The present article is the continuation of a series of papers intended to show the importance of Gelfand Triplets, also called Rigged Hilbert Spaces (RHS) in the description of a variety of situations in ordinary as well as relativistic Quantum Mechanics. Along the present article, we intend to give some examples of these applications. For the benefit of the reader, we review in this Section some introductory material with appropriate references.

Let us recall that a RHS is a triplet of spaces [1,2]: where: i.) the space $\mathcal{H}$ is an infinite dimensional complex separable Hilbert space. The separability is important as it implies that complete orthonormal sets, also called discrete orthonormal basis, are countably infinite; ii.) the space $\mathrm{\Phi }$ is a dense1 subspace of $\mathcal{H}$. It is endowed with a finer topology (has more open sets) than the Hilbert space topology; iii.) finally, the space ${\mathrm{\Phi }}^{\times }$ is the space of all ant-linear continuous functionals2 on $\mathrm{\Phi }$, endowed with a particular topology compatible with duality (we do not intend to explain this technicality, just give later an example).

Let $\varphi$ be an arbitrary vector in the Hilbert space $\mathcal{H}$. Then, we may define a unique anti-linear continuous functional on $\mathrm{\Phi }$, $F_{\varphi }\in {\mathrm{\Phi }}^{\times }$, just by where $\langle -|-\rangle$ denotes the scalar product on $\mathcal{H}$, where we assume linearity on the right and ant-linearity on the left. Continuity of $F_{\varphi }$ on $\mathrm{\Phi }$, for each $\varphi \in \mathcal{H}$ can be proven. This gives a one to one mapping (although never onto), $\varphi ⟼F_{\varphi }$, between $\mathcal{H}$ and ${\mathrm{\Phi }}^{\times }$. Usually, we identify $F_{\varphi }$ with $\varphi$, which justifies the inclusion $\mathcal{H}\subset {\mathrm{\Phi }}^{\times }$.

RHS have been used in Physics to:

1.- Give a rigorous meaning of the Dirac formulation of Quantum Mechanics [3,4,5,6,7,8,9]. In particular, one gives meaning to the Dirac spectral representation for self-adjoint unbounded operators on Hilbert spaces. This result has a particular interest, which may justify that we give a brief account of it in here. Let A be a self adjoint operator on the Hilbert space $\mathcal{H}$, with domain $\mathcal{D}\left(A\right)$. Let $\mathrm{\Phi }\subset \mathcal{H}\subset {\mathrm{\Phi }}^{\times }$ a RHS such that: i.) $\mathrm{\Phi }\subset \mathcal{D}\left(A\right)$; ii.) $A\mathrm{\Phi }\subset \mathrm{\Phi }$, which means that for any $\varphi \in \mathrm{\Phi }$, it results that $A\varphi \in \mathrm{\Phi }$. iii.) A is continuous on $\mathrm{\Phi }$ (in general, not on $\mathcal{H}$). Then, it may be extended as a linear continuous operator on the anti-dual ${\mathrm{\Phi }}^{\times }$ of $\mathrm{\Phi }$, by means of the duality formula: where we are using the same notation to denote the original operator A and its extension to the anti-dual. Note that continuity is not necessary if we just want to extend the operator with linearity.

We say that $F_{\lambda }\in {\mathrm{\Phi }}^{\times }$ is a generalized eigenvector of A with eigenvalue $\lambda$ if $A\mathrm{\Phi }\subset \mathrm{\Phi }$, so that A can be extended to ${\mathrm{\Phi }}^{\times }$, $A{F}_{\lambda }=\lambda F_{\lambda }$, where A denotes its extension to ${\mathrm{\Phi }}^{\times }$. Equivalently, $\langle A\varphi |F_{\lambda }\rangle =\lambda \langle \varphi |F_{\lambda }\rangle$, $\forall \varphi \in \mathrm{\Phi }$.

Then, the Dirac spectral representation comes after the celebrated Gelfand-Maurin Theorem, which states the following:

iv.) Then, for any pair $\psi ,\phi \in \mathrm{\Phi }$, we have the following spectral representation

If we omit the arbitrary $\psi ,\phi \in \mathrm{\Phi }$, we obtain the following formal decomposition:

Finally, the identity $I:\mathrm{\Phi }⟼{\mathrm{\Phi }}^{\times }$ may be expressed as

2.- Gamow vectors represent the purely exponential decaying part of a quantum resonance. Experiments have shown that, during almost all times of observation, the decay of unstable quantum states is exponential. There are deviations for very short and very large times, which are difficult to detect, particularly the latter ones. However, the Hamiltonian that describe the process is self adjoint and self adjoint operators produces unitary time evolutions that do not decay. The only possibility to describe an exponential time decay is by means of enlarging the Hamiltonian to the dual on a rigged Hilbert space. This has been well studied [10,11,12,13]. Hamiltonians admit spectral decompositions in terms of Gamow states and continuum states, as typically happens with the Friedrichs model [14].

3.- There have been multiple attempts to describe irreversibility in quantum processes. RHS have played an essential role in some of these attempts [15,16,17,18,19,20].

4.- Classical chaotic systems are studied through their power spectrum that often shows singularities called the Pollicot Ruelle resonances. As in the case of Hamiltonians with resonances, Koopman and Frobenius-Perron operators admit respective spectral decompositions in terms of these resonances. A complete formal study of this situation has been done in the RHS language [21,22,23,24].

5.- If on a RHS one replaces the Hilbert space by a Liouville space, which is also a Hilbert space, one gets a structure suitable to study certain problems of Quantum Statistical Mechanics including generalized states and singular structures [25]. However, this option has not been widely used.

6.- Axiomatic Theory of Quantum Fields includes interesting objects that fit into a formulation by RHS, such as Wightman functionals, Borchers algebras, generalized states and others [26,27]. Nevertheless, this line of construction of Quantum fields has been nearly abandoned some time ago.

7.- Distributions are usually functionals on a space ${\mathrm{\Phi }}^{\times }$ of a RHS where the space $\mathrm{\Phi }$ is a space of test functions. Distributions have many applications in Physics either Quantum or Classical. Analogously, differential equations of interest in Physics have often distributions as solutions.

8.- White noise and other stochastic processes may admit a description in terms of RHS [28,29].

9.- Last but not least. One of the ideas that lead to the notion of RHS was the analysis of Lie algebras of operators describing symmetries in Quantum Physics. Up to the knowledge of the authors, specific examples thereof have not been very much developed until recently. See [30] and quotations thereof. Near this context, stability properties of some RHS under the fractional Fourier transform have applied to signal theory [31].

The mentioned recent examples of applications of RHS in symmetries have shown that RHS are the suitable structures that include at the same time, discrete and continuous basis, Lie algebras of symmetries represented by continuous operators and special functions. Each particular example contains a different choice of these ingredients. For pedagogical and methodological reasons, our description is based on two RHS, one abstract and the other a representation of the former by functions. Both RHS are unitarily equivalent in the following sense: Let $\mathrm{\Phi }\subset \mathcal{H}\subset {\mathrm{\Phi }}^{\times }$ be the abstract RHS (the nature of vectors are not specified) and let $\mathcal{G}$ a Hilbert space of functions, say of the type $L^2(\Delta )$. Since the second is also a separable infinite dimensional Hilbert space, there is a unitary mapping connecting both: $U:\mathcal{H}⟼\mathcal{G}$. Then, one may define $\mathrm{\Psi }:=U\mathrm{\Phi }$, the image of $\mathrm{\Phi }$ by U into $\mathcal{G}$. The topology on $\mathrm{\Phi }$ may be transported by U into $\mathrm{\Psi }$. Then, we have a new RHS: $\mathrm{\Psi }\subset \mathcal{G}\subset {\mathrm{\Psi }}^{\times }$. The operator U may be extended to a one to one mapping from ${\mathrm{\Phi }}^{\times }$ onto ${\mathrm{\Psi }}^{\times }$, also conserving the topologies, by means of a duality formula: where $U^{†}$ is the adjoint of U. This extension of U has all good properties. The total construction can be seen in the following diagram:

Obviously, $U^{†}$ makes the inverse job.

Although in general, it studies systems without a continuous spectrum, the SUSY method to construct Hamiltonians with a discrete spectrum similar to the spectrum of a given Hamiltonian implies the use of ladder operators, which are in general unbounded. With the help of a suitable construction of RHS, one can make continuous these ladder operators.

Gelfand triplets are the suitable mathematical framework so as to include some important notions used in standard non-relativistic quantum mechanics for a particular system, such as: Special functions, discrete and continuous basis, generators of symmetry groups given as continuous essentially self adjoint operators and last but not least, the Hamiltonian of this system, also as a continuous operator. This is not possible on the standard formulation of quantum mechanics on Hilbert space. Thus, one may complete the basic formulation of quantum Theory under the basis proposed by Dirac [32].

The first and most celebrated example of this construction is provided by the one dimensional harmonic oscillator, where the main ingredients are:

There exists some other examples, see [30]. In the present contribution, we are illustrating some of the properties that have emerged for the harmonic oscillator and listed above for another solvable model such as the one dimensional Pöschl-Teller potential [34]. Its Hamiltonian has the form (where we have omitted irrelevant constants and $\mathsf{\ell }>0$ is fixed)

Normalized solutions of the Schrödinger equation, $H_{\mathsf{\ell }}{\psi }_n=E_{\mathsf{\ell }}^n{\psi }_{\mathsf{\ell }}^n$ are of the form where $P_{\beta }^{\alpha }\left(z\right)$ are the Legendre functions3.

Functions ${\psi }_{\mathsf{\ell }}^n\left(x\right)$, $n=0,1,2,\cdots$ with fixed $\mathsf{\ell }>0$, form an orthonormal basis (complete orthonormal set) of $L^2[-\pi /2,\pi /2]$. Each function $f\left(x\right)\in L^2[-\pi /2,\pi /2]$ admits a span of the form $f\left(x\right)={\sum }_{n=0}^{\infty }{a}_n{\psi }_{\mathsf{\ell }}^n\left(x\right)$, where the series converges on the $L^2$-norm and this norm is given for the function $f\left(x\right)$ as ${\left|\right|f\left|\right|}^2={\sum }_{n=0}^{\infty }{\left|a_n\right|}^2$.

Then, let us construct the space ${\mathrm{\Phi }}_{\mathsf{\ell }}$ of all functions $f\left(x\right)\in L^2[-\pi /2,\pi /2]$, satisfying the following condition:

Note that $p_{\mathsf{\ell }}^s\left(f\right)$, $\mathsf{\ell }>0$ and $s=0,1,2,\cdots$, is a countably infinite set of norms, hence seminorms, on ${\mathrm{\Phi }}_{\mathsf{\ell }}$, with $p_{\mathsf{\ell }}^0\left(f\right)=\left|\right|f\left|\right|$. By construction, all ${\mathrm{\Phi }}_{\mathsf{\ell }}$, $\mathsf{\ell }>0$, are algebraically and topologically isomorphic to the one dimensional Schwartz space $\mathcal{S}$. For each $\mathsf{\ell }>0$, we have a Gelfand triplet:

Exactly as in the case of the Harmonic oscillator, the Pöschl-Teller Hamiltonian (12) defines ladder operators, which relate the functions in the orthonormal basis. They are constructed as follows: Let

These transformations have a subindex showing on which eigenfunction they act. Thus, we have

Then, we define the action on the eigenfunctions of the basis of the creation $B^{+}$ and the annihilation operator $B^{-}$ as (we omit the subindex ℓ on the operators)

Note that $B^{-}{\psi }_{\mathsf{\ell }}^0=\mathbf{0}$. Operators $B^{\pm }$ are unbounded on $L^2[-\pi /2,\pi /2]$ and, therefore, not continuous and not defined on the whole $L^2[-\pi /2,\pi /2]$. However, $B^{\pm }$ may be extended with continuity to ${\mathrm{\Phi }}_{\mathsf{\ell }}$. Let us define the action of $B^{\pm }$ on $f\left(x\right)\in {\mathrm{\Phi }}_l$, $f\left(x\right)={\sum }_{n=0}^{\infty }{a}_n{\psi }_{\mathsf{\ell }}^n$, as:

We need to show that the action of $B^{\pm }$ as in (19) is well defined and is in ${\mathrm{\Phi }}_{\mathsf{\ell }}$. First of all note that $2\mathsf{\ell }+n=n+1+(2\mathsf{\ell }-1)\le (n+1)+|2\mathsf{\ell }-1|$. Thus, with $C_{\mathsf{\ell }}=\sqrt{1+|2\mathsf{\ell }-1|}$. Thus, $p_{\mathsf{\ell }}^s\left(B^{+}f\right)\le C_{\mathsf{\ell }}{p}_{\mathsf{\ell }}^{s+1}\left(f\right)$ for all $f\left(x\right)\in {\mathrm{\Phi }}_{\mathsf{\ell }}$, which shows that $B^{+}f\in {\mathrm{\Phi }}_{\mathsf{\ell }}$. In addition, this inequality also shows the continuity of $B^{+}$ on ${\mathrm{\Phi }}_{\mathsf{\ell }}$. This comes from the following result [33]:

Let $\mathrm{\Phi }$ a locally convex space for which the topology is given by the family of seminorms ${\left\{p_i\right\}}_{i\in I}$, where I is an index set. Let $A:\mathrm{\Phi }⟼\mathrm{\Phi }$ a linear mapping. A is continuous on $\mathrm{\Phi }$ if and only if for each seminorm $p_i$, there exists a positive constant $C_i$ and $n\left(i\right)$ seminorms (the seminorms and the number $n\left(i\right)$ depends on the seminorm $p_i$) such that

The same result is true if the image space is another locally convex space, $\mathrm{\Psi }$, different from $\mathrm{\Phi }$. Then, the seminorms $p_i$ on the left hand side of the inequality (21) should be replaced by the seminorms defining the topology on $\mathrm{\Psi }$. Same if the image space is the field of complex numbers $\mathbb{C}$ or any normed space. In this case, we have only one seminorm, which is the norm.

Thus, after (21), $B^{+}$ is continuous on ${\mathrm{\Phi }}_{\mathsf{\ell }}$. A similar proof goes for the consistency and continuity of $B^{-}$ on ${\mathrm{\Phi }}_{\mathsf{\ell }}$.

Analogously, we may extend the Hamiltonian $H_{\mathsf{\ell }}$ as in (12) to a continuous operator on ${\mathrm{\Phi }}_{\mathsf{\ell }}$ using the following definition valid for any $f\left(x\right)\in {\mathrm{\Phi }}_{\mathsf{\ell }}$ (recall that $H_{\mathsf{\ell }}{\psi }_{\mathsf{\ell }}^n\left(x\right)={(\mathsf{\ell }+n)}^2{\psi }_{\mathsf{\ell }}^n\left(x\right)$):

The proof goes as in the previous cases. Furthermore, since $H_{\mathsf{\ell }}$ is a symmetric operator and $(H_{\mathsf{\ell }}\pm iI){\psi }_{\mathsf{\ell }}^n=[{(\mathsf{\ell }+n)}^2+i]{\psi }_{\mathsf{\ell }}^n$, so that the image of ${\mathrm{\Phi }}_{\mathsf{\ell }}$ by $(H_{\mathsf{\ell }}\pm iI)$ is dense in $L^2(-\pi /2,\pi /2)$, it results that H is essentially self adjoint on ${\mathrm{\Phi }}_{\mathsf{\ell }}$. Then, $H_{\mathsf{\ell }}$ is self adjoint on its maximal domain and is positive. Therefore, it admits a unique positive self adjoint extension, $B:=\sqrt{H_{\mathsf{\ell }}}$. The operators $B^{\pm },B$ satisfy the commutation relations of the generators of the Lie algebra $so(2,1)$, which are

This the elements of the algebra $so(2,1)$ as well as those of its enveloping algebra are continuously defined on ${\mathrm{\Phi }}_{\mathsf{\ell }}$ and continuously extendable to ${\mathrm{\Phi }}_{\mathsf{\ell }}^{\times }$.

In the standard non-relativistic quantum mechanics, super-symmetry (SUSY) is a procedure that serves to find from a given Hamiltonian with discrete spectrum to another one with a similar or equal spectrum. In some cases, the new Hamiltonian gives rise to a second one, then the second one to a third one and so on, making an infinite sequence of SUSY transformations. If we depart from a Hamiltonian with an infinite number of bound states, we may produce an infinite sequence of Hamiltonians with the same property of having an infinite number of bound states. This is not always the case, although we have examples thereof. Let us explain first how SUSY works under the mentioned circumstances and, then, we construct a Gelfand triplet suitable for the whole scheme.

The point of departure is the factorization method [35,36] that we briefly sketch here. Let H be a Hamiltonian with an infinite number of bound states, which, in addition, it may be factorized by an operator B and its formal adjoint $B^{+}$ as

A typical example is the Hamiltonian of the harmonic oscillator, where B and $B^{+}$ are the annihilation and creation operators, respectively, and $\lambda ={\lambda }^{\prime }=1/2$ (writing the harmonic oscillator Hamiltonian with a factor $1/2$ in front of the derivative). In general, this decomposition is not unique and there exists $A_0$ and its formal adjoint $A_0^{+}$ such that where $\beta \left(x\right)$ fulfils a solvable Riccati equation. However, the Hamiltonian is different from $H_0$. Many examples appear in the literature. We cite a few of them only [35,36,39,40,41,47,48]. The Hamiltonians $H_0$ and $H_1$ satisfy the following intertwining relation:

In addition, if $H_0{\psi }_0^n=E_n{\psi }_0^n$, the sequence of vectors are orthonormal and satisfy the relations $H_1{\psi }_n^1=E_n{\psi }_n^1$, $n=1,2,\cdots$. Under some conditions that we are supposing to be fulfilled here, the process may go to the infinity as we may depict in the following diagram:

The above diagram requires an explanation. Let us start with the vertical lines. The sequence of vectors $\left\{{\psi }_0^n\right\}$, $n=0,1,2,\cdots$, are the normalized eigenvectors of $H_0$. They are an orthonormal basis that span a Hilbert space, ${\mathcal{H}}_0$. $B_{0,n}^{+}$ transform ${\psi }_0^n$ into ${\psi }_0^{n+1}$, etc. The sequence of vectors $\left\{{\psi }_1^n\right\}$, $n=1,2,\cdots$ are the normalized eigenvectors of $H_1$. They are an orthonormal basis that span a Hilbert space, ${\mathcal{H}}_1$. $B_{1,n}^{+}$ transform ${\psi }_1^n$ into ${\psi }_1^{n+1}$, etc, and so on. Thus, we have a sequence of separable infinite dimensional Hilbert spaces $\{{\mathcal{H}}_0,{\mathcal{H}}_1,{\mathcal{H}}_2,\cdots \}$ with their respective discrete basis (complete orthonormal sets).

Horizontal lines. Although the generalization of (41) is ${\psi }_{\mathsf{\ell }}^n=\left(A_{\mathsf{\ell }-1}^{+}{\psi }_{n-1}^{\mathsf{\ell }-1}\right)/\sqrt{E_{n-1}+2\lambda }$, we have omitted the square roots in the diagram for simplicity. Otherwise this diagram would be excessively burdened with notation. We assume that all vectors (indeed eigenfunctions) ${\psi }_{\mathsf{\ell }}^n$ are normalized.

Next, let us consider all Hilbert spaces ${\mathcal{H}}_{\mathsf{\ell }}$, $\mathsf{\ell }=0,1,2,\cdots$ as independent4. This allows for the construction of the orthogonal direct sum ${\oplus }_{\mathsf{\ell }=0}^{\infty }{\mathcal{H}}_{\mathsf{\ell }}$. This sum is well defined as a Hilbert space [33] with orthonormal basis $\left\{{\psi }_{\mathsf{\ell }}^n\right\}$ with $n,\mathsf{\ell }=0,1,2,\cdots$.

The sequence of Hamiltonians $H_{\mathsf{\ell }}$ are defined as $H_{\mathsf{\ell }}:=A_{\mathsf{\ell }-1}^{+}{A}_{\mathsf{\ell }-1}-\lambda$. These Hamiltonians are iso-spectral in the sense that $H_{\mathsf{\ell }}{\psi }_{\mathsf{\ell }}^n=E_n{\psi }_{\mathsf{\ell }}^n$, $n=\mathsf{\ell },\mathsf{\ell }+1,\cdots$.

For any $f_{\mathsf{\ell }}\in {\mathcal{H}}_{\mathsf{\ell }}$, we have the span $f_{\mathsf{\ell }}={\sum }_{n=\mathsf{\ell }}^{\infty }{a}_n^{\mathsf{\ell }}{\psi }_n^{\mathsf{\ell }}$. Let us consider the space ${\mathcal{S}}_{\mathsf{\ell }}$ of all functions $f_{\mathsf{\ell }}$ such that

We have discussed before in the present article the properties of ${\mathcal{S}}_{\mathsf{\ell }}$ endowed with the seminorms $p_{\mathsf{\ell }}^s\left(f_{\mathsf{\ell }}\right)$ as defined in (43). In particular ${\mathcal{S}}_{\mathsf{\ell }}\subset {\mathcal{H}}_{\mathsf{\ell }}\subset {\mathcal{S}}_{\mathsf{\ell }}^{\times }$ is a Gelfand triplet for each $\mathsf{\ell }=0,1,2,\cdots$. Then, define

On ${\mathrm{\Phi }}_{\mathsf{\ell }}$, we define the following set of seminorms: If $f_k\in {\mathcal{S}}_k$, $k=0,1,2\cdots ,\mathsf{\ell }$,

With these set of seminorms, ${\mathrm{\Phi }}_{\mathsf{\ell }}$, $\mathsf{\ell }=0,1,2,\cdots$, is a Frèchet space and the triplets are Gelfand triplets for all $\mathsf{\ell }=0,1,2,\cdots$. Note that

Now, take the identity $I_{\mathsf{\ell }}:{\mathrm{\Phi }}_{\mathsf{\ell }}⟼{\mathrm{\Phi }}_{\mathsf{\ell }+1}$, such that $I_{\mathsf{\ell }}(f_0+f_1+\cdots f_{\mathsf{\ell }})=f_0+f_1+\cdots +f_{\mathsf{\ell }}+\mathbf{0}$, with $f_k\in {\mathrm{\Phi }}_k$. Each of the identities $I_{\mathsf{\ell }}$, $\mathsf{\ell }=0,1,2,\cdots$, are continuous mappings, since

Needless to say that $\mathrm{\Phi }={\cup }_{\mathsf{\ell }=0}^{\infty }{\mathrm{\Phi }}_{\mathsf{\ell }}$ is a linear space. Then, let us endow it with the strict inductive limit topology5 produced on $\mathrm{\Phi }$ by the family ${\left\{{\mathrm{\Phi }}_{\mathsf{\ell }}\right\}}_{\mathsf{\ell }\in \left\{0\right\}\cup \mathbb{N}}$. One usually calls LF the spaces which are strict inductive limits of Frèchet spaces (from “Limit Frèchet”). Thus $\mathrm{\Phi }$ is a Frèchet space.

Now, take $f\left(x\right)\in \mathrm{\Phi }$. After the definition of $\mathrm{\Phi }$, there exists ${\mathrm{\Phi }}_k$ such that $f\left(x\right)\in {\mathrm{\Phi }}_k$. Then,

Recalling diagram (42), let us define the linear operator $A^{+}$ on $\mathrm{\Phi }$ as, for any $f\left(x\right)\in \mathrm{\Phi }$, we have

Obviously, $A^{+}{\mathrm{\Phi }}_{\mathsf{\ell }}⟼{\mathrm{\Phi }}_{\mathsf{\ell }+1}$, $\mathsf{\ell }=0,1,2,\cdots$, and is a linear mapping.

In order to show the topological properties of A, as well as those of the operators included in the diagram (42) and also of Hamiltonians $H_{\mathsf{\ell }}$, $\mathsf{\ell }=0,1,2,\cdots$, we need to specify which model are using. In this presentation, we consider two of them, those for which $H_0$ is: i.) the standard one dimensional Harmonic oscillator and ii.) the standard Pöscl-Teller one dimensional Hamiltonian.