Gelfand Triplets, Ladder Operators and Coherent States

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Gelfand Triplets, Ladder Operators and Coherent States

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A peer-reviewed article of this preprint also exists.

Maria Blazquez^‡,

Manuel Gadella^*,‡,

Gerardo Jimenez Trejo^‡

Maria Blazquez^‡,

Manuel Gadella^*,‡,

Gerardo Jimenez Trejo^‡

^‡ These authors contributed equally to this work.

This version is not peer-reviewed

Submitted:

27 September 2024

Posted:

29 September 2024

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Abstract

In the present paper and inspired with 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 of their parameters, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre-Gauss functions with negative value 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.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. Introduction and Motivation

As 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 observable 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^{- i t H}

, is well defined and is unitary and, therefore, preserves the probabilities defined by wave functions and 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 involve many subtleties that do not appear in the matrix calculus or on its direct generalization, the calculus with bounded operators. The simple idea 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 dense1 domain

D (A) \subset H

, where

H

is a separable infinite dimensional Hilbert space. Contrarily to what happens for bounded operators (i.e. matrices on a finite dimensional Hilbert space), the relation

〈 ψ | A φ 〉 = 〈 A ψ | φ 〉, \forall ϕ, φ \in D (A),

(1)

does not imply the self-adjointness of A. Just that its adjoint

A^{†}

extends2A,

A ≺ A^{†}

. Self-adjointness of A implies not only

A ≺ A^{†}

, but also that

D (A) \equiv D (A^{†})

, 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 an unbounded self adjoint operator A, there is always a dense subdomain thereof,

Φ

, endowed with a topology finer3 than the Hilbert space topology, on where 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 provides 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 particularly interesting 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 to construct a sequence of Hamiltonians each one with a similar spectrum than the precedent. Here, Hamiltonians and the ladder operators which factorize them are unbounded.

Then, our question is. Can we find a domain of the Hilbert space and a topology on this domain for which Hamiltonian and ladder operators which 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 pass through a mathematically concept widely used in quantum physics, such that the notion of Gelfand triplet also known as rigged Hilbert space (RHS). Let us define this notion.

A Gelfand triplet or rigged Hilbert space is a tern of linear spaces over the complex field [8,9,10,11]:

Φ \subset H \subset Φ^{\times},

(2)

where, i.)

H

is an infinite dimensional separable Hilbert space; ii.)

Φ

is a dense subspace of

H

endowed with its own topology which is finer than the topology that

Φ

has inherited from4

H

; iii.)

Φ^{\times}

is the space of continuous antilinear5 functionals6 on

Φ

endowed with a topology compatible7 with the dual pair

{Φ, Φ^{\times}}

. Often,

Φ^{\times}

is called the antidual space of

Φ

The space

Φ

is a locally convex space [12], with a topology defined by a set of seminorms8 of which one has to be the Hilbert space norm defined on

Φ

as a subspace of

H

. In our particular context, the topology on

Φ

will be defined by a countably infinite set of seminorms, so that the space

Φ

be metrizable9.

A sequence

{ϕ_{n}} \in Φ

converges to

ϕ \in Φ

if and only if for any seminorm p giving the topology on

Φ

, it happens that

p (ϕ_{n}) ⟼ p (ϕ)

. If

Φ

is metrizable, a linear10 mapping

F : Φ ⟼ C

is continuous if and only if for any convergent sequence

ϕ_{n} ⟼ ϕ

Φ

, we have that

F (ϕ_{n}) ⟼ F (ϕ)

. Similarly, an operator A on

Φ

is continuous if

A ϕ_{n} ⟼ A ϕ

Nevertheless if

Φ

were a locally convex space, other more operative method to check the continuity of linear or antilinear mappings is in order. If

F : Φ ⟼ 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}, \dots, p_{n}}

from those which define the topology on

Φ

such that for all

ϕ \in Φ

, we have:

| F (ϕ) | \leq C {p_{1} (ϕ) + p_{2} (ϕ) + \dots + p_{n} (ϕ)} .

(3)

Analogously, if

A : Φ ⟼ Φ

is linear, it is continuous if and only if, for any seminorm

p_{i}

defining the topology on

Φ

there is a positive constant

C > 0

and k seminorms out of those which define the topology on

Φ

such that for all

ϕ \in Φ

[13],

p_{i} (A ϕ) \leq C {p_{i_{1}} (ϕ) + p_{i_{2}} (ϕ) + \dots + p_{i_{k}} (ϕ)} .

(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 Φ ⟼ Ψ

, where

Φ

and

Ψ

are different locally convex spaces. If

{p_{i}}

and

{q_{j}}

are the families of seminorms that produce the topologies on

Φ

and

Ψ

, respectively, then (4) is now written as

q_{j} (A ϕ) \leq C_{j} {p_{i_{j_{1}}} (ϕ) + p_{i_{j_{2}}} (ϕ) + \dots + p_{i_{j_{k}}} (ϕ)},

(5)

for all seminorm

q_{j}

Ψ

, 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

Φ^{\times}

. This is a locally convex topology, for which the seminorms are defined like this: For each

ϕ \in Φ

, we define a seminorm

p_{ϕ}

Φ^{\times}

, such that for any

F \in Φ^{\times}

, one has,

p_{ϕ} (F) : = | F (ϕ) | .

(6)

Finally, it is convenient to use

〈 F | ϕ 〉

instead of

F (ϕ)

to denote the action of

F \in Φ^{\times}

ϕ \in Φ

. As a matter of fact, the weak topology11 induces on

H

a topology weaker (has less open sets) than the Hilbert space topology. Note that any

ψ \in H

gives an

F_{ψ} \in Φ^{\times}

defined as

〈 F_{ψ} | ϕ 〉 = F_{ψ} (ϕ) : = 〈 ψ | ϕ 〉, \forall ϕ \in Φ,

(7)

where

〈 - | - 〉

denotes the scalar product on

H

. The functional

F_{ψ}

is obviously linear on

Φ

. Furthermore, it is also continuous since

| 〈 F_{ψ} | ϕ 〉 | = | 〈 ψ | ϕ 〉 | \leq | | ψ | | | | ϕ | | = C p_{0} (ϕ), \forall ϕ \in Φ,

(8)

with

C = | | ψ | |

. Being given

ψ \in H

, the

F_{ψ} \in Φ^{\times}

is unique. This mapping

i (ψ) = F_{ψ}

, for all

ψ \in H

is antilinear and continuous12. 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 (ψ) = ψ

, whatever

ψ

is),

Φ ⟼ H ⟼ Φ^{\times}

1.1. 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

S

is the linear space of all functions

f (x) : R ⟼ C

, where

R

is the real line, such that:

i.) Any

f (x) \in S

is indefinitely differentiable at all points of the real line

R

ii.) Any

f (x) \in S

converges to zero at the infinite faster than the inverse of any polynomial. This means that for any (complex, x real) polynomial

P (x)

and any

f (x) \in S

, one has that

lim_{x \to \pm \infty} \frac{f (x)}{P (x)} = 0 .

(9)

All functions

f (x) \in S

are in

L^{2} (R)

. Furthermore,

S

is dense in

L^{2} (R)

with the topology of the latter.

We may endow

S

with a locally convex metrizable topology into three equivalent forms, although we choose here one particularly interesting for our purposes. This comes after an interesting characterization of the Schwartz space

S

in terms of the Hermite functions,

{ψ_{n} (x)}

to be defined in the next Section. Hermite functions form an orthonormal basis (complete orthonormal set) for the Hilbert space

L^{2} (R)

, so that any

f (x) \in L^{2} (R)

takes the form

f (x) = \sum_{n = 0}^{\infty} a_{n} ψ_{n} {(x), | | f (x) | |}^{2} = \sum_{n = 0}^{\infty} {| a_{n} |}^{2} < \infty .

(10)

Then,

f (x) \in L^{2} (R)

is in

S

if and only if for the sequnce

{a_{n}}

in (10), one has that for all

k = 0, 1, 2, \dots

,[13]

\sum_{n = 0}^{\infty} {(n + 1)}^{2 k} {| a_{n} |}^{2} < \infty .

(11)

Then, let us define the following set of norms

p_{k} (-) = | | - {| |}_{k}

for all

f (x) \in S

p_{k}^{2} (f) : = {| | f (x) | |}_{k}^{2} : = \sum_{n = 0}^{\infty} {(n + 1)}^{2 k} {| a_{n} |}^{2}, k = 0, 1, 2, \dots .

(12)

These norms (which are also seminorms) define the topology on

S

. Note that i.) Since

k = 0

is a possible value, the list includes the norm on

L^{2} (R)

. This implies that the topology on

S

is finer than the Hilbert space topology on

L^{2} (R)

; ii.) Since the number of norms (seminorms) is countably infinite, the space

S

is metrizable. We may add that

S

is Frèchet space having the property of nuclearity13. Thus,

S \subset L^{2} (R) \subset S^{\times}

(13)

is a Gelfand triplet or RHS. The antidual is isomorphic algebraic and topologically to the space of tempered distributions, usually defined as continuous linear functionals on

S

This paper is organized as follows: In the next Section, we discuss the well known case of the Harmonic oscillator on the Schwartz space and the continuity of all operators involved on it, with also a mention to usual coherent states. Section 3 introduces the notion of Laguerre-Gauss special functions as orthonormal bases of spaces of type

L^{2} (R^{+})

with mention of an optical model which serves as a physical motivation for this mathematical construction. Functions of each orthonormal bases are characterized by a fixed value of a constant

l = 0, 1, 2, \dots

. For each one of the values of l, we define some ladder operators and give their intertwining relations with a Hamiltonian derived from the optical model. In Section 4, we introduce for each value of

l = 0, 1, 2, \dots

a Gelfand triplet. Then, the Hamiltonian and ladder operators are continuous as linear mappings on these triplets. This is the main Section of the present article, in which we define new set of Gelfand triplets in order to fit as continuous linear functionals those Laguerre-Gauss functions with negative value of l. On Section 5, we give a brief presentation of various coherent states constructed on our spaces spanned by the Laguerre-Gauss functions. We finish with some concluding remarks and two appendices.

2. The Harmonic Oscillator

Although the contents of this Section are well known, we consider that a pedagogical account of them will help to 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 = \frac{p^{2}}{2 m} + \frac{1}{2} m ω^{2} q^{2}, ω : = \sqrt{k / m} .

(14)

This Hamiltonian has a pure non-degenerate discrete spectrum with infinite values given by

E_{n} = ℏ ω (n + 1 / 2)

n = 0, 1, 2, \dots

. They respective eigenfunctions are the normalized Hermite functions:

ψ_{n} (x) : = \frac{1}{2^{n} n!} (\frac{m ω}{π ℏ}) e^{- \frac{m ω x^{2}}{2 ℏ}} H_{n} (\sqrt{\frac{m ω}{ℏ}} x), n = 0, 1, 2, \dots,

(15)

where

H_{n} (y)

are the Hermite polynomials. The annihilation and creation operators are, respectively, given by

a : = \sqrt{\frac{m ω}{2 ℏ}} (q + \frac{i}{m ω} p), a^{†} : = \sqrt{\frac{m ω}{2 ℏ}} (q - \frac{i}{m ω} p) .

(16)

These operators are usually called, the ladder operators. Note that q and p are the multiplication and differentiation operators, respectively, i.e.,

q ψ (x) = x ψ (x)

and

p ψ (x) = - i ℏ ψ^{'} (x)

, where the prime denotes derivation with respect to x. In terms of the ladder operators, the Hamiltonian (14) is written as

H = ℏ ω (N + \frac{1}{2}), N : = a^{†} a .

(17)

The operator N is the number operator. When using the ladder operators is somehow convenient a change in the notation for the sake of simplicity. Henceforth, we shall use

| n 〉 : = ψ_{n} (x)

, 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:

The Hilbert space on which Hamiltonian and ladder operator act is $L^{2} (R)$ .
The Hermite functions form an orthonormal basis (also called complete orthonormal set) in $L^{2} (R)$ . Then, the subspace of $L^{2} (R)$ of (finite) linear combinations of Hermite functions is dense in $L^{2} (R)$ .
Hermite functions are Schwartz functions.
The Hamiltonian and the ladder operators are unbounded operators. Hence, they do not act on the whole $L^{2} (R)$ , 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 $L^{2} (R)$ . 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

S

. Let

ψ (x) \in S

and write

ψ (x) = \sum_{n = 0}^{\infty} a_{n} ψ_{n} (x) = \sum_{n = 0}^{\infty} a_{n} | n 〉 .

(19)

Since

ψ (x) \in S

, the coefficients

{a_{n}}

must satisfy (12). Let us define

a ψ

, the action of the operator a on

ψ (x)

a ψ : = \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

S

. For any norm

p_{k} (-)

as in (12),

k = 0, 1, 2, \dots

, we have for any

ψ \in S

that (

m = n + 1

)

\begin{matrix} p_{k} (a ψ) = \sum_{n = 0}^{\infty} b_{n}^{2} {(n + 1)}^{2 k} = \sum_{n = 0}^{\infty} | a_{n + 1} |^{2} {(2 n + 1)}^{2 k + 1} = \sum_{m = 1}^{\infty} {| a_{m} |}^{2} m^{2 k + 1} \\ \leq \sum_{m = 0}^{\infty} {| a_{m} |}^{2} {(m + 1)}^{2 k + 2} = p_{k + 1} (ψ), \end{matrix}

(21)

where the convergence of these sums for all k show that (20) is well defined. Now, let us define the action of

a^{†}

ψ (x) \in S

a^{†} ψ : = \sum_{n = 0}^{\infty} a_{n} \sqrt{n + 1} | n + 1 〉 .

(22)

Thus, for

k = 0, 1, 2, \dots

p_{k} (a^{†} ψ) = \sum_{n = 0}^{\infty} b_{n}^{2} {(n + 1)}^{2 k} = \sum_{n = 0}^{\infty} | a_{n} |^{2} {(n + 1)}^{2 k + 1} \leq \sum_{n = 0}^{\infty} {| a_{n} |}^{2} {(n + 1)}^{2 k + 2} = p_{k + 1} (ψ),

(23)

which shows both that (22) is well defined and that

a^{†}

is continuous on

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} (H ψ) = ℏ^{2} ω^{2} \sum_{n = 0}^{\infty} {(n + 1 / 2)}^{2} | a_{n} |^{2} {(n + 1)}^{2 k} \leq ℏ^{2} ω^{2} \sum_{n = 0}^{\infty} {| a_{n} |}^{2} {(n + 1)}^{2 k + 2} = ℏ^{2} ω^{2} p_{k + 1} (ψ) .

(24)

A note about the coherent states. Let

α \in C

arbitrary but fixed. Its coherent state is given by

| α 〉 : = e^{- \frac{1}{2} {| α |}^{2}} \sum_{n = 0}^{\infty} \frac{α^{n}}{\sqrt{n!}} | n 〉 .

(25)

Coherent states are eigenvectors of the annihilation operator, so that

a | α 〉 = α | α 〉

. Coherent states evolve classically. In the present case,

| α 〉 \in S

for all

α \in C

. To prove it, we just have to show that for all

k = 0, 1, 2, \dots

\sum_{n = 0}^{\infty} \frac{{| α |}^{2 n}}{n!} {(n + 1)}^{2 k} < \infty .

(26)

Then, we just need to show that the general term in the series (26) goes to zero at the infinity faster than

1 / n

. To see that this is true, just note that

lim_{n \to \infty} n^{2} \frac{{| α |}^{2 n}}{n!} {(n + 1)}^{2 k} = 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.

3. 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 [15]. 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:

[- (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) + \frac{Ω^{2}}{2} ρ^{2}] U = i \frac{\partial}{\partial z} U, Ω^{2} ρ^{2} < < 1,

(28)

where

Ω

is a constant and

ρ = \sqrt{x^{2} + y^{2}}

. This model describes the z-propagation of electromagnetic waves through media with square refractive index

1 - Ω^{2} ρ^{2}

. This model has been studied in [16,17,18,19,20] and its physical motivation is not particularly relevant to our purposes.

Which really concerns to 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 as a two dimensional model in spatial coordinates. The plane

(x, y)

may be also described by its polar coordinates

(ρ, θ)

. Then, (28) admits as solutions functions of the type (see [16]):

U_{l, p} = C_{l, p} (z, θ) Φ_{| l |, p} (ρ) l \in Z, p = 0, 1, 2, \dots .

(29)

Under some conditions [16], 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, θ)

, our concern goes to

Φ_{| l |, p} (ρ)

, which are of the form [16]:

Φ_{| l |, p} (ρ) = \frac{2 {(- 1)}^{p}}{w_{0}} \sqrt{\frac{Γ (p + 1)}{Γ (| l | + p + 1)}} {[\frac{\sqrt{2} ρ}{w_{0}}]}^{| l |} exp (- \frac{ρ^{2}}{w_{0}^{2}}) L_{p}^{(| l |)} (\frac{2 ρ^{2}}{w_{0}^{2}}),

(30)

where

w_{0}

is a constant and

L_{p}^{(| l |)} (x)

are the associated Laguerre polynomials of degree p and order

| l |

. These special functions are the mentioned Laguerre-Gauss functions. The most important properties of these functions that concerns to us are [15,16]:

For each fixed value of l, $Φ_{| l |, p} (ρ)$ form an orthonormal basis for $L^{2} (R^{+}, ρ d ρ)$ . Note that ordinary Laguerre functions

$M_{n}^{α} (y) = \sqrt{\frac{Γ (n + 1)}{Γ (n + α + 1)}} y^{α / 2} e^{- y / 2} L_{n}^{α} (y),$

(31)

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

$[- \frac{w_{0}^{2}}{2} (\frac{\partial^{2}}{\partial ρ^{2}} + \frac{1}{ρ} \frac{\partial}{\partial ρ} - \frac{l^{2}}{ρ^{2}}) + 2 \frac{ρ^{2}}{w_{0}^{2}} - 2 β_{| l |, p}] Φ_{| l |, p} (ρ) = 0,$

(32)

with

$β_{| l |, p} = | l | + 2 p + 1 .$

(33)

Next, we define the following functions:

ψ_{| l |, p} (ρ) : = ρ^{1 / 2} Φ_{| l |, p} (ρ) .

(34)

After the aforementioned properties of

Φ_{| l |, p} (ρ)

, it is straightforward to show that, for each fixed value of l, the sequence

{ψ_{| l |, p} (ρ)}

p = 0, 1, 2, \dots

, is an orthonormal basis in

L^{2} (R^{+})

. Functions

ψ_{| l |, p} (ρ)

are also called Laguerre-Gauss.

Now for a fixed value of the integer,

l \in Z

and

ρ \geq 0

, let us consider the following operator on

L^{2} (R^{+})

[21]:

H_{l} : = \frac{w_{0}^{2}}{2} [- \frac{d^{2}}{d ρ^{2}} + \frac{l^{2} - \frac{1}{4}}{ρ^{2}}] + 2 \frac{ρ^{2}}{w_{0}^{2}} .

(35)

After (32) and (34), we obviously have for

l \in Z

and

p = 0, 1, 2, \dots

H_{l} ψ_{| l |, p} (ρ) = 2 β_{| l |, p} ψ_{| l |, p} (ρ) .

(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

ψ_{| l |, p} (ρ)

p = 0, 1, 2, \dots

, forms an orthonormal basis in

L^{2} (R^{+})

, 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} (R^{+})

containing this basis for l. In particular, each of the

H_{l}

l \in Z

, is essentially self adjoint on the subspaces

Φ_{l}

to be defined later.

Then, (35) defines a family of Hamiltonians on

L^{2} (R^{+})

indexed by

Z

(or rather by

| l | = 0, 1, 2, \dots

, since

H_{l}

depends on

l^{2}

). In any case, Hamiltonians of this family,

l \in Z

, can be factorized in terms of the following operators [7]:

A_{l}^{-} : = (\frac{d}{d ρ} - \frac{l + \frac{1}{2}}{ρ} + ρ), A_{l}^{+} : = (- \frac{d}{d ρ} - \frac{l + \frac{1}{2}}{ρ} + ρ),

(37)

and

B_{l}^{-} : = (\frac{d}{d ρ} + \frac{l - \frac{1}{2}}{ρ} + ρ), B_{l}^{+} : = (- \frac{d}{d ρ} + \frac{l - \frac{1}{2}}{ρ} + ρ) .

(38)

These objects as defined on (37) and (38) are the Laguerre-Gauss ladder operators. This factorization of

H_{l}

l \in Z

, can be performed into four different modes and this goes as follows:

H_{l} = B_{l}^{+} B_{l}^{-} - ε_{l - 2} = B_{l + 1}^{-} B_{l + 1}^{+} - ε_{l} = A_{l}^{+} A_{l}^{-} + ε_{l} = A_{l - 1}^{-} A_{l - 1}^{+} + ε_{l - 2},

(39)

where

{ε_{l}}

is a sequence of real numbers. Some other relations are the following:

\begin{matrix} A_{l}^{-} H_{l} = (H_{l + 1} + 2) A_{l}^{-}, H_{l} A_{l}^{+} = A_{l}^{+} (H_{l + 1} + 2), \\ B_{l}^{-} H_{l} = (H_{l - 1} + 2) B_{l}^{-}, H_{l} B_{l}^{+} = B_{l}^{+} (H_{l - 1} + 2) . \end{matrix}

(40)

4. Gelfand Triplets and Continuous Operators

Let

φ (ρ) \in L^{2} (R^{+})

arbitrarily chosen. Then, for fixed l, we know that

φ (ρ) = \sum_{p = 0}^{\infty} α_{l, p} ψ_{| l |, p} (ρ),

(41)

where

{α_{l, p}}

is a sequence of complex numbers such that

\sum_{p = 0}^{\infty} {| α_{l, p} |}^{2} < \infty .

(42)

The convergence of the series in (41) is in the norm of

L^{2} (R^{+})

, as is well known.

For fixed

l \geq 0

, let us construct now a Gelfand triplet in the following form: Let us choose the subspace

Φ_{l}

L^{2} (R^{+})

of all vectors

φ (p)

as in (41) such that

p_{s, | l |}^{2} (φ) : = \sum_{p = 0}^{\infty} | α_{l, p} |^{2} {(| l | + 2 p + 1)}^{2 s} < \infty, s = 0, 1, 2, 3, \dots .

(43)

Then,

p_{s, | l |}

s = 0, 1, 2 \dots

is a family of seminorms14 that produce the topology on

Φ_{l}

. In Appendix B, we shall prove some topological properties of this space for

l > 0

Note that

Φ_{l}

contains all the functions on the orthonormal basis

{ψ_{| l |, p} (ρ)}_{p = 0}^{\infty}

, so that it is dense in

L^{2} (R^{+})

. Since the choice

s = 0

in (43) gives the norm on

L^{2} (R^{+})

, the topology induced on

Φ_{l}

by the Hilbert space norm is coarser than the topology on

Φ_{l}

. Hence the canonical injection

j : Φ_{l} ⟼ L^{2} (R^{+})

j (φ) = φ

\forall φ \in Φ_{l}

is continuous. Therefore, if

Φ_{l}^{\times}

is the antidual space of

Φ_{l}

, we conclude that

Φ_{l} \subset L^{2} (R^{+}) \subset Φ_{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

Φ_{l}

, which means that

H_{l} ψ \in Φ_{l}

for any

φ \in Φ_{l}

, and that it is a continuous operator on

Φ_{l}

. First of all, we need to define the action of

H_{l}

on each of the

φ \in Φ_{l}

, with the requirement that it extends (36). This definition should go as

H_{l} \sum_{p = 0}^{\infty} α_{l, p} ψ_{| l |, p} (ρ) = \sum_{p = 0}^{\infty} α_{l, p} (| l | + 2 p + 1) ψ_{| l |, p} (ρ) .

(45)

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

p_{s, | l |}^{2} (H_{l} φ) = \sum_{p = 0}^{\infty} | α_{l, p} |^{2} {(| l | + 2 p + 1)}^{2 (s + 1)} = p_{s + 1}^{2} (φ) .

(46)

This shows that the coefficients

α_{l, p} (| l | + 2 p + 1)

H_{l} φ

verify relations (43) if

φ \in Φ_{l}

, so that

H_{l} φ \in Φ_{l}

and

H_{l}

leaves

Φ_{l}

invariant for all

l \in Z

. In addition, after (8) one concludes that

H_{l}

is continuous on

Φ_{l}

. Note that none of the

H_{l}

is a bounded (continuous) operator on

L^{2} (R^{+})

4.1. 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 \geq 0

. Laguerre-Gauss functions for negative values of l can be defined, although 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

ρ^{| l |} d ρ

could be used.

The action of the ladder operators (37) and (38) on the basis functions

ψ_{| l |, p} (ρ)

has been well studied [16,22]. For

l \geq 0

, we have that

\begin{matrix} A_{l}^{-} ψ_{l, p} (ρ) = 2 \sqrt{p} ψ_{l + 1, p - 1} (ρ), A_{l - 1}^{+} ψ_{l, p} (ρ) = 2 \sqrt{p + 1} ψ_{l - 1, p + 1} (ρ) (l \neq 0) \\ B_{l}^{-} ψ_{l, p} (ρ) = 2 \sqrt{p + l} ψ_{l - 1, p - 1} (ρ) (l \neq 0), B_{l + 1}^{+} ψ_{l, p} (ρ) = 2 \sqrt{p + l + 1} ψ_{l + 1, p} (ρ) . \end{matrix}

(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{matrix} A_{l}^{-} & : & Φ_{l} ⟼ Φ_{l + 1}, \end{matrix}

(48)

\begin{matrix} A_{l - 1}^{+} & : & Φ_{l} ⟼ Φ_{l - 1}, (l \neq 0), \end{matrix}

(49)

\begin{matrix} B_{l}^{-} & : & Φ_{l} ⟼ Φ_{l - 1}, (l \neq 0), \end{matrix}

(50)

\begin{matrix} B_{l + 1}^{+} & : & Φ_{l} ⟼ Φ_{l + 1} . \end{matrix}

(51)

The proof is simple. First, take

φ \in Φ_{l}

, so that taking into account (41), we have

(A_{l}^{-} ψ) (ρ) = A_{l}^{-} \sum_{p = 0}^{\infty} α_{l, p} ψ_{l, p} (ρ) = \sum_{p = 0}^{\infty} α_{l, p} 2 \sqrt{p} ψ_{l + 1, p - 1} (ρ) .

(52)

Then,

p_{s, | l + 1 |}^{2} (A_{l}^{-} φ) = 4 \sum_{p = 0}^{\infty} p | α_{l, p} |^{2} {(| l + 1 | + 2 p + 1)}^{2 s},

(53)

with

s = 0, 1, 2, \dots

. Clearly, (henceforth, we omit the modulus for l since we are assuming that

l \geq 0

)

p \leq (l + 1 + 2 p + 1),

(54)

and

\begin{matrix} l + 1 + 2 p + 1 = l + 2 p + 1 + 1 \leq 2 (l + 2 p + 1), \end{matrix}

(55)

so that, (53) becomes

p_{s, l + 1}^{2} (A_{l}^{-} φ) \leq 8 \sum_{p = 0}^{\infty} {| α_{l, p} |}^{2} {(l + 2 p + 1)}^{2 (s + 1)} = 8 p_{s + 1, l}^{2} (φ), \forall φ \in Φ_{l} .

(56)

Equation (56) shows that

1.- The series in (53) converges for all values of

s = 0, 1, 2, \dots

for

l + 1

, which implies that

A_{l}^{-} φ

belongs to

Φ_{l + 1}

for all

φ \in Φ_{l}

2.- In accordance to (5), the mapping

A_{l}^{-} : Φ_{l} ⟼ Φ_{l + 1}

is continuous with the topologies defined on both spaces.

The definition for the remainder ladder operators on

Φ_{l}

is similar. In fact,

A_{l - 1}^{+} φ : = 2 \sum_{p = 1}^{\infty} \sqrt{p + 1} α_{l, p} ψ_{l - 1, p + 1} (ρ),

(57)

B_{l}^{-} φ : = 2 \sum_{p = 1}^{\infty} \sqrt{p + l} α_{l, p} ψ_{l - 1, p - 1} (ρ),

(58)

and

B_{l + 1}^{+} φ : = 2 \sum_{p = 0}^{\infty} \sqrt{p + l + 1} α_{l, p} ψ_{l + 1, p} (ρ) .

(59)

The proof of the continuity of these operators is made in analogy with the proof we have given for

A_{l}^{-}

. Take for instance,

\begin{matrix} p_{s, l + 1}^{2} (B_{l + 1}^{+} φ) = 4 \sum_{p = 0}^{\infty} (p + l + 1) {| α_{l, p} |}^{2} {(l + 1 + 2 p + 1)}^{2 s} \\ \leq 8 \sum_{p = 0}^{\infty} | α_{l, p} |^{2} {(l + 2 p + 1)}^{2 s + 3} \leq 8 \sum_{p = 0}^{\infty} {| α_{l, p} |}^{2} {(l + 2 p + 1)}^{2 (s + 2)} = 8 p_{s, l}^{2} (φ), \forall φ \in Φ_{l} . \end{matrix}

(60)

This proves, first that

B_{l + 1}^{+}

maps

Φ_{l}

into

Φ_{l + 1}

and, then, that this map is continuous with respect to the topologies of the initial and final spaces. Same kind of arguments goes for

A_{l - 1}^{+}

and

B_{l}^{-}

4.2. The Laguerre-Gauss Functions with Negative Value of l

Let us go back to the family of functions in (30), were for simplicity we choose

w_{0}^{2} = 2

. This choice simplifies the notation and does not affect to the mathematical discussion under process. Then, the functions in (30) for fixed

l \geq 0

and

p = 0, 1, 2, \dots

form an orthonormal basis for

L^{2} (R^{+}, ρ d ρ)

. Then, for any

ψ (ρ) \in L^{2} (R^{+}, ρ d ρ)

, we have the following span:

ψ (ρ) = \sum_{p = 0}^{\infty} a_{l, p} Φ_{l, p} (ρ), \sum_{p = 0}^{\infty} {| a_{l, p} |}^{2} < \infty,

(61)

where the first sum in (61) converges in the norm of

L^{2} (R^{+}, ρ d ρ)

Now, let

Ψ_{l}

l \geq 0

, the space of all

ψ (ρ) \in L^{2} (R^{+}, ρ d ρ)

, such that for fixed

l \geq 0

, we have that

q_{s, l}^{2} (ψ) : = \sum_{p = 0}^{\infty} {| a_{l, p} |}^{2} {[(l + 2 p + 1)!]}^{2 s} < \infty, s = 0, 1, 2, \dots .

(62)

No matter how small the spaces

Ψ_{l}

could be as subspaces of

L^{2} (R^{+}, ρ d ρ)

, they are dense in

L^{2} (R^{+}, ρ d ρ)

with the Hilbert space topology, since all

Ψ_{l}

contains a orthonormal basis of

L^{2} (R^{+}, ρ d ρ)

. In addition, the seminorm in (62)

q_{l, 0}

is nothing else than the Hilbert space norm, so that the canonical injection

i : Ψ_{l} ⟼ L^{2} (R^{+}, ρ d ρ)

i (ψ) = ψ

is continuous for all values of

l = 0, 1, 2, \dots

. We have a new series of Gelfand triplets given by

Ψ_{l} \subset L^{2} (R^{+}, ρ d ρ) \subset Ψ_{l}^{\times}, l = 0, 1, 2, \dots .

(63)

Let us go back to the beginning of Section 4. Also for

l \geq 0

and a subspace,

{\tilde{Φ}}_{l}

, of vectors

ψ (ρ) \in L^{2} (R^{+}, ρ d ρ)

, we may define the set of norms (43), wich for an “abuse de langage” shall denote as

p_{s, l} (ψ)

, exactly as in the case described there15. This gives a new Gelfand triplet:

{\tilde{Φ}}_{l} \subset L^{2} (R^{+}, ρ d ρ) \subset {\tilde{Φ}}_{l}^{\times} .

(64)

Furthermore, it is obvious that for

l = 0, 1, 2, \dots

Ψ_{l} \subset {\tilde{Φ}}_{l}

. Let us call

j_{l} : Ψ_{l} ⟼ {\tilde{Φ}}_{l}

to the corresponding canonical injection. Then for any

ψ \in Ψ_{l}

as in (61), we have that

p_{s, l} (j_{l} (ψ)) = \sum_{p = 0}^{\infty} | a_{l, p} |^{2} {(l + 2 p + 1)}^{2 s} \leq \sum_{p = 0}^{\infty} {| a_{l, p} |}^{2} {[(l + 2 p + 1)!]}^{2 s} = q_{s, l} (ψ),

(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:

Ψ_{l} \subset {\tilde{Φ}}_{l} \subset L^{2} (R^{+}, ρ d ρ) \subset {\tilde{Φ}}_{l}^{\times} \subset Ψ_{l}^{\times}, l = 0, 1, 2, \dots .

(66)

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

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

l \geq 0

)

Φ_{- l, p} (ρ) = {(- 1)}^{p} \sqrt{\frac{Γ (p + 1)}{Γ (l + p + 1)}} ρ^{- l} exp (- ρ^{2} / 2) L_{p}^{l} (ρ^{2}) .

(67)

Note that the associate Laguerre polynomial

L_{m}^{α} (x)

makes sense only if

α \in (- 1, \infty)

, so that we cannot use

α = - l

. The objective is to show that

Φ_{- l, p} (ρ) \in Ψ_{l}^{\times}

l = 0, 1, 2, \dots

First of all, let us prove that if

ψ (ρ) \in Ψ_{l}

, then, the first series in (61) converges also uniformly. To this end, let us consider the following relation concerning Laguerre functions:

e^{- x} x^{k / 2} L_{n}^{k} (x) = \frac{1}{n!} \int_{0}^{\infty} e^{- t} t^{n + k / 2} J_{k} (2 \sqrt{t x}) d t,

(68)

where

J_{k} (t)

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} (t)

is given by

J_{k} (y) = \frac{1}{π} \int_{0}^{π} cos (k τ - y sin τ) d τ,

(69)

so that

| J_{k} (y) | \leq 1

, k being an integer number. For us,

k = l

and

n = p

. Then,

| e^{- x} x^{l / 2} L_{p}^{l} (x) | \leq \frac{1}{p!} \int_{0}^{\infty} e^{- t} t^{p + l / 2} d t = \frac{Γ (p + l / 2 + 1)}{p!} .

(70)

Taking

x = ρ^{2}

, we have that

| e^{- ρ^{2}} ρ^{l} L_{p}^{l} (ρ^{2}) | \leq \frac{Γ (p + l / 2 + 1)}{p!} .

(71)

ψ (ρ) \in Ψ_{l}

as in (61), one has the following series of inequalities for each

l = 0, 1, 2, \dots

\begin{matrix} | | ψ (ρ) | | = |\sum_{p = 0}^{\infty} a_{l, p} Φ_{l, p} (ρ)| \leq \sum_{p = 0}^{\infty} | a_{l, p} | | Φ_{l, p} (ρ) | \leq \sum_{p = 0}^{\infty} | a_{l, p} | \sqrt{\frac{Γ (p + 1)}{Γ (l + p + 1)}} \frac{Γ (p + l / 2 + 1)}{p!} \\ \leq \sum_{p = 0}^{\infty} | a_{l, p} | (l + 2 p + 1)! \leq \sum_{p = 0}^{\infty} | a_{l, p} | {[(l + 2 p + 1)!]}^{2} \frac{1}{(l + 2 p + 1)!} \\ \leq \sqrt{\sum_{p = 0}^{\infty} {| a_{l, p} |}^{2} {[(l + 2 p + 1)!]}^{4}} \times \sqrt{\sum_{p = 0}^{\infty} \frac{1}{{[(l + 2 p + 1)!]}^{2}}} = C_{l} [q_{2, l} (ψ)], \end{matrix}

(72)

where

C_{l} : = \sqrt{\sum_{p = 0}^{\infty} \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} | | Φ_{l, p} (ρ) |

converges uniformly for

ρ \in [0, \infty)

16.

Now, let

ψ (ρ)

be an arbitrary function in

Ψ_{l}

. The action of

Φ_{- l, q} (ρ)

as in (66),

q = 0, 1, 2, \dots

, as a functional on

Ψ_{l}

should be for all

ψ (ρ) \in Ψ_{l}

\begin{matrix} \int_{0}^{\infty} ψ (ρ) Φ_{- l, q} (ρ) ρ d ρ = \int_{0}^{\infty} (\sum_{p = 0}^{\infty} a_{l, p} Φ_{l, p} (ρ)) Φ_{- l, q} (ρ) ρ d ρ \\ = \sum_{p = 0}^{\infty} a_{l, p} \int_{0}^{\infty} Φ_{l, p} (ρ) Φ_{- l, q} (ρ) ρ d ρ, \end{matrix}

(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} Φ_{l, p} (ρ) Φ_{- l, q} (ρ) ρ d ρ = \sqrt{\frac{Γ (p + 1) Γ (q + 1)}{Γ (l + p + 1) Γ (l + q + 1)}} \int_{0}^{\infty} e^{- ρ^{2}} L_{p}^{l} (ρ^{2}) L_{q}^{l} (ρ^{2}) ρ d ρ .

(74)

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

L_{p}^{l} (ρ^{2}) = \frac{1}{p!} \sum_{s = 0}^{p} \frac{p!}{s!} (\begin{matrix} p + l \\ p - s \end{matrix}) {(- ρ^{2})}^{s},

(75)

so that

\begin{matrix} |\int_{0}^{\infty} e^{- ρ^{2}} L_{p}^{l} (ρ^{2}) L_{q}^{l} (ρ^{2}) ρ d ρ| \\ \leq \sum_{s = 0}^{p} \frac{1}{s!} \frac{(p + l)!}{(p - s)! (l + s)!} \sum_{r = 0}^{p^{'}} \frac{1}{r!} \frac{(p^{'} + l)!}{(p^{'} - r)! (l + r)!} \int_{0}^{\infty} e^{- ρ^{2}} ρ^{2 (r + s)} ρ d ρ \end{matrix}

(76)

Let us make the change of variables

ρ^{2} = t

in the integral. It gives

\frac{1}{2} \int_{0}^{\infty} e^{- t} t^{r + s} d t = \frac{1}{2} Γ (r + s + 1) = \frac{1}{2} (r + s)!,

(77)

so that (75) is smaller or equal to

\sum_{s = 0}^{p} \sum_{r = 0}^{p^{'}} (p + l)! (p^{'} + l)! (s + r)! .

(78)

Now, we have two possibilities. Either

p^{'} > p

, so that

(s + r)! < (2 p^{'})!

p^{'} \leq p

, in which case

(s + r)! \leq (2 p)!

. Also note that

r s \leq (r + s)!

, so that if

p^{'} > p

, then (78) is smaller or equal to

(p + l)! (p^{'} + l)! (2 p^{'})! (r s) \leq (p + l)! {(p^{'} + l)! {[(2 p^{'})!]}^{2}} \leq (l + 2 p + 1)! {(p^{'} + l)! {[(2 p^{'})!]}^{2}} .

(79)

Note that the term between the brackets does not depend on p and only of l and

p^{'}

which are fixed here. Then, if

F_{p^{'}}

represents the functional on

Ψ_{l}

defined by

Φ_{- l, p^{'}} (ρ)

with (73), we have for all

ψ \in Ψ_{l}

\begin{matrix} | F_{q} (ψ) | = |\int_{0}^{\infty} ψ (ρ) Φ_{- l, q} (ρ) ρ d ρ| \leq \sum_{p = 0}^{\infty} | a_{l, p} | q! |\int_{0}^{\infty} e^{- ρ^{2}} L_{p}^{l} (ρ^{2}) L_{q}^{l} (ρ^{2}) ρ d ρ| \\ \leq C_{q, l} \sum_{p = 0}^{q - 1} | a_{l, p} | (l + 2 p + 1)! + \sum_{p = q}^{\infty} | a_{l, p} | q! |\int_{0}^{\infty} e^{- ρ^{2}} L_{p}^{l} (ρ^{2}) L_{q}^{l} (ρ^{2}) ρ d ρ|, \end{matrix}

(80)

with

C_{q, l} : = q! (q + l)! {[(2 q)!]}^{2}

. We analyze both summands in the last row of (80) separately. For the first one, we have:

\begin{matrix} C_{q, l} \sum_{p = 0}^{q - 1} | a_{l, p} | (l + 2 p + 1)! = C_{p^{'}, l} \sum_{p = 0}^{q - 1} | a_{l, p} | {((l + 2 p + 1)!)}^{2} \frac{1}{(l + 2 p + 1)!} \\ \leq C_{q, l} \sqrt{\sum_{p = 0}^{q - 1} | a_{l, p} {|^{2} (l + 2 p + 1)!)}^{4}} \sqrt{\sum_{p = 0}^{q - 1} \frac{1}{{((l + 2 p + 1)!)}^{2}}} \\ \leq C_{q, l} \sqrt{\sum_{p = 0}^{\infty} | a_{l, p} {|^{2} (l + 2 p + 1)!)}^{4}} \sqrt{\sum_{p = 0}^{\infty} \frac{1}{{((l + 2 p + 1)!)}^{2}}} = C_{q, l} C_{l} [q_{2, l} (ψ)], \end{matrix}

(81)

where the second inequality is nothing else that the Cauchy-Schwarz inequality and the meaning of

C_{l}

is obvious. For the second term in the last row of (80) (

p \geq p^{'}

), we just have to replace

{[(2 p^{'})!]}^{2}

in (80) by

{[(2 p)!]}^{2}

, which is smaller or equal to

{[(l + 2 p + 1)!]}^{2}

and

(p^{'})!

p! \leq (l + 2 p + 1)!

. Then if

K_{p^{'}, l} : = (p^{'} + l)!

and following similar arguments as just before, we have that

\begin{matrix} \sum_{p = q}^{\infty} | a_{l, p} | q! |\int_{0}^{\infty} e^{- ρ^{2}} L_{p}^{l} (ρ^{2}) L_{q}^{l} (ρ^{2}) ρ d ρ| \leq K_{q, l} \sum_{p = q}^{\infty} | a_{l, p} | {[(l + 2 p + 1)!]}^{4} \\ \leq K_{q, l} \sum_{p = 0}^{\infty} | a_{l, p} | {[(l + 2 p + 1)!]}^{4} \leq \sqrt{\sum_{p = 0}^{\infty} {| a_{l, p} |}^{2} {[(l + 2 p + 1)!]}^{10}} \sqrt{\frac{1}{{[(l + 2 p + 1)!]}^{2}}} \\ \leq {\tilde{K}}_{q, l} [q_{4, l} (ψ)], \end{matrix}

(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

ψ \in Ψ_{l}

q_{2, l} (ψ) \leq q_{4, l} (ψ)

, we finally arrive to

| F_{q} (ψ) | \leq 2 K [q_{4, l} (ψ)], \forall ψ \in Ψ_{l},

(83)

which proves the continuity of

F_{q}

Ψ_{l}

. This goes for all

q = 0, 1, 2, \dots

, which ends the proof of our statement.

The need of introducing such an strong topology is due to some difficulties producing by the terms in the first sum of (76) and 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.

4.3. 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

Ψ_{l}

. It is noteworthy that equations (48-) also hold with the spaces

Ψ_{l}

with continuity. Just need to check this result with the new version of (48), since the other cases go similarly. The equivalent formula to (48) is

A_{l}^{-} : Ψ_{l} ⟼ Ψ_{l + 1}

(84)

Now, take

ψ (ρ)

as in (61) and assume that

ψ (ρ) \in Ψ_{l}

. Then (

l \geq 0

) after (52), we have for

s = 0, 1, 2, \dots

q_{s, l + 1}^{2} (A_{l}^{-} ψ) = 4 \sum_{p = 0}^{\infty} p {| α_{l, p} |}^{2} {[(l + 1 + 2 p + 1)!]}^{2 s} .

(85)

Clearly,

(l + 2 p + 2)! = (l + 2 p + 2) [(l + 2 p + 1)!] = 2 (l / 2 + p) [(l + 2 p + 1)!] .

(86)

Since

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

(87)

we have that

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

(88)

so that

q_{s, l + 1}^{2} (A_{l}^{-} ψ) \leq 2^{2 s + 2} \sum_{p = 0}^{\infty} {| α_{l, p} |}^{2} {[(l + 2 p + 1)!]}^{4 s + 2} = 2^{2 s + 2} q_{2 s + 1, l}^{2} (ψ) \forall ψ \in Φ_{l} .

(89)

This shows, at the same time, that

A_{l}^{-} ψ \in Ψ_{l}

, for all

ψ \in Ψ_{l}

and that the mapping,

A_{l}^{-}

as in (83) is continuous. Similar proofs will go for the equivalent of mappings (-).

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 is not only enough but it also looks more appropriate the use of the topology introduced by the norms (43). The topology given by (62) makes the spaces

Ψ_{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

Φ_{- l, q} (ρ)

4.4. The Laguerre-Gauss Modes

Along the present subsection, we intend to recall a discussion already presented in [16]. That paper 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 [16] as

U_{l, p} (θ, z, ρ) = \frac{e^{i l θ}}{2 π} exp [- i z β_{| l |, p}] Φ_{| l |, p} (ρ),

(90)

where

β_{| l |, p}

is given by (33). For each fixed value of

z \in R

, the functions

U_{l, p} (θ, z, ρ)

l \in Z

and

p = 0, 1, 2, \dots

form an orthonormal basis on

L^{2} (R^{2})

. 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_{- π}^{π} \int_{0}^{\infty} U_{l, p} U_{l^{'}, p} ρ d ρ d θ = δ_{l, l^{'}} δ_{p, p^{'}},

(91)

and completeness

\sum_{l \in Z} \sum_{p = 0}^{\infty} U_{l, p} (θ, ρ) U_{l, p} (θ^{'}, ρ^{'}) = δ (θ - θ^{'}) δ (ρ - ρ^{'})

(92)

relations hold.

In [16] was defined the following operator for each fixed value of

l \in Z

(we have chosen

w_{0}^{2} = 2

in the formulas given in [16]):

L_{l} : = ρ^{- 1 / 2} L_{l} ρ^{1 / 2}, with L_{l} = \frac{1}{2} [- \frac{\partial^{2}}{\partial ρ^{2}} + \frac{(l + 1 / 2) (l - 1 / 2)}{ρ^{2}}] + ρ^{2}

(93)

It is proven in [16] that the LG modes

U_{l, p} (θ, ρ)

p = 0, 1, 2, \dots

are eigenfunctions of

L_{l}

for each fixed value of

l = 0, 1, 2, \dots

with eigenvalue

\frac{1}{2} β_{| l |, p}

, i.e.,

L_{l} U_{l, p} = \frac{1}{2} β_{| l |, p} U_{l, p}, p = 0, 1, 2, \dots, l \in Z .

(94)

Let us call

H_{l}

the subspace of

L^{2} (R^{2})

by the functions

U_{l, p} (θ, ρ)

p = 0, 1, 2, \dots

and

l \in Z

fixed, which form an orthonormal basis for

H_{l}

. Obviously

L^{2} (R^{2}) = \oplus_{l \in Z} H_{l}

. The functions

U_{l, p} (θ, ρ)

differ from the functions

Φ_{| l |, p} (ρ)

in (30) just for a phase and a multiplicative constant. Thus, they could be identified by an “abuse de langage” and hence

H_{l}

and

L^{2} (R^{+}, ρ d ρ)

for each value of l. Any

ψ_{l} (θ, ρ) \in H_{l}

with fixed l has the form:

ψ_{l} (θ, ρ) = \sum_{p = 0}^{\infty} a_{l, p} U_{l, p} (θ, ρ) with \sum_{p = 0}^{\infty} {| a_{l, p} |}^{2} < \infty .

(95)

Then, let us consider the subspace of functions

ψ_{l}

as in (95) such that

\sum_{p = 0}^{\infty} | a_{l, p} |^{2} {(| l | + 2 p + 1)}^{2 s} < \infty, s = 0, 1, 2, \dots .

(96)

This is a subspace of

H_{l}

which may be identify with

{\tilde{Φ}}_{l}

after the aforementioned “abuse de langage”, so that

{\tilde{Φ}}_{l} \subset H_{l} \subset {\tilde{Φ}}_{l}^{\times}

is a RHS for all values of l. If we define the action of

L_{l}

on each

ψ_{l} \in {\tilde{Φ}}_{l}

, for fixed l, as

L_{l} ψ_{l} (θ, ρ) : = \frac{1}{2} \sum_{p = 0}^{\infty} a_{l, p} (| l | + 2 p + 1) U_{l, p} (θ, ρ),

(97)

we immediately see using the arguments at the beginning of the present Section that (90) is well defined and that

L_{l}

is continuous on

{\tilde{Φ}}_{l}

Ladder operators,

L_{l}^{\pm}

, have been also defined in [16] and this definition can be extended to

Φ_{l}

as follows:

L_{l}^{\pm} ψ_{l} (θ, ρ) = \sum_{p = 0}^{\infty} a_{l, p} U_{l, p \pm 1}, with a_{l, - 1} = 0 .

(98)

Same arguments prove that

L_{l}^{\pm}

are continuous on

{\tilde{Φ}}_{l}

This construction may be extended trivially to

L^{2} (R^{2})

. In fact, if

ψ (θ, ρ) \in L^{2} (R^{2})

, we have

ψ (θ, ρ) = \sum_{l = - \infty}^{\infty} \sum_{p = 0}^{\infty} a_{l, p} U_{l, p} (θ, ρ), with \sum_{l = - \infty}^{\infty} \sum_{p = 0}^{\infty} {| a_{l, p} |}^{2} < \infty .

(99)

Then, select the subspace

Φ

of all vectors,

ψ (θ, ρ)

, in

L^{2} (R^{2})

such that, for all

l \in Z

and

p = 0, 1, 2, \dots

\sum_{l = - \infty}^{\infty} \sum_{p = 0}^{\infty} | a_{l, p} |^{2} {(| l | + 2 p + 1)}^{2 s} < \infty, s = 0, 1, 2, \dots .

(100)

Since

Φ

contains all the LG modes

{U_{l, p}}

l \in Z

p = 0, 1, 2, \dots

, it is dense in

L^{2} (R^{2})

. Since

s = 0

gives the Hilbert space topology, then the topology on

Φ

given by the seminorms for which their squares are given in (100), the topology on

Φ

is finer that the Hilbert space topology. Consequently,

Φ \subset L^{2} (R^{2}) \subset Φ^{\times}

(101)

is a RHS. On

Φ

are continuous the following operators:

[⨁_{l = - \infty}^{\infty} L_{l}] ψ (θ, ρ) : = \frac{1}{2} \sum_{l = - \infty}^{\infty} \sum_{p = 0}^{\infty} a_{l, p} (| l | + 2 p + 1) U_{l, p} (θ, ρ),

(102)

and

[⨁_{l = - \infty}^{\infty} L_{l}^{\pm}] ψ (θ, ρ) : = \sum_{l = - \infty}^{\infty} \sum_{p = 0}^{\infty} a_{l, p} U_{l, p \pm 1} (θ, ρ),

(103)

with

a_{l, - 1} = 0

for all value of l. Proofs are as in previous examples.

5. Coherent States and Continuous Generators for the Algebra $s u (1, 1)$

Let us give a brief analysis of different types of coherent states 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 [16]. These are ladder operators defined on

Φ_{l}

for fixed values of

l > 0

. On the basis vectors of

Φ_{l}

these operators act as follows [7]:

A^{+} ψ_{l, p} : = \sqrt{p + 1} ψ_{l, p + 1} (ρ), A^{-} ψ_{l, p} : = \sqrt{p} ψ_{l, p - 1} (ρ),

(104)

so that their action on an arbitrary vector in

Φ_{l}

l > 0

, of the form (41) with (43) takes the form:

A^{+} φ (ρ) : = \sum_{p = 0}^{\infty} a_{l, p} \sqrt{p + 1} ψ_{l, p + 1} (ρ), A^{-} φ (ρ) : = \sum_{p = 1}^{\infty} a_{l, p} \sqrt{p} ψ_{l, p - 1} (ρ) .

(105)

Following similar techniques than those given in Section 4.1, we easily show that both ladder operators

A^{\pm}

are well defined and continuous on each of the

Φ_{l}

l = 1, 2, \dots

Since, we are focusing our attention on coherent states, we are mainly interested on the operator

A^{-}

, since coherent states are eigenvectors of the annihilation operator with arbitrary eigenvalue. Recall that coherent states are the eigenstates of the annihilation operator, so that if

η^{α} (ρ)

is a coherent state

A^{-} η^{α} (ρ) = α η^{α} (ρ)

with

α \in C

. Clearly coherent states must satisfy the following relation:

A^{-} η^{α} (ρ) = \sum_{p = 0}^{\infty} b_{l, p} \sqrt{p} ψ_{l, p - 1} (ρ) = α \sum_{p = 0}^{\infty} b_{l, p} ψ_{l, p} (ρ) .

(106)

A straightforward calculation gives

η^{α} (ρ) \equiv η_{l}^{α} (ρ) = \sum_{p = 0}^{\infty} \frac{α^{p}}{\sqrt{p!}} e^{- {| α |}^{2} / 2} ψ_{l, p} (ρ) .

(107)

Thus, coherent states are defined for each value of

l = 1, 2, \dots

. Furthermore,

Proposition.- For fixed

α \in C

and

l = 1, 2, \dots

η_{l}^{α} (ρ) \in Φ_{l}

Proof.- We just need to show that

\sum_{p = 0}^{\infty} \frac{{| α |}^{2 p}}{p!} e^{- {| α |}^{2}} {(l + 2 p + 1)}^{2 s} < \infty, s = 0, 1, 2, \dots .

(108)

Note that the term

e^{- {| α |}^{2}}

in (108) is a constant and, hence, irrelevant. To show this, it is sufficient to prove that for large values of p we have17

\frac{{| α |}^{2 p}}{p!} {(l + 2 p + 1)}^{2 s} < \frac{1}{p^{2}} s = 0, 1, 2, \dots .

(109)

Note that, according to the Stirling formula,

p! \sim \sqrt{2 π p} {(\frac{p}{e})}^{p},

(110)

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

\frac{a^{p}}{\sqrt{2 π p} p^{p}} {(l + 2 p + 1)}^{2 s} p^{2} < 1, a = {| α |}^{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 + 2 p + 1)}^{2 s} p^{2} < p^{p / 2},

(113)

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

These coherent states are often referred as Glauber-Klauder-Sudarshan coherent states. Note that the coherent state

η_{l}^{α} (ρ) \notin Ψ_{l}

, since for

s = 3

\sum_{p = 0}^{\infty} \frac{{| α |}^{2 p}}{{(p!)}^{2}} {[(l + 2 p + 1)!]}^{6} = \infty .

(114)

Next, let us consider a new set of ladder operators,

A_{l}^{\pm}

, formally defined as

A_{l}^{\pm} : = (ρ^{2} \mp ρ \frac{d}{d ρ} - (l + 2 p \mp \frac{1}{2})) .

(115)

Their action on the basis of functions

ψ_{l, p} (ρ)

is given by

A_{l}^{+} ψ_{l, p - 1} (ρ) = \sqrt{p (l + p)} ψ_{l, p} (ρ), A_{l}^{-} ψ_{l, p} (ρ) = \sqrt{p (l + p)} ψ_{l, p - 1} (ρ) .

(116)

Their extension to any vector

φ (ρ) \in Φ_{l}

is straightforward. This extensions are well defined and continuous on

Φ_{l}

As in the previous case, we may here define coherent states for each positive value of l,

ψ_{l, α}^{BG} (ρ)

, as

A_{l}^{-} ψ_{l, α}^{BG} (ρ) = α ψ_{l, α}^{BG} (ρ)

\forall α \in C

. They are called the Barut-Girardello coherent states. They have the following form:

ψ_{l, α}^{BG} (ρ) = \sum_{p = 0}^{\infty} \frac{α^{p}}{\sqrt{p! (l + p)!}} \frac{α^{l / 2}}{\sqrt{I_{l} (2 α)}} ψ_{l, p} (ρ),

(117)

where

I_{l} (x)

is the modified Bessel function. Again,

ψ_{l, α}^{BG} (ρ) \in Φ_{l}

and

ψ_{l, α}^{BG} (ρ) \notin Ψ_{l}

Finally, we have the Perelomov coherent states. Their construction requieres some analysis. First of all, let us define a new operator

A_{0}

2 A_{l}^{0} ψ_{l, p} (ρ) : = [A_{l}^{-}, A_{l}^{+}] ψ_{l, p} (ρ) = (l + 2 p + 1) ψ_{l, p} (ρ) .

(118)

Clearly,

A_{l}^{0}

may be extended to

Φ_{l}

with continuity. Note that the operators

A_{l}^{\pm}

and

A_{l}^{0}

give a system of generators on each of the

Φ_{l}

of the Lie algebra

s u (1, 1)

, since

[A_{l}^{0}, A_{l}^{\pm}] = \pm A_{l}^{\pm}, [A_{l}^{-}, A_{l}^{+}] = 2 A_{l}^{0} .

(119)

In consequence, each of the

Φ_{l}

supports an irreducible representation of the algebra

s u (1, 1)

given by continuous operators with the topology on

Φ_{l}

. Note that the operators on the corresponding enveloping algebras are also continuous on each of the

Φ_{l}

. Their canonical extension to the duals

Φ_{l}^{\times}

are also continuous with any topology on

Φ_{l}^{\times}

compatible with duality.

In order to construct the Perolomov coherent states, let us consider for each

l > 0

the ground state

ψ_{l, 0} (ρ)

and define

ξ_{l} (ρ) : = D (λ, t) ψ_{l, 0} (ρ),

(120)

where

λ

is a complex number and

D (λ, t) : = e^{t (λ A_{l}^{+} - λ^{*} A_{l}^{-})} = e^{α A_{l}^{+}} e^{β A_{l}^{0}} e^{γ 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{matrix} \frac{d}{d t} D (λ, t) = (λ A_{l}^{+} - λ^{*} A_{l}^{-}) K_{+} K_{0} K_{-} \\ = K_{+} K_{0} γ^{'} A_{l}^{-} K_{-} + α^{'} A_{l}^{+} K_{+} K_{0} K_{-} + K_{+} A_{l}^{0} β^{'} A_{l}^{0} K_{0} K_{-} . \end{matrix}

(122)

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

λ = α^{'} - α β^{'} + α^{2} e^{- β} γ^{'}, β^{'} = 2 α γ^{'}, λ^{*} = - e^{- β} γ^{'},

(123)

where the star denotes conjugation. If we now define

ζ : = \frac{λ}{| λ |} tanh | λ |,

(124)

we have

α = ζ, β = {log (1 - | ζ |}^{2}), γ = - ζ^{*} .

(125)

Thus, we have all parameters in terms of

ζ

. After some algebra, we can obtain the explicit expressions for the Perolomov coherent states for each value of

l > 0

ξ_{l}^{ζ} (ρ) = \sum_{p = 0}^{\infty} \frac{ζ^{p}}{\sqrt{p!}} {log (1 - | ζ |}^{2})^{(l + 1)} ψ_{l, p} (ρ) .

(126)

Analogously,

ξ_{l}^{ζ} (ρ) \in Φ_{l}

and

ξ_{l}^{ζ} (ρ) \notin Ψ_{l}

for

l = 1, 2, \dots

. Nevertheless, after (65), it is clear that

η_{l}^{α} (ρ), ψ_{l, α}^{BG} (ρ), ξ_{l}^{ζ} (ρ) \in Ψ_{l}^{\times}

, for all positive values of l.

5.1. 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 α = d x d y

shows that for it fixed

l > 0

, one has

\begin{matrix} \frac{1}{π} \int_{C} | η_{l}^{α} (ρ) 〉 〈 η_{l}^{α} (ρ) | d α = \sum_{p, q = 0}^{\infty} | ψ_{l, p} (ρ) 〉 〈 ψ_{l, q} (ρ) | \frac{1}{π} \int_{C} \frac{{(α^{*})}^{p}}{\sqrt{p!}} \frac{{(α)}^{q}}{\sqrt{q!}} e^{- {| α |}^{2}} d α \\ = \sum_{p, q = 0}^{\infty} | ψ_{l, p} (ρ) 〉 〈 ψ_{l, q} (ρ) | \frac{p!}{\sqrt{p! q!}} δ_{p, q} = \sum_{p = 0}^{\infty} | ψ_{l, p} (ρ) 〉 〈 ψ_{l, p} (ρ) | = I, \end{matrix}

(127)

where I is the identity on

L^{2} (R^{+})

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} (R^{+})

. It is, in addition, a representation of the identity mapping,

I_{l}

, from

Φ_{l}

into its dual

Φ_{l}^{\times}

. In fact, let us take an arbitrarily fixed

φ_{l} (ρ) \in Φ_{l}

and consider

\frac{1}{π} \int_{C} | η_{l}^{α} (ρ) 〉 〈 η_{l}^{α} (ρ) | φ_{l} (ρ) 〉 d α = \sum_{p = 0}^{\infty} | ψ_{l, p} (ρ) 〉 〈 ψ_{l, p} (ρ) | φ_{l} (ρ) 〉 = φ_{l} (ρ) .

(128)

Thus,

I_{l} φ_{l} = φ_{l}

. Let us apply the first term in (128) to an any

ϕ_{l} (ρ) \in Φ_{l}

. We have,

\frac{1}{π} \int_{C} 〈 ϕ_{l} (ρ) | η_{l}^{α} (ρ) 〉 〈 η_{l}^{α} (ρ) | φ_{l} (ρ) 〉 d α = \sum_{p = 0}^{\infty} 〈 ϕ_{l} (ρ) | ψ_{l, p} (ρ) 〉 〈 ψ_{l, p} (ρ) | φ_{l} (ρ) 〉 = 〈 ϕ_{l} | φ_{l} 〉 .

(129)

Therefore,

I_{l} φ_{l}

acts on each of the

ϕ_{l} (ρ) \in Φ_{l}

as a linear functional,

F_{φ_{l}}

, so that

F_{φ_{l}} (ϕ_{l} (ρ)) \equiv 〈 ϕ_{l} | I_{l} φ_{l} 〉 = 〈 ϕ_{l} | φ_{l} 〉

. Thus,

| F_{φ_{l}} (ϕ_{l} (ρ)) | = | 〈 ϕ_{l} | φ_{l} 〉 | \leq | | ϕ_{l} | | | | φ_{l} | | = C_{l} p_{l, 0} (ϕ_{l}), \forall ϕ_{l} (ρ) \in Φ_{l},

(130)

where

C_{l} \equiv | | φ_{l} | |

. Note that

φ_{l} (ρ) \in Φ_{l}

has been chosen to be fixed, which proves that

F_{φ_{l}} \in Φ_{l}^{\times}

and this is true for each

φ_{l} (ρ) \in Φ_{l}

. It is customary to identify

F_{φ_{l}}

with

φ_{l}

and this is the true meaning of the identity

I_{l} : Φ_{l} ⟼ Φ_{l}^{\times}

The weak topology on

Φ_{l}^{\times}

is given by the set of seminorms,

q_{ψ_{l}}

, which are given by

q_{ψ_{l}} (φ_{l}) : = | F_{ψ_{l}} (φ_{l}) | = | 〈 ψ_{l} | φ_{l} 〉 |

\forall φ_{l} \in Φ_{l}

. Each vector

ψ_{l} \in Φ_{l}

defines a unique seminorm

q_{ψ_{l}}

, the correspondence

ψ_{l} ⟼ q_{ψ_{l}}

is one to one and vectors

ψ_{l}

run out the space

Φ_{l}

. Then, for all

φ_{l} (ρ) \in Φ_{l}

one has

q_{ψ_{l}} (I_{l} φ_{l}) = q_{ψ_{l}} (F_{φ_{l}}) = | 〈 ψ_{l} | φ_{l} 〉 | = K_{l} | | φ_{l} | | = K_{k} p_{l, 0} (φ_{l}),

(131)

which shows the continuity of the canonical identity mapping

I_{l} : Φ_{l} ⟼ Φ_{l}^{\times}

. Note that in this argument, we may find also the proof that the identity mapping

I : L^{2} (R^{+}) ⟼ Φ_{l}^{\times}

is also continuous. Needless to say that canonical identity maps are linear.

A resolution of the identity for (117) has been obtained in [23]. It has the form for each

l > 0

\int_{C} | ψ_{l, α}^{BG} (ρ) 〉 〈 ψ_{l, α}^{BG} (ρ) | d σ (α) = \sum_{p = 0}^{\infty} | ψ_{l, p} (ρ) 〉 〈 ψ_{l, p} (ρ) | = I .

(132)

Here,

d σ (α) = σ (r) r d r d θ

r : = | α |

, is a measure on the complex plane. The function

σ (r)

is a product of a constant times a power of r times a Bessel function of third kind with 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

s u (1, 1)

exists [24]. This would requiere a discussion which does not fit within the context of the present article.

6. 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 know example considers the Heisenberg-Weyl Lie algebra. This algebra includes the momentum and poistion 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 [25]. This example has been the guide to analyze other systems in which Legendre, Laguerre, Zernike or Jacobi spacial functions take the place of Hermite functions [26,27]. All the mentioned special functions are very important in a wide range of quantum problems.

Recent works on Quantum Optics reveal to 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 values of the parameter l.

A refinement on the topology of test functions shows that Laguerre-Gauss functions with negative values of l,

Φ_{- l, q} (ρ)

, can be looked as functionals on the space of test functions labelled 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 II and another one with a more restrictive topology and, therefore, smaller.

We also show that different types of coherent states can be constructed with the help of the Laguerre-Gauss special functions and that belong to the test spaces with topology similar to the Schwartz space, although do not belong to the more restrictive subspace. In the derivation of the Perelomov coherent states, we include a representation of

s u (1, 1)

by continuous operators.

Author Contributions

All authors have equally contributed to the present research, including conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing—original draft preparation, writing—review and editing, visualization, supervision, project administration and funding acquisition. All authors have read and agreed to the published version of the manuscript.

Funding

Financial support is acknowledged to the the Spanish MCIN with funding from the European Union Next Generation EU, PRTRC17.11, and the Consejería de Educación from the JCyL through the QCAYLE project, as well as MCIN projects PID2020-113406GB-I00 and RED2022-134301-T. The work of M. Blazquez and G. Jimenez Trejo was partially supported by the Junta de Castilla y León (Project BU229P18), Consejo Nacional de Humanidades, Ciencias y Tecnologías (Project A1-S-24569 and CF 19-304307) and Instituto Politécnico Nacional (Project SIP20242277). M. Blazquez and G. Jimenez Trejo thanks to Consejo Nacional de Humanidades, Ciencias y technologíasfor the PhD scholarship assigned to CVU 885124 and CVU 994641, respectively.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data has been created.

Acknowledgments

M. Blazquez and G. Jimenez Trejo thanks to Prof S. Cruz y Cruz and Quantiita by their support and invaluable help in reading and commenting this work.

Conflicts of Interest

The authors declare no conflict of interests.

Appendix A. Frames

In this Appendix, we wish to make the following comment:

Resolutions of the identity for coherent states show that coherent states are continuous frames. This is a rather trivial statement, provided we take into account the definition of continuous frames on Hilbert spaces. Then, let us give the technical definition thereof [28]. See also [29].

Definition.- Let

(X, μ)

be a measure space where

μ

is a

σ

-finite positive measure. A weakly measurable function

ζ : x \in X ⟼ ζ_{x} \in H

is a continuous frame in the Hilbert space

H

, if there exists two positive constants

A, B \geq 0

such that

{A | | f | |}^{2} \leq \int_{X} | 〈 f | ζ_{x} 〉 |^{2} d μ \leq {B | | f | |}^{2}, \forall f \in H .

(A1)

One of the most interesting examples of continuous frames are the so called Gelfand bases [28]. Perhaps the most typical example of Gelfand base comes out the Gelfand Maurin decomposition theorem mentioned in the Introduction. Although we do not intend to go into details which go far beyond the scope of the present article, let us give a brief presentation thereof:

Theorem (Gelfand-Maurin).- Let A be a self adjoint unbounded operator on a separable infinite dimensional Hilbert space

H

. Then, there is a Gelfand triplet

Φ \subset H \subset Φ^{\times}

such that

i.)

A Φ \subset Φ

and A is continuous on

Φ

. Thus, A can be continuously extended to

Φ^{\times}

. The extension is given by the so called duality formula:

〈 A φ | F 〉 = 〈 φ | A F 〉, \forall φ \in Φ, \forall F \in Φ^{\times} .

(A2)

ii.) Let

σ (A) \subset R

the whole spectrum of A. Then, there exists a

σ

-finite measure,

d μ (λ)

, on

σ (A)

such that for almost all

λ \in σ (A)

, with respect to

d μ (λ)

, there is a functional

F_{λ} \in Φ^{\times}

such that

A F_{λ} = λ F_{λ}

. This means that

F_{λ}

is an eigenvector of A in

Φ^{\times}

with eigenvalue

λ

. For

λ

in the continuous spectrum of A,

F_{λ}

is not normalized, i.e., a vector in

H

, and then

F_{λ}

of often mentioned as the generalized eigenvector of A with generalized eigenvalue

λ

iii.) The following spectral decomposition holds: For all pairs

φ, ψ \in Φ

n = 0, 1, 2, \dots

, we have that

〈 φ | A^{n} ψ 〉 = \int_{σ (A)} λ^{n} 〈 φ | F_{λ} 〉 〈 F_{λ} | ψ 〉 d μ (λ),

(A3)

where

〈 F_{λ} | ψ 〉

is the action of the functional

F_{λ}

on the vector

ψ \in Φ

and

〈 φ | F_{λ} 〉 = {〈 F_{λ} | φ 〉}^{*}

, where the star denotes complex conjugate.

Once we have stated this result, consider (A3) and take

n = 0

. Consequently, we have for all

φ \in Φ

the following expression:

{| | φ | |}^{2} = \int_{σ (A)} 〈 φ | F_{λ} 〉 〈 F_{λ} | φ 〉 d μ (λ) = \int_{σ (A)} {| 〈 F_{λ} | φ 〉 |}^{2} d μ (λ) .

(A4)

Clearly, (A4) has the form (A1) with the constants

A = B = 1

and

φ \in Φ

. In this case, we have a continuous frame, which is not defined on the Hilbert space

H

, but instead on the locally convex space

Φ

Nevertheless, we have another example of continuous frame, which fits with our previous discussion on coherent states. To state it, we just need to construct the resolution of the identity for each set of coherent states. Take for instance the resolution of the identity given in terms of coherent states (127). For every

f (ρ) \in L^{2} (R^{+}, d x)

, we may write

{| | f | |}^{2} = \frac{1}{π} \int_{C} 〈 f (ρ) | η_{l}^{α} (ρ) 〉 〈 η_{l}^{α} (ρ) | f (ρ) 〉 d α = \int_{C} {| 〈 f (ρ) | η_{l}^{α} (ρ) 〉 |}^{2} d α / π .

(A5)

Observe that (A5) is identical to (A1), with

H \equiv L^{2} (R^{+}, d x)

X \equiv C

x \equiv α

ζ_{x} \equiv η_{l}^{α} (ρ)

(here we have one frame for each value of

l > 0

d μ \equiv d α / π

and

A = B = 1

Appendix B. A Comment on the Topology of the Φ_l with l > 0

In the present Appendix, we shall show that each of the spaces

Φ_{l}

l > 0

, defined in Section 4 is isomorphic algebraic and topological to the Schwartz space

S

, given in the Introduction. Then, after a definition of the notion of unitary equivalence of rigged Hilbert spaces, we shall show that the triplets

S \subset L^{2} (R) \subset S^{\times}

and

Φ_{l} \subset L^{2} (R^{+}) \subset Φ_{l}^{\times}

for all

l > 0

are unitarily equivalent.

The isomorphism goes as follows: Consider an arbitrary function

f (ρ) \in Φ_{l}

f (ρ) = \sum_{p = 0}^{\infty} a_{p} ψ_{l, p} (ρ),

(A6)

and consider the mapping

T_{l} : Φ_{l} ⟼ S

, where

S

is the Schwartz space, given by

T_{l} [f (ρ)] : = \sum_{p = 0}^{\infty} a_{p} ψ_{n} (x),

(A7)

where

ψ_{n} (x)

are the normalized Hermite functions (15). Obviously,

T_{l}

is linear from

Φ_{l}

L^{2} (R)

. Let us prove that the image by

T_{l}

of any

f (ρ) \in Φ_{l}

is in

S

and that this mapping is one to one and continuous. As customary along the present paper, we call

p_{n}

n = 0, 1, 2 \dots

the norms on

S

and

p_{s, l}

s = 0, 1, 2, \dots

, l fixed, the norms on

Φ_{l}

. Then, for

s = 0, 1, 2, \dots

p_{s} (T_{l} f) = \sum_{n = 0}^{\infty} | a_{n} |^{2} {(n + 1)}^{2 s} \leq \sum_{n = 0}^{\infty} {| a_{n} |}^{2} {(l + 2 n + 1)}^{2 s} = p_{s, l} (f),

(A8)

which shows that

T_{l} f

is in

S

due that the first series in (A8) converges, since the second one converges because

f \in Φ_{l}

. In addition, (A8) shows the continuity of

T_{l}

. In addition,

T_{l}

is one to one and onto by construction.

Then, there exists the inverse

T_{l}^{- 1}

, of

T_{l}

. Let us prove that

T_{l}^{- 1}

is continuous. Let us consider first

{(l + 2 p + 1)}^{2 s} = {(l - 1 + 2 p + 2)}^{2 s} = {(l - 1 + 2 (p + 1))}^{2 s} = \sum_{k = 0}^{2 s} \frac{(2 s)!}{k! (2 s - k)!} {(l - 1)}^{k} 2^{2 s - k} {(p + 1)}^{2 s - k} .

(A9)

Let

g (x) : = \sum_{p = 0}^{\infty} a_{p} ψ_{p} (x) \in S

. Then,

\begin{matrix} p_{l, s} (T_{l}^{- 1} g) = \sum_{p = 0}^{\infty} | a_{p} |^{2} {(l + 2 p + 1)}^{2 s} = \sum_{k = 0}^{2 s} \frac{(2 s)!}{k! (2 s - k)!} {(l - 1)}^{k} 2^{2 s - k} \sum_{p = 0}^{\infty} {| a_{p} |}^{2} {(p + 1)}^{2 s - k} \\ = \sum_{k = 0}^{\infty} C_{2 s - k} p_{2 s - k} (g), \forall g (x) \in S, \end{matrix}

(A10)

where,

C_{k} : = \frac{(2 s)!}{k! (2 s - k)!} {(l - 1)}^{k} 2^{2 s - k}, k = 0, 1, 2, \dots, 2 s .

(A11)

Note that the second identity in (A10) is legitimate since it comes from a series of positive terms. observe that for

l = 1

, only the term

k = 0

remains in the sum on k.

In conclusion, we have shown the algebraic isomorphism and topological homeomorphism between each of the

Φ_{l}

l > 0

, and the Schwartz space

S

References

Gelfand, I.M.; Vilenkin, N.Y. Generalized Functions: Applications of Harmonic Analysis; Academic Press, New York, 1964.
Maurin, K. General Eigenfunction Expansions and Unitary Representation of Topological Groups; Polish Scientific Publishers, Warszawa, 1968.
Infeld, L; Hull, T. E., The factorization method. Rev. Mod. Phys., 1951, 23(1), 21–68. [Google Scholar] [CrossRef]
Cooper, F.; Khare, A. ; Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific, Singapur, 2001. [CrossRef]
Mielnik, B. Factorization method and new potentials with the oscillator spectrum, 1984, J. Math. Phys., 19 (3), 239-250. [CrossRef]
Fernández, D.J. New hydrogen-like potentials, 1984, Lett. Math. Phys., 8 (4), 337-343. [CrossRef]
Rosas-Ortiz, O. , Exactly solvable hydrogen-like potentials and the factorization method. J. Phys. A: Math. Gen, 1998, 31(50), 10163–10179. [Google Scholar] [CrossRef]
Bohm. A., The rigged Hilbert spaces in quantum physics, ICTP, Report No 4, Trieste (1965).
Roberts, J.E. Rigged Hilbert spaces in quantum mechanics, Commun. Math. Phys., 1966, 3, 98–119. [Google Scholar] [CrossRef]
Antoine, J.P. Dirac formalism and symmetry problems in quantum mechanics. I. General Dirac formalism, J. Math. Phys., 1969, 10(1), 53–69. [Google Scholar] [CrossRef]
Melsheimer, O. Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory. J. Math. Phys., 1973, 15(7), 902–916. [Google Scholar] [CrossRef]
Horváth, J. Topological Vector Spaces and Distributions, Addison-Wesley, Reading Massachusetts, 1966.
Reed, M.; Simon, B. Functional Analysis, Academic Press, New York, 1972.
Pietsch, A. Nuclear Locally Convex Spaces, Springer Verlag, Berlin 1972.
Karimi, E.; Boyd, R.W.; De La Hoz, P.; De Guise, H.; Rehácek, J.; Hradil, Z.; Aiello, A.; Leuchs, G.; Sánchez-Soto, L.L. Radial quantum number of Laguerre-Gauss modes. Phys. Rev A, 2014, 89, 063813. [Google Scholar] [CrossRef]
Cruz y Cruz, S.; Gress, Z.; Jiménez-Macías, P.; Rosas-Ortiz, O. Bessel-Gauss beams of arbitrary integer order: Propagation profile, coherence properties and quality factor. Photonics, 2023, 10, 1162. [Google Scholar] [CrossRef]
Morales Rodríguez, M.P.; Magaña Loaiza, O.S.; Pérez-García, B.; Nieto-Calzada, L.M.; Marroquín-Gutiérrez, F.; Rodríguez-Lara, B.M. Coherent states of the Laguerre-Gauss modes, Optics Letters, 2024, 49 96), 1489-1492. [CrossRef]
Cruz y Cruz, S.; Gress, Z.; Jimenez-Macias, P.; Rosas-Ortiz, O. Laguerre-Gaussian Wave Propagation in Parabolic Media. In Geometric Methods in Physics XXXVIII; Trends in Mathematics; Birkhäuser: Cham, Switzerland, 2020. [Google Scholar] [CrossRef]
Cruz y Cruz, S.; Gress, Z. Group approach to the paraxial propagation of Hermite-Gaussian modes in a parabolic medium. Ann. Phys., 2017, 383, 257–277. [Google Scholar] [CrossRef]
Gress, Z.; Cruz y Cruz, S. Hermite Coherent States for Quadratic Refractive Index Optical Media. In Integrability, Supersymmetry and Coherent States; CRM Series in Mathematical Physics; Springer Nature Switzerland AG: Gewerbestrasse, Switzerland, 2019. [Google Scholar] [CrossRef]
Rosas Ortiz, O.; Cruz y Cruz, S.; Enriquez, M. SU(1, 1) and SU(2) approaches to the radial oscillator: Generalized coherent states and squeezing of variances, Annals of Physics, 2016, 373, 346-373. [CrossRef]
Jimenez Trejo, G; Agarwal, P.; Cruz y Cruz, S. Localized Paraxial beams as optical realizations of generalized coherent states. 2024, Preprint.
Barut, A.O.; Girardello, L. New coherent states associated with non-compact groups, Commun. Math. Phys., 1971, 21, 41–55. [Google Scholar] [CrossRef]
A. Perelomov, Generalized Coherent States and Their Applications, Springer Verlag, Berlin, Heidelberg 1986.
Celeghini, E.; Gadella, M.; del Olmo, M.A. Applications of rigged Hilbert spaces in quantum mechanics and signal processing. J. Math. Phys., 2016, 57, 072105. [Google Scholar] [CrossRef]
Celeghini, E.; Gadella, M.; del Olmo, M.A. Zernike functions, rigged Hilbert spaces and potential applications. J. Math. Phys., 2019, 60, 083508. [Google Scholar] [CrossRef]
Celeghini, E.; Gadella, M.; del Olmo, M.A. Groups, Jacobi Functions and rigged Hilbert spaces. J. Math. Phys., 2020, 61, 033508. [Google Scholar] [CrossRef]
Trapani, C.; Triolo, S.; Tschinke, F. Distribution Frames and Bases, J. Four. Anal. Appl., 2019, 25(4), 2109–2140. [Google Scholar] [CrossRef]
Antoine, J.P.; Trapani, C. Operators in rigged Hilbert spaces, Gelfand bases and generalized eigenvalues, Mathematics, 2023, 11, 195. [CrossRef]

1	The domain of an operator should be dense, otherwise its adjoint is not well defined.
2	This means that $D (A) \subset D (A^{†})$ and that for all $ψ \in D (A)$ , $A ψ = A^{†} ψ$ .
3	A topology is finer than another on the same set if it has more open sets.
4	In particular, this implies that the canonical injection, $i (φ) = φ$ , $\forall φ \in Φ$ is continuous.
5	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.
6	A mapping $F : Φ ⟼ C$ is an antilinear functional on $Φ$ if for any pair $ϕ, φ \in Φ$ and any pair of complex numbers $α, β \in C$ , one has that $F (α ϕ + β φ) = α^{} F (ϕ) + β^{} F (φ),$ where the asterisk denotes complex conjugation. In addition, if F were continuous with respect to the topologies on $Φ$ and on $C$ , then, $F \in Φ^{\times}$ . Note that $Φ^{\times}$ has a natural structure as a linear space.
7	This notion of compatibility is rather technical and usually we shall choose on $Φ^{\times}$ its weak topology to be defined later.
8	A seminorm is a mapping $p : Φ ⟼ C$ , such that for all $ϕ \in Φ$ , i.) $p (ϕ) \geq 0$ ; ii.) for all $α \in C$ , $p (α ϕ) = \| α \| p (ϕ)$ ; iii.) for all $ϕ, φ \in Φ$ , $p (ϕ + φ) \leq p (ϕ) + p (φ)$ . Thus, a seminorm is like a norm in which we admit that $p (ϕ) = 0$ with $ϕ \neq 0$ . In particular any norm is a seminorm.
9	We always may choose $Φ$ to be complete under its topology. A complete metrizable locally convex space is a Frèchet space.
10	Linearity is here irrelevant, but we shall use linear mappings only.
11	As any other compatible with duality such as the strong or the McKey topologies.
12	The antilinearity is obvious. To show continuity, let us choose an arbitrary $φ \in Φ$ . Then, $p_{φ} (i (ψ)) = p_{φ} (F_{ψ}) = 〈 F_{ψ} \| φ 〉 = 〈 ψ \| φ 〉 \leq \| \| φ \| \| \| \| ψ \| \| \leq C \| \| ψ \| \|,$ which proves the continuity of $i : H ⟼ Φ^{\times}$ .
13	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 $i : S ⟼ L^{2} (R)$ admits a spectral decomposition similar to those of compact operators [14].
14	These are indeed norms. Recall that norms are also seminorms.
15	Obviously, the space ${\tilde{Φ}}_{l}$ is algebraic and topologically isomorphic to $Φ_{l}$ for $l \geq 0$ .
16	Similarly, we may prove exactly the same result if instead the norms $q_{s, l}$ , we had used the norms $p_{s, l}$ .
17	Since $\sum_{p = 1}^{\infty} \frac{1}{p^{2}} < \infty$ .

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Gelfand Triplets, Ladder Operators and Coherent States

Abstract

1. Introduction and Motivation

1.1. The Schwartz Space

2. The Harmonic Oscillator

3. On Laguerre-Gaussian Ladder Operators

4. Gelfand Triplets and Continuous Operators

4.1. The Ladder Operators

4.2. The Laguerre-Gauss Functions with Negative Value of l

4.3. On the Continuity of Ladder Operators

4.4. The Laguerre-Gauss Modes

5. Coherent States and Continuous Generators for the Algebra s u ( 1 , 1 )

5.1. Resolutions of the Identity

6. Concluding Remarks

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Appendix A. Frames

Appendix B. A Comment on the Topology of the Φl with l > 0

References

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5. Coherent States and Continuous Generators for the Algebra $s u (1, 1)$

Appendix B. A Comment on the Topology of the Φ_l with l > 0