ the ground state ${\psi }_{l,0}\left(\rho \right)$ and define

$${\xi }_l\left(\rho \right):=D(\lambda ,t){\psi }_{l,0}\left(\rho \right),$$

(120)

where $\lambda$ is a complex number and

$$D(\lambda ,t):=e^{t(\lambda {\mathcal{A}}_l^{+}-{\lambda }^{*}{\mathcal{A}}_l^{-})}=e^{\alpha {\mathcal{A}}_l^{+}}{e}^{\beta {\mathcal{A}}_l^0}{e}^{\gamma {\mathcal{A}}_l^{-}}=K_{+}{K}_0{K}_{-},$$

(121)

where the last identity in (121) is just the definition of $K_{\pm }$ and $K_0$ and the second one is a consequence of the Baker–Campbell–Hausdorff formula. In order to go further, let us derive with respect to the parameter t so that

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}D(\lambda ,t)=(\lambda {\mathcal{A}}_l^{+}-{\lambda }^{*}{\mathcal{A}}_l^{-})K_{+}{K}_0{K}_{-}\\ \hfill =K_{+}{K}_0{\gamma }^{\prime }{\mathcal{A}}_l^{-}{K}_{-}+{\alpha }^{\prime }{\mathcal{A}}_l^{+}{K}_{+}{K}_0{K}_{-}+K_{+}{\mathcal{A}}_l^0{\beta }^{\prime }{\mathcal{A}}_l^0{K}_0{K}_{-}.\end{array}$$

(122)

Then, using the commutation relations and some algebra, we arrive to

$$\lambda ={\alpha }^{\prime }-\alpha {\beta }^{\prime }+{\alpha }^2{e}^{-\beta }{\gamma }^{\prime },{\beta }^{\prime }=2\alpha {\gamma }^{\prime },{\lambda }^{*}=-e^{-\beta }{\gamma }^{\prime },$$

(123)

where the star denotes conjugation. If we now define

$$\zeta :={\displaystyle \frac{\lambda }{\left|\lambda \right|}}tanh\left|\lambda \right|,$$

(124)

we have

$$\alpha =\zeta ,\beta ={log(1-|\zeta |}^2),\gamma =-{\zeta }^{*}.$$

(125)

Thus, we have all parameters in terms of $\zeta$. After some algebra, we can obtain the explicit expressions for the Perolomov coherent states for each value of $l>0$ as

$${\xi }_l^{\zeta }\left(\rho \right)=\sum _{p=0}^{\infty }{\displaystyle \frac{{\zeta }^p}{\sqrt{p!}}}{log(1-|\zeta |}^2{)}^{(l+1)}{\psi }_{l,p}\left(\rho \right).$$

(126)

Analogously, ${\xi }_l^{\zeta }\left(\rho \right)\in {\Phi }_l$ and ${\xi }_l^{\zeta }\left(\rho \right)\notin {\Psi }_l$ for $l=1,2,\dots$. Nevertheless, after (65), it is clear that ${\eta }_l^{\alpha }\left(\rho \right),{\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right),{\xi }_l^{\zeta }\left(\rho \right)\in {\Psi }_l^{\times }$, for all positive values of l.

Resolutions of the Identity

In this subsection, we briefly analyze the meaning and some properties of the resolutions of the identity given by the coherent states (107). A standard calculation with $d\alpha =dxdy$ shows that for fixed $l>0$, one has

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\pi }}{\int }_{\mathbb{C}}\left|{\eta }_l^{\alpha }\left(\rho \right)⟩⟨{\eta }_l^{\alpha }\left(\rho \right)\right|d\alpha =\sum _{p,q=0}^{\infty }\left|{\psi }_{l,p}\left(\rho \right)⟩⟨{\psi }_{l,q}\left(\rho \right)\right|{\displaystyle \frac{1}{\pi }}{\int }_{\mathbb{C}}{\displaystyle \frac{{\left({\alpha }^{*}\right)}^p}{\sqrt{p!}}}{\displaystyle \frac{{\left(\alpha \right)}^q}{\sqrt{q!}}}{e}^{-{\left|\alpha \right|}^2}d\alpha \\ \hfill =\sum _{p,q=0}^{\infty }\left|{\psi }_{l,p}\left(\rho \right)⟩⟨{\psi }_{l,q}\left(\rho \right)\right|{\displaystyle \frac{p!}{\sqrt{p!q!}}}{\delta }_{p,q}=\sum _{p=0}^{\infty }\left|{\psi }_{l,p}\left(\rho \right)⟩⟨{\psi }_{l,p}\left(\rho \right)\right|=I,\end{array}$$

(127)

where I is the identity on $L^2\left({\mathbb{R}}^{+}\right)$.

Note that the resolution of the identity given by the first integral in (127) not only represents an identity on the Hilbert space $L^2\left({\mathbb{R}}^{+}\right)$. It is, in addition, a representation of the identity mapping, $I_l$, from ${\Phi }_l$ into its dual ${\Phi }_l^{\times }$. In fact, let us take an arbitrarily fixed ${\phi }_l\left(\rho \right)\in {\Phi }_l$ and consider

$${\displaystyle \frac{1}{\pi }}{\int }_{\mathbb{C}}|{\eta }_l^{\alpha }\left(\rho \right)⟩⟨{\eta }_l^{\alpha }\left(\rho \right)|{\phi }_l\left(\rho \right)⟩d\alpha =\sum _{p=0}^{\infty }|{\psi }_{l,p}\left(\rho \right)⟩⟨{\psi }_{l,p}\left(\rho \right)|{\phi }_l\left(\rho \right)⟩={\phi }_l\left(\rho \right).$$

(128)

Thus, $I_l{\phi }_l={\phi }_l$. Let us apply the first term in (128) to any ${\varphi }_l\left(\rho \right)\in {\Phi }_l$. We have

$${\displaystyle \frac{1}{\pi }}{\int }_{\mathbb{C}}⟨{\varphi }_l\left(\rho \right)|{\eta }_l^{\alpha }\left(\rho \right)⟩⟨{\eta }_l^{\alpha }\left(\rho \right)|{\phi }_l\left(\rho \right)⟩d\alpha =\sum _{p=0}^{\infty }⟨{\varphi }_l\left(\rho \right)|{\psi }_{l,p}\left(\rho \right)⟩⟨{\psi }_{l,p}\left(\rho \right)|{\phi }_l\left(\rho \right)⟩=⟨{\varphi }_l|{\phi }_l⟩.$$

(129)

Therefore, $I_l{\phi }_l$ acts on each of the ${\varphi }_l\left(\rho \right)\in {\Phi }_l$ as a linear functional, $F_{{\phi }_l}$, so that $F_{{\phi }_l}\left({\varphi }_l\left(\rho \right)\right)\equiv ⟨{\varphi }_l|I_l{\phi }_l⟩=⟨{\varphi }_l|{\phi }_l⟩$. Thus,

$$|F_{{\phi }_l}\left({\varphi }_l\left(\rho \right)\right)|=|⟨{\varphi }_l|{\phi }_l⟩|\le ||{\varphi }_l\left|\right|\left|\right|{\phi }_l\left|\right|=C_l{p}_{l,0}\left({\varphi }_l\right),\forall {\varphi }_l\left(\rho \right)\in {\Phi }_l,$$

(130)

where $C_l\equiv \left|\right|{\phi }_l\left|\right|$. Note that ${\phi }_l\left(\rho \right)\in {\Phi }_l$ has been chosen to be fixed, which proves that $F_{{\phi }_l}\in {\Phi }_l^{\times }$ and this is true for each ${\phi }_l\left(\rho \right)\in {\Phi }_l$. It is customary to identify $F_{{\phi }_l}$ with ${\phi }_l$ and this is the true meaning of the identity $I_l:{\Phi }_l⟼{\Phi }_l^{\times }$.

The weak topology on ${\Phi }_l^{\times }$ is given by the set of seminorms, $q_{{\psi }_l}$, which are given by $q_{{\psi }_l}\left({\phi }_l\right):=|F_{{\psi }_l}\left({\phi }_l\right)|=|⟨{\psi }_l|{\phi }_l⟩|$, $\forall {\phi }_l\in {\Phi }_l$. Each vector ${\psi }_l\in {\Phi }_l$ defines a unique seminorm $q_{{\psi }_l}$, the correspondence ${\psi }_l⟼q_{{\psi }_l}$ is one to one and vectors ${\psi }_l$ run out the space ${\Phi }_l$. Then, for all ${\phi }_l\left(\rho \right)\in {\Phi }_l$, one has

$$q_{{\psi }_l}\left(I_l{\phi }_l\right)=q_{{\psi }_l}\left(F_{{\phi }_l}\right)=|⟨{\psi }_l|{\phi }_l⟩|=K_l\left|\right|{\phi }_l\left|\right|=K_k{p}_{l,0}\left({\phi }_l\right),$$

(131)

which shows the continuity of the canonical identity mapping $I_l:{\Phi }_l⟼{\Phi }_l^{\times }$. Note that in this argument, we may also find the proof that the identity mapping $I:L^2\left({\mathbb{R}}^{+}\right)⟼{\Phi }_l^{\times }$ is also continuous. Needless to say, canonical identity maps are linear.

A resolution of the identity for (117) was obtained in [31]. It has the form for each $l>0$,

$${\int }_{\mathbb{C}}\left|{\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)⟩⟨{\psi }_{l,\alpha }^{\mathrm{BG}}\left(\rho \right)\right|d\sigma \left(\alpha \right)=\sum _{p=0}^{\infty }\left|{\psi }_{l,p}\left(\rho \right)⟩⟨{\psi }_{l,p}\left(\rho \right)\right|=I.$$

(132)

Here, $d\sigma \left(\alpha \right)=\sigma \left(r\right)rdrd\theta$, $r:=\left|\alpha \right|$ is a measure on the complex plane. The function $\sigma \left(r\right)$ is a product of a constant times a power of r times a Bessel function of a third kind with an argument proportional to r. Properties of this resolution of the identity are similar, as in the previous case.

We do not have a resolution of the identity for the coherent states of the form (126). Nevertheless, a resolution of the identity for the coherent states obtained for the representation of the discrete series of $su(1,1)$ exists [29]. This would require discussion, which does not fit within the context of the present article.

7. Concluding Remarks

In a recent series of articles, it has been shown that Gelfand triplets, also named as rigged Hilbert spaces, are the precise mathematical framework that includes several tools currently used in Quantum Mechanics. These are discrete and continuous bases, a representation of Lie algebras by continuous operators and discrete bases given by specific special functions.

A former and well-known example considers the Heisenberg–Weyl Lie algebra. This algebra includes the momentum and position operators along the harmonic oscillator ladder operators, which are continuous on the Schwartz space (and not on Hilbert space). Here, we take as special functions the normalized Hermite functions [12]. This example has been a guide to analyze other systems in which Legendre, Laguerre, Zernike or Jacobi spacial functions take the place of Hermite functions [13,14]. All the mentioned special functions are very important in a wide range of quantum problems.

Recent works on Quantum Optics reveal the importance of the Laguerre–Gauss special functions. Following the ideas of previous works, we construct suitable Gelfand triplets in which ladder operators, Hamiltonians and other operators are continuous on spaces spanned by Laguerre–Gauss functions for each fixed positive value of the parameter l.

A refinement on the topology of test functions shows that Laguerre–Gauss functions with negative values of l, ${\Phi }_{-l,q}\left(\rho \right)$, can be looked at as functionals on the space of test functions labeled by $l=|-l|$. Thus, we have two types of Gelfand triplets, one in which the topology of the space of test functions is similar to the topology of the Schwartz space, as proven in Appendix B, and another one with a more restrictive and, therefore, smaller topology.

We also show that different types of coherent states can be constructed with the help of the Laguerre–Gauss special functions that belong to the test spaces with a topology similar to the Schwartz space, although they do not belong to the more restrictive subspace. In the derivation of the Perelomov coherent states, we include a representation of $su(1,1)$ by continuous operators.

Future work on this field includes the formalization of other operators coming from factorization, such as ladder operators that appear in Supersymmetric Quantum Mechanics, in terms of Gelfand triplets. Further work will go to the description of continuous basis in terms of frames, a possibility that comes after the discussion in Appendix A.