RHS and Quantum Mechanics: Some Extra Examples

Abstract

Rigged Hilbert spaces (RHSs) are the right mathematical context that include many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS, discrete and continuous bases, as well as an abstract basis and the basis of special functions and representations of Lie algebras of symmetries are used by continuous operators. This is not possible in Hilbert spaces. In the present paper, we study a model showing all these features, based on the one-dimensional Pöschl–Teller Hamiltonian. Also, RHS supports representations of all kinds of ladder operators as continuous mappings. We give an interesting example based on one-dimensional Hamiltonians with an infinite chain of SUSY partners, in which the factorization of Hamiltonians by continuous operators on RHS plays a crucial role.

1. Introduction, Materials and Methods

The present article is the continuation of a series of papers intended to show the importance of Gelfand Triplets, also called Rigged Hilbert Spaces (RHSs) in the description of a variety of situations in ordinary as well as relativistic Quantum Mechanics. Throughout 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 an RHS is a triplet of spaces [1,2]:

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

(1)

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 $\Phi$ is a dense (This means that in any neighborhood of any vector $\varphi \in \mathcal{H}$, there always exists a vector $\phi \in \Phi$. The infinite dimensional character of the Hilbert space $\mathcal{H}$ ensures the existence of dense subspaces in $\mathcal{H}$, different from $\mathcal{H}$) subspace of $\mathcal{H}$. It is endowed with a finer topology (has more open sets) than the Hilbert space topology. (iii) Finally, the space ${\Phi }^{\times }$ is the space of all anti-linear continuous functionals (These are mappings $F:\Phi ⟼\mathbb{C}$, $\mathbb{C}$ being the field of complex numbers, continuous with respect to the topologies on $\Phi$ and $\mathbb{C}$. Antilinearity means that if $F\in {\Phi }^{\times }$, $\psi ,\phi \in \Phi$ and $\alpha ,\beta \in \mathbb{C}$ are complex numbers, then $F(\alpha \psi +\beta \phi )={\alpha }^{*}F\left(\psi \right)+{\beta }^{*}F\left(\phi \right)$, where the star denotes complex conjugation) on $\Phi$, endowed with a particular topology compatible with duality (we do not intend to explain this technicality, just give an example later).

In the present paper, we shall use two kinds of topologies on the space $\Phi$. One is provided by a countable sequence of norms, $\left|\right|-{\left|\right|}_n$, $n=0,1,2,\cdots$, one of them being the Hilbert space norm. Then, $\Phi$ is a metrizable space, with metrics given by the distance:

$$d(\varphi ,\phi ):=\sum _{n=0}^{\infty }{\displaystyle \frac{1}{2^n}}{\displaystyle \frac{\left|\right|\varphi -{\phi \left|\right|}_n}{1+\left|\right|\varphi -{\phi \left|\right|}_n}},\forall \varphi ,\phi \in \Phi .$$

(2)

These spaces are often called countably normed spaces [1]. A sequence $\left\{{\psi }_n\right\}$ converges on a countably normed space if and only if it converges with respect to all its norms. Analogously, a sequence is a Cauchy sequence if it is for all norms. If all Cauchy sequences are convergent, then the space is complete. A complete metrizable locally convex space (A locally convex space is a topological vector space such that the origin has a fundamental system of convex neighborhoods. All spaces that appear along the present article are locally convex [3]) is a Frèchet space (Here, norms could be replaced by seminorms. The difference between a norm and a seminorm is that the seminorm of a vector could be zero even if this vector is not zero. Any topology on a locally convex topological vector space is given by a family of seminorms).

The second topology on $\Phi$ to be considered here is a strict inductive limit of metrizable spaces [3], to be described later.

Now, let $\varphi$ be an arbitrary vector in the Hilbert space $\mathcal{H}$. Then, we may define a unique anti-linear continuous functional on $\Phi$, $F_{\varphi }\in {\Phi }^{\times }$, just by

$$F_{\varphi }\left(\phi \right):=\langle \phi |\varphi \rangle ,\forall \phi \in \Phi ,$$

(3)

where $\langle -|-\rangle$ denotes the scalar product on $\mathcal{H}$, where we assume linearity on the right, and anti-linearity on the left. The continuity of $F_{\varphi }$ on $\Phi$ 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 ${\Phi }^{\times }$. Usually, we identify $F_{\varphi }$ with $\varphi$, which justifies the inclusion $\mathcal{H}\subset {\Phi }^{\times }$.

An interesting result is the following. Let A be a self-adjoint operator on the Hilbert space $\mathcal{H}$, with domain $\mathcal{D}\left(A\right)$. Let $\Phi \subset \mathcal{H}\subset {\Phi }^{\times }$ RHS such that (i) $\Phi \subset \mathcal{D}\left(A\right)$; (ii) $A\Phi \subset \Phi$, which means that for any $\varphi \in \Phi$, it results that $A\varphi \in \Phi$; (iii) A is continuous on $\Phi$ (in general, not on $\mathcal{H}$). Then, it may be extended as a linear continuous operator on the anti-dual ${\Phi }^{\times }$ of $\Phi$, by means of the duality formula:

$$\langle A\varphi |F\rangle =\langle \varphi |AF\rangle ,\forall \varphi \in \Phi ,\forall F\in {\Phi }^{\times },$$

(4)

where we are using the same notation to denote the original operator A and its extension to the antidual. Note that continuity is not necessary if we just want to extend the operator with linearity. Also, self-adjointness is not strictly necessary, being that the condition of Hermiticity on $\Phi$, i.e., $\langle \varphi |A\phi \rangle =\langle A\varphi |\phi \rangle$, $\forall \varphi ,\phi \in \Phi$, is sufficient.

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

Next, let us briefly recall some of the most relevant uses of RHS (Gelfand triplets):

• First of all, it gives a rigorous meaning of the Dirac formulation of Quantum Mechanics [4,5,6,7,8,9,10]. In particular, one gives meaning to the Dirac spectral representation for self-adjoint unbounded operators on Hilbert spaces. We may sketch this result as follows. Let A be a self-adjoint operator on an infinite-dimensional separable Hilbert space. Then, there exists an RHS $\Phi \subset \mathcal{H}\subset {\Phi }^{\times }$, such that (i) $A\Phi \subset \Phi$ and A is continuous on $\Phi$. (ii) There exists a measure $d\mu \left(\lambda \right)$ defined on the spectrum, $\sigma \left(A\right)$ of A. For almost all $\lambda \in \sigma \left(A\right)$, with respect to the measure $d\mu \left(\lambda \right)$, there is a $|\lambda \rangle \in \Phi$, such that $A|\lambda \rangle =\lambda |\lambda \rangle$, so that $|\lambda \rangle$ is a generalized eigenvector of A with a generalized eigenvalue on the spectrum of A, which includes the continuous spectrum of A, either absolutely or singular continuous. (iii) For any pair $\psi ,\phi \in \Phi$, we have the following spectral representation ($\langle \lambda |\phi \rangle ={\langle \phi |\lambda \rangle }^{*}$):

  $$\langle \psi |A^n\phi \rangle ={\int }_{\sigma \left(A\right)}{\lambda }^n\langle \psi |\lambda \rangle \langle \lambda |\phi \rangle d\mu \left(\lambda \right),n=0,1,2,\cdots .$$

  (5)
• Other uses have been the following: (i) analysis of quantum unstable states or quantum resonances [11,12,13,14,15]; (ii) study of quantum irreversible processes [16,17,18,19,20,21]; (iii) analysis of some classical chaotic systems [22,23,24,25]; (iv) Quantum Statistical Mechanics [26]; (v) Axiomatic Theory of Quantum Fields [27,28]; (vi) white noise and other stochastic processes [29,30].
• Last but not least, one of the ideas that led to the notion of RHS was the analysis of Lie algebras of operators describing symmetries in Quantum Physics. To the knowledge of the authors, specific examples thereof have not been widely developed until recently. See [31] and quotations thereof. In this context, stability properties of some RHSs under the fractional Fourier transform have been applied to signal theory [32].

The mentioned recent examples of applications of RHS in symmetries have shown that RHSs are suitable structures that include, at the same time, discrete and continuous bases, and Lie algebras of symmetries are 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 RHSs, one abstract and the other a representation of the former by functions. Both RHSs are unitarily equivalent in the following sense. Let $\Phi \subset \mathcal{H}\subset {\Phi }^{\times }$ be the abstract RHS (the nature of vectors is not specified) and let $\mathcal{G}$ be 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 $\Psi :=U\Phi$, the image of $\Phi$ by U into $\mathcal{G}$. The topology on $\Phi$ may be transported by U into $\Psi$. Then, we have a new RHS: $\Psi \subset \mathcal{G}\subset {\Psi }^{\times }$. The operator U may be extended to a one-to-one mapping from ${\Phi }^{\times }$ onto ${\Psi }^{\times }$, also conserving the topologies by means of a duality formula:

$$\langle U^{†}\phi |G\rangle =\langle \phi |UG\rangle ,\forall \phi \in \Phi ,\forall G\in {\Psi }^{\times }$$

(6)

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:

$$\begin{array}{cccccc}\hfill & \Phi \hfill & \subset \hfill & \mathcal{H}& \subset \hfill & {\Phi }^{\times }\hfill \\ \hfill & U\downarrow \hfill & \hfill & U\downarrow & \hfill & U\downarrow \hfill \\ \hfill & \Psi \hfill & \subset \hfill & \mathcal{G}& \subset \hfill & {\Psi }^{\times }\hfill \end{array}.$$

(7)

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 these ladder operators continuous.

2. Special Functions, Lie Algebras and Bases

Gelfand triplets are the suitable mathematical framework to include some important notions used in standard non-relativistic quantum mechanics for a particular system, such as special functions, discrete and continuous bases, 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 a Hilbert space. Thus, one may complete the basic formulation of Quantum Theory under the basis proposed by Dirac [33].

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

• Let $\mathcal{S}$ be the one-dimensional Schwartz space of all functions $f\left(x\right):\mathbb{R}⟼\mathbb{C}$, which are indefinitely differentiable at all points of the real line $\mathbb{R}$ and such that all these functions and their derivatives approach zero at the infinite faster than the inverse of a polynomial of arbitrary degree. With its usual topology [34], $\mathcal{S}$ is a locally convex Frèchet space (complete and metrizable), so that

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

  (8)

  is a Gelfand triplet or RHS.
• The normalized Hermite functions are all in $\mathcal{S}$ and form an orthonormal basis (complete orthonormal set) of $L^2\left(\mathbb{R}\right)$. Thus, there is a discrete basis of Special Functions that span both $\mathcal{S}$ and $L^2\left(\mathbb{R}\right)$.
• The Lie algebra associated with the Harmonic oscillator is the Heisenberg–Weyl Lie algebra. In the one-dimensional case under consideration, the generators of this algebra are the identity operator I, the multiplication operator $Qf\left(x\right)=xf\left(x\right)$, and the derivation operator $Pf\left(x\right)=-i{f}^{\prime }\left(x\right)$, where the prime denotes derivation with respect to the variable x. Both Q and P are self-adjoint operators on suitable domains in $L^2\left(\mathbb{R}\right)$ and both are essentially self-adjoint on $\mathcal{S}$. Furthermore, both Q and P are continuous on $\mathcal{S}$ with the topology of the latter. Therefore, all the elements of the covering algebra spanned by I, Q and P are continuous on $\mathcal{S}$. This includes the Hamiltonian of the harmonic oscillator. In addition, all elements of the envelope algebra may be extended to continuous operators on ${\mathcal{S}}^{\times }$, when we endow the latter with any topology compatible with the dual pair $\{\mathcal{S},{\mathcal{S}}^{\times }\}$. It is important to remark that creation and annihilation operators are also continuous on $\mathcal{S}$ and ${\mathcal{S}}^{\times }$.
• The operator Q and P along the RHS satisfy the Gelfand–Maurin Theorem [1], so that each one has generalized eigenvectors that satisfy Equation (5). In particular, for Q, we have that for $\psi \left(x\right),\phi \left(x\right)\in \mathcal{S}$ and $n=0,1,2,\cdots$,

  $$\langle \psi |Q^n\phi \rangle ={\int }_{-\infty }^{\infty }{x}^n\langle \psi |x\rangle \langle x|\phi \rangle dx,$$

  (9)

  with $Q|x\rangle =x|x\rangle \in {\mathcal{S}}^{\times }$, $x\in \mathbb{R}$. If we take $n=0$ and omit the arbitrary $\psi \in \mathcal{S}$, we obtain the following formal expression for each $\phi \in \mathcal{S}$:

  $$\phi ={\int }_{-\infty }^{\infty }|x\rangle \langle x|\phi \rangle dx.$$

  (10)
  This is a span of $\phi \in \mathcal{S}$ in terms of the eigenfunctionals ${\left\{\right|x\rangle \}}_{x\in \mathbb{R}}$. Due to this span, we call a continuous basis the set of functionals ${\left\{\right|x\rangle \}}_{x\in \mathbb{R}}$.
  In addition, there are some relations between discrete and continuous bases [32].

There exist some other examples; see [31,35]. In the present contribution, we illustrate some of the properties that have emerged for the harmonic oscillator and list above for another solvable model such as the one-dimensional Pöschl–Teller potential [36]. Its Hamiltonian has the following form (where we have omitted irrelevant constants and $\mathcal{l}>0$ is fixed):

$$H_{\mathcal{l}}:=-{\displaystyle \frac{d^2}{d{x}^2}}+{\displaystyle \frac{\mathcal{l}(\mathcal{l}-1)}{{cos}^{2}x}},-{\displaystyle \frac{\pi }{2}}<x<{\displaystyle \frac{\pi }{2}},\mathcal{l}>0.$$

(11)

Normalized solutions of the Schrödinger equation, $H_{\mathcal{l}}{\psi }_n=E_{\mathcal{l}}^n{\psi }_{\mathcal{l}}^n$, are of the form

$${\psi }_{\mathcal{l}}^n\left(x\right)=\sqrt{{\displaystyle \frac{(\mathcal{l}+n)\Gamma (2\mathcal{l}+n)}{n!}}}\sqrt{cosx}{P}_{\mathcal{l}+n-1/2}^{1/2-\mathcal{l}}\left(sinx\right),E_{\mathcal{l}}^n={(\mathcal{l}+n)}^2,$$

(12)

where $P_{\beta }^{\alpha }\left(z\right)$ are the Legendre functions (These functions have the form:

$$P_{\beta }^{\alpha }\left(z\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\left[{\displaystyle \frac{1+z}{1-z}}\right]}^{\alpha /2}{}_{2}F_1\left(-\beta ,\beta +1,1-\alpha ,{\displaystyle \frac{1-z}{2}}\right),$$

where ${}_{2}F_1$ is the hypergeometric function).

Functions ${\psi }_{\mathcal{l}}^n\left(x\right)$, $n=0,1,2,\cdots$, with fixed $\mathcal{l}>0$, 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 $f\left(x\right)={\sum }_{n=0}^{\infty }{a}_n{\psi }_{\mathcal{l}}^n\left(x\right)$, 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 ${\Phi }_{\mathcal{l}}$ of all functions $f\left(x\right)\in L^2[-\pi /2,\pi /2]$, satisfying the following condition:

$${\left[p_{\mathcal{l}}^s\left(f\right)\right]}^2=\sum _{n=0}^{\infty }{\left|a_n\right|}^2{(n+1)}^{2s}<\infty .$$

(13)

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

$${\Phi }_{\mathcal{l}}\subset L^2[-\pi /2,\pi /2]\subset {\Phi }_{\mathcal{l}}^{\times }.$$

(14)

Exactly as in the case of the Harmonic oscillator, the Pöschl–Teller Hamiltonian (11) defines ladder operators, which relate the functions in the orthonormal basis. They are constructed as follows. Let

$$B_n^{+}:={\left({\displaystyle \frac{E_{\mathcal{l}}^n}{E_{\mathcal{l}}^{n-1}}}\right)}^{1/4}(-cosx{\partial }_x+\sqrt{E_{\mathcal{l}}^{n-1}}\sin x);B_n^{-}:={\left({\displaystyle \frac{E_{\mathcal{l}}^{n-1}}{E_{\mathcal{l}}^n}}\right)}^{1/4}(cosx{\partial }_x+\sqrt{E_{\mathcal{l}}^n}\sin x).$$

(15)

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

$$B_n^{-}{\psi }_{\mathcal{l}}^n=\sqrt{n(2\mathcal{l}+n-1)}{\psi }_{\mathcal{l}}^{n-1},B_n^{+}{\psi }_{\mathcal{l}}^{n-1}=\sqrt{n(2\mathcal{l}+n-1)}{\psi }_{\mathcal{l}}^n.$$

(16)

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)

$$B^{-}{\psi }_{\mathcal{l}}^n=B_n^{-}{\psi }_{\mathcal{l}}^n,B^{+}{\psi }_{\mathcal{l}}^n=B_n^{+}{\psi }_{\mathcal{l}}^n,n=0,1,2,\cdots .$$

(17)

Note that $B^{-}{\psi }_{\mathcal{l}}^0=\mathbf{0}$. 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 ${\Phi }_{\mathcal{l}}$. Let us define the action of $B^{\pm }$ on $f\left(x\right)\in {\Phi }_l$, $f\left(x\right)={\sum }_{n=0}^{\infty }{a}_n{\psi }_{\mathcal{l}}^n$, as:

$$B^{+}f\left(x\right):=\sum _{n=0}^{\infty }{a}_n\sqrt{(n+1)(2\mathcal{l}+n)}{\psi }_{\mathcal{l}}^{n+1},B^{-}f\left(x\right):=\sum _{n=0}^{\infty }{a}_n\sqrt{n(2\mathcal{l}+n-1)}{\psi }_{\mathcal{l}}^{n-1}.$$

(18)

We need to show that the action of $B^{\pm }$ as in (18) is well defined and is in ${\Phi }_{\mathcal{l}}$. First of all, note that $2\mathcal{l}+n=n+1+(2\mathcal{l}-1)\le (n+1)+|2\mathcal{l}-1|$. Thus,

$$\begin{array}{c}\hfill {\left[p_{\mathcal{l}}^s\left(B^{+}f\right)\right]}^2=\sum _{n=0}^{\infty }{\left|a_n\right|}^2(n+1)(2\mathcal{l}+n){(n+1)}^{2s}\\ \hfill \le \sum _{n=0}^{\infty }|a_n{|}^2{(n+1)}^2{(n+1)}^{2s}+|2\mathcal{l}-1|\sum _{n=0}^{\infty }{\left|a_n\right|}^2(n+1){(n+1)}^{2s}\\ \hfill \le (1+|2\mathcal{l}-1\left|\right)\sum _{n=0}^{\infty }{\left|a_n\right|}^2{(n+1)}^{2(s+1)}=C_{\mathcal{l}}^2{\left[p_{\mathcal{l}}^{s+1}\left(f\right)\right]}^2,\forall f\left(x\right)\in {\Phi }_{\mathcal{l}},\end{array}$$

(19)

with $C_{\mathcal{l}}=\sqrt{1+|2\mathcal{l}-1|}$. Thus, $p_{\mathcal{l}}^s\left(B^{+}f\right)\le C_{\mathcal{l}}{p}_{\mathcal{l}}^{s+1}\left(f\right)$ for all $f\left(x\right)\in {\Phi }_{\mathcal{l}}$, which shows that $B^{+}f\in {\Phi }_{\mathcal{l}}$. In addition, this inequality also shows the continuity of $B^{+}$ on ${\Phi }_{\mathcal{l}}$. This comes from the following result [34]:

Let $\Phi$ be 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 $A:\Phi ⟼\Phi$ be a linear mapping. A is continuous on $\Phi$ 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 $n\left(i\right)$ depend on the seminorm $p_i$) such that

$$p_i\left(Af\right)\le C_i\{p_{1\left(i\right)}\left(f\right)+p_{2\left(i\right)}\left(f\right)+\cdots +p_{n\left(i\right)}\left(f\right)\},\forall f\in \Phi .$$

(20)

The same result is true if the image space is another locally convex space, $\Psi$, different from $\Phi$. Then, the seminorms $p_i$ on the left-hand side of the inequality (20) should be replaced by the seminorms defining the topology on $\Psi$. The same is true 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 (20), $B^{+}$ is continuous on ${\Phi }_{\mathcal{l}}$. A similar proof goes for the consistency and continuity of $B^{-}$ on ${\Phi }_{\mathcal{l}}$.

Analogously, we may extend the Hamiltonian $H_{\mathcal{l}}$ as in (11) to a continuous operator on ${\Phi }_{\mathcal{l}}$ using the following definition, valid for any $f\left(x\right)\in {\Phi }_{\mathcal{l}}$ (recall that $H_{\mathcal{l}}{\psi }_{\mathcal{l}}^n\left(x\right)={(\mathcal{l}+n)}^2{\psi }_{\mathcal{l}}^n\left(x\right)$):

$$H_{\mathcal{l}}f\left(x\right):=\sum _{n=0}^{\infty }{a}_n{(\mathcal{l}+n)}^2{\psi }_{\mathcal{l}}^n\left(x\right).$$

(21)

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

$$[B,B^{\pm }]=\pm B^{\pm },[B^{-},B^{+}]=2B.$$

(22)

This, the element of the algebra $so(2,1)$, as well as those of its enveloping algebra, is continuously defined on ${\Phi }_{\mathcal{l}}$ and continuously extendable to ${\Phi }_{\mathcal{l}}^{\times }$.

Some Further Properties: Continuous Basis

Here, we investigate the existence of a continuous basis for the model under discussion. For technical reasons, we are restricting ourselves to half-integer values of ℓ, and for simplicity, let us assume that $\mathcal{l}=1/2$. Then, the Legendre functions that appear in the right-hand side of (12) are just the ordinary Legendre polynomials that admit the following upper bound:

$$|P_n\left(x\right)|\le \sqrt{{\displaystyle \frac{2}{\pi }}}.$$

(23)

We know that the convergence of the series in the span $f\left(x\right)={\sum }_{n=0}^{\infty }{a}_n{\psi }_{1/2}^n\left(x\right)$, $f\left(x\right)\in L^2[-\pi /2,\pi /2]$, makes sense in the norm topology. This convergence does not imply almost everywhere pointwise convergence. However, if $f\left(x\right)\in {\Phi }_{1/2}$, this convergence proceeds in the uniform and the absolute sense, and hence pointwise. The proof proceeds as follows. Take the series:

$$\begin{array}{c}\hfill \left|f\left(x\right)\right|=\left|\sum _{n=0}^{\infty }{a}_n{\psi }_{1/2}^n\left(x\right)\right|\le \sum _{n+0}^{\infty }|a_n||{\psi }_{1/2}^n\left(x\right)|\le \sqrt{{\displaystyle \frac{2}{\pi }}}\sum _{n=0}^{\infty }\left|a_n\right|\sqrt{n+1/2}\\ \hfill \le \sqrt{{\displaystyle \frac{2}{\pi }}}\sum _{n=0}^{\infty }\left|a_n\right|\sqrt{n+1}{\displaystyle \frac{{(n+1)}^2}{{(n+1)}^2}}\le \sqrt{{\displaystyle \frac{2}{\pi }}}\sqrt{\sum _{n=0}^{\infty }{\left|a_n\right|}^2{(n+1)}^4}\sqrt{\sum _{n=0}^{\infty }{\displaystyle \frac{1}{{(n+1)}^3}}}=C{p}_{1/2}^2\left(f\right),\end{array}$$

(24)

where the meaning of C is obvious and the seminorm $p_{1/2}^2\left(f\right)$ has been defined in (14). The last inequality is the Cauchy–Schwarz inequality. The uniform and absolute convergence of the series ${\sum }_{n=0}^{\infty }{a}_n{\psi }_{1/2}^n\left(x\right)$ is then a consequence of the Weierstrass M-Theorem.

Next, and as already mentioned in the introduction, let $\mathcal{H}$ be an abstract infinite-dimensional separable Hilbert space and $U:L^2[-\pi /2,\pi /2]⟼\mathcal{H}$ be unitary. As outlined in the introduction, there exists an abstract Gelfand triple, $\Psi \subset \mathcal{H}\subset {\Phi }^{\times }$, such that

$$\begin{array}{cccccc}\hfill & {\Phi }_{1/2}\hfill & \subset \hfill & L^2[-\pi /2,\pi /2]& \subset \hfill & {\Phi }_{1/2}^{\times }\hfill \\ \hfill & U\downarrow \hfill & \hfill & U\downarrow & \hfill & U\downarrow \hfill \\ \hfill & \Psi \hfill & \subset \hfill & \mathcal{H}& \subset \hfill & {\Psi }^{\times }\hfill \end{array}.$$

(25)

Take $f\in \Psi$ and define a functional $|x\rangle$ such that $\langle f|x\rangle :=f^{*}\left(x\right)=U^{-1}f$. This functional is obviously antilinear. Its continuity follows from (24), since

$$\left|\langle f|x\rangle \right|=|f^{*}\left(x\right)|\le C{p}_{1/2}^2\left(f\right),\forall f\in \Psi ,$$

(26)

and the comment on the paragraph follows from (21). Note that U and its inverse $U^{-1}$ are bijective and bicontinuous among the spaces, as marked in (25). In the following, we take the complex conjugate $\langle x|f\rangle :={\langle f|x\rangle }^{*}=f\left(x\right)$. Now, take two arbitrary vectors $f,g\in \Psi$ and consider $f\left(x\right)=U^{-1}f$ and $g\left(x\right)=U^{-1}g$, which are both in ${\Phi }_{1/2}$. Since a unitary operator preserves the scalar product, we have that

$$\langle g|f\rangle ={\int }_{-\pi /2}^{\pi /2}{g}^{*}\left(x\right)f\left(x\right)dx={\int }_{-\pi /2}^{\pi /2}\langle g|x\rangle \langle x|f\rangle dx.$$

(27)

Then, omitting the arbitrary $g\in \Psi$, we have for all $f\in \Psi$ the following formal expression:

$$f={\int }_{-\pi /2}^{\pi /2}|x\rangle \langle x|f\rangle dx={\int }_{-\pi /2}^{\pi /2}f\left(x\right)|x\rangle dx,$$

(28)

so that every $f\in \Psi$ can be formally written in terms of the continuous functionals ${\left\{\right|x\rangle \}}_{x\in [-\pi /2,\pi /2]}$. This construction preserves linearity on $\Psi$. Hence, the set of functionals is often called a continuous basis on $\Psi$.

Next, define $|n\rangle :=U\left[{\psi }_{1/2}^n\left(x\right)\right]$, $n=0,1,2,\cdots$. Obviously, ${\left\{\right|n\rangle \}}_{n\in \left\{0\right\}\cup \mathbb{N}}$ is an orthonormal basis in $\mathcal{H}$, which is also in $\Psi$. As mentioned before, an orthonormal basis is often denoted as a discrete basis. Let us see that there exists a formal relation between discrete and continuous bases. We know that ${\sum }_{n=0}^{\infty }|n\rangle \langle n|=I$, where I is the identity on $\mathcal{H}$, and this series converges in the strong operator sense.

Then, if $f\in \Psi$, we have formally that $\langle f|x\rangle ={\sum }_{n=0}^{\infty }\langle f|n\rangle \langle n|x\rangle$, so that, omitting the arbitrary $f\in \Psi$, we have the following formal relation (note that $\langle n|x\rangle ={\psi }_{1/2}^n\left(x\right)$ after the above definition and taking into account that the functions ${\psi }_{1/2}^n\left(x\right)$ are real):

$$|x\rangle =\sum _{n=0}^{\infty }|n\rangle \langle n|x\rangle =\sum _{n=0}^{\infty }{\psi }_{1/2}^n\left(x\right)|n\rangle ,$$

(29)

which gives the continuous basis in terms of the discrete basis. Then, from (28), we have

$$|n\rangle ={\int }_{-\pi /2}^{\pi /2}{\psi }_{1/2}^n\left(x\right)|x\rangle dx,n=0,1,2,\cdots ,$$

(30)

which is the inversion formula of (29).

In the case of the harmonic oscillator, the generalized basis ${\left\{\right|x\rangle \}}_{x\in [-\pi /2,\pi /2]}$ is given by a complete set of eigenfunctionals of the multiplication operator $Qf\left(x\right)=xf\left(x\right)$. In the present case, we cannot guarantee that $xf\left(x\right)\in {\Phi }_{1/2}$ if $f\left(x\right)\in {\Phi }_{1/2}$. Nevertheless, there is a way out. Taking into account that

$$\sin x{P}_n\left(\sin x\right)={\displaystyle \frac{n+1}{2n+1}}{P}_{n+1}\left(\sin x\right)+{\displaystyle \frac{n}{2n+1}}{P}_{n-1}\left(\sin x\right),$$

(31)

we have that the same relation is fulfilled by the eigenfunctions ${\psi }_{1/2}^n\left(x\right)$. Then, define the multiplication operator $\sin Q$ as $\left(\sin Q\right)f\left(x\right)=\sin xf\left(x\right)$. The operator $\sin Q$ is obviously bounded on $L^2[-\pi /2,\pi /2]$. It is also well defined and continuous on ${\Phi }_{1/2}$. Its action on any function $f\left(x\right)\in {\Phi }_{1/2}$ is given by

$$\begin{array}{c}\hfill \left(\sin Q\right)f\left(x\right)=\sum _{n=0}^{\infty }{a}_n\sin x{\psi }_{1/2}^n\left(x\right)\\ \hfill =\sum _{n=0}^{\infty }{a}_n{\displaystyle \frac{n+1}{2n+1}}\sin x{\psi }_{1/2}^{n+1}\left(x\right)+\sum _{n=0}^{\infty }{a}_n{\displaystyle \frac{n}{2n+1}}\sin x{\psi }_{1/2}^{n-1}\left(x\right).\end{array}$$

(32)

Then, the stability of ${\Phi }_{1/2}$ by $\sin Q$ as well as the continuity of the latter is a simple exercise using (32).

Now, define $T:=U^{-1}\left[\sin Q\right]U$ on $\Psi$ and take $f,g\in \Psi$. We have

$$\langle g|Tf\rangle ={\int }_{-\pi /2}^{\pi /2}{g}^{*}\left(x\right)\left(\sin x\right)f\left(x\right)dx={\int }_{-\pi /2}^{\pi /2}\sin x\langle g|x\rangle \langle x|f\rangle dx,$$

(33)

so that omitting the arbitrary $g,f\in {\Phi }_{1/2}$, we have

$$T={\int }_{-\pi /2}^{\pi /2}\sin x|x\rangle \langle x|dx,$$

(34)

which may be looked at as a sort of spectral decomposition of $\sin Q$.

As a final property on the trigonometric Pöschl–Teller, we would like to remark that the Pöschl–Teller coherent states are given by ($z\in \mathbb{C}$ fixed)

$${\psi }_z(x,t)=e^{-{\left|z\right|}^2}\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{\sqrt{n!}}}{e}^{-i{(\mathcal{l}+n)}^{2}t}{\psi }_{\mathcal{l}}^n\left(x\right)\in {\Phi }_{\mathcal{l}},$$

(35)

which is trivial.

3. Gelfand Triplets Associated with SUSY

In the standard non-relativistic quantum mechanics, super-symmetry (SUSY) is a procedure that serves to find from a given Hamiltonian with a discrete spectrum 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 [37,38] that we briefly sketch here. Let H be a Hamiltonian with an infinite number of bound states, which, in addition, may be factorized by an operator B and its formal adjoint $B^{+}$ as

$$H_0=-{\displaystyle \frac{d^2}{d{x}^2}}+V_0\left(x\right)=B{B}^{+}-\lambda =B^{+}B+{\lambda }^{\prime }.$$

(36)

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

$$H_0=A_0{A}_0^{+}-\lambda ,A_0:={\displaystyle \frac{d}{dx}}+\beta \left(x\right),A_0^{+}=-{\displaystyle \frac{d}{dx}}+\beta \left(x\right),$$

(37)

where $\beta \left(x\right)$ fulfills a solvable Riccati equation. However, the Hamiltonian

$$H_1:=A_0^{+}{A}_0-\lambda =-{\displaystyle \frac{d^2}{d{x}^2}}+V_1\left(x\right),$$

(38)

is different from $H_0$. Many examples appear in the literature. We cite a few of them only [37,38,39,40,41,42,43,44,45]. The Hamiltonians $H_0$ and $H_1$ satisfy the following intertwining relation:

$$H_1{A}_0^{+}=A_0^{+}{H}_0.$$

(39)

In addition, if $H_0{\psi }_0^n=E_n{\psi }_0^n$, the sequence of vectors

$${\psi }_n^1:={\displaystyle \frac{A_0^{+}{\psi }_{n-1}^0}{\sqrt{E_{n-1}+2\lambda }}},n=1,2,3,\cdots$$

(40)

is orthonormal and satisfies 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 tend to infinity, as we may depict in the following diagram:

$$\begin{array}{ccccccccc}\hfill & {\psi }_0^0\hfill & \stackrel{A_0^{+}}{\to }\hfill & {\psi }_1^1& \stackrel{A_1^{+}}{\to }\hfill & {\psi }_2^2\hfill & \stackrel{A_2^{+}}{\to }\hfill & {\psi }_3^3\hfill & \cdots \hfill \\ \hfill & B_{0,0}^{+}\downarrow \hfill & \hfill & B_{1,1}^{+}\downarrow & \hfill & B_{2,2}^{+}\downarrow \hfill & \hfill & B_{3,3}^{+}\downarrow \hfill & \cdots \hfill \\ \hfill & {\psi }_1^0\hfill & \stackrel{A_0^{+}}{\to }\hfill & {\psi }_2^1& \stackrel{A_1^{+}}{\to }\hfill & {\psi }_3^2\hfill & \stackrel{A_2^{+}}{\to }\hfill & {\psi }_4^3\hfill & \cdots \hfill \\ \hfill & B_{0,1}^{+}\downarrow \hfill & \hfill & B_{1,2}^{+}\downarrow & \hfill & B_{2,3}^{+}\downarrow \hfill & \hfill & A_{3.4}^{+}\downarrow \hfill & \cdots \hfill \\ \hfill & {\psi }_2^0\hfill & \stackrel{A_0^{+}}{\to }\hfill & {\psi }_3^1& \stackrel{A_1^{+}}{\to }\hfill & {\psi }_4^2\hfill & \stackrel{A_2^{+}}{\to }\hfill & {\psi }_5^3\hfill & \cdots \hfill \\ \hfill & \cdots \hfill & \cdots \hfill & \cdots & \cdots \hfill & \cdots \hfill & \cdots \hfill & \hfill \end{array}.$$

(41)

The above diagram requires an explanation. Let us start with the vertical lines. The sequence of vectors $\left\{{\psi }_n^0\right\}$, $n=0,1,2,\cdots$ represents the normalized eigenvectors of $H_0$. They are an orthonormal basis that spans a Hilbert space, ${\mathcal{H}}_0$. $B_{0,n}^{+}$ transforms ${\psi }_n^0$ into ${\psi }_0^{n+1}$, etc. The sequence of vectors $\left\{{\psi }_n^1\right\}$, $n=1,2,\cdots$ represents the normalized eigenvectors of $H_1$. They are an orthonormal basis that spans a Hilbert space, ${\mathcal{H}}_1$. $B_{1,n}^{+}$ transforms ${\psi }_n^1$ into ${\psi }_{n+1}^1$, etc. 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 (40) is ${\psi }_n^{\mathcal{l}}=\left(A_{\mathcal{l}-1}^{+}{\psi }_{n-1}^{\mathcal{l}-1}\right)/\sqrt{E_{n-1}+2\lambda }$, 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) ${\psi }_n^{\mathcal{l}}$ are normalized.

Next, let us consider all Hilbert spaces ${\mathcal{H}}_{\mathcal{l}}$, $\mathcal{l}=0,1,2,\cdots$ as independent (We make this Ansatz even being aware that the operators $A_n^{+}$ are transformations between functions and that the spaces ${\mathcal{H}}_{\mathcal{l}}$ could even be identical or one a subspace of the other. We shall further comment on this point). This allows for the construction of the orthogonal direct sum ${\oplus }_{\mathcal{l}=0}^{\infty }{\mathcal{H}}_{\mathcal{l}}$. This sum is well defined as a Hilbert space [34] with an orthonormal basis $\left\{{\psi }_n^{\mathcal{l}}\right\}$ with $n,\mathcal{l}=0,1,2,\cdots$.

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

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

$${\left[p_{\mathcal{l}}^s\left(f_{\mathcal{l}}\right)\right]}^2:=\sum _{n=\mathcal{l}}^{\infty }{\left|a_n^{\mathcal{l}}\right|}^2{(n+1)}^{2s}<\infty ,s=0,1,2\cdots .$$

(42)

We have discussed before in the present article the properties of ${\mathcal{S}}_{\mathcal{l}}$ endowed with the seminorms $p_{\mathcal{l}}^s\left(f_{\mathcal{l}}\right)$ as defined in (42). In particular, ${\mathcal{S}}_{\mathcal{l}}\subset {\mathcal{H}}_{\mathcal{l}}\subset {\mathcal{S}}_{\mathcal{l}}^{\times }$ is a Gelfand triplet for each $\mathcal{l}=0,1,2,\cdots$. Then, define

$${\Phi }_{\mathcal{l}}:=\underset{k=0}{\overset{\mathcal{l}}{⨁}}{\mathcal{S}}_k.$$

(43)

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

$${\left[p_{\mathcal{l}}^{s_0,s_1,\cdots ,s_{\mathcal{l}}}(f_0+f_1+\cdots +f_{\mathcal{l}})\right]}^2:={\left[p_0^{s_0}\left(f_0\right)\right]}^2+{\left[p_1^{s_1}\left(f_1\right)\right]}^2+\cdots +{\left[p_{\mathcal{l}}^{s_{\mathcal{l}}}\left(f_{\mathcal{l}}\right)\right]}^2.$$

(44)

With this set of seminorms, ${\Phi }_{\mathcal{l}}$, $\mathcal{l}=0,1,2,\cdots$, is a Frèchet space and the triplets

$${\Phi }_{\mathcal{l}}\subset \underset{k=0}{\overset{\mathcal{l}}{⨁}}{\mathcal{H}}_k\subset {\Phi }_{\mathcal{l}}^{\times },$$

(45)

are Gelfand triplets for all $\mathcal{l}=0,1,2,\cdots$. Note that

$${\Phi }_0\subset {\Phi }_1\subset {\Phi }_2\subset \cdots \subset {\Phi }_{\mathcal{l}}\subset \cdots \subset \bigcup _{\mathcal{l}=0}^{\infty }{\Phi }_{\mathcal{l}}=:\Phi .$$

(46)

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

$${\left[p_{\mathcal{l}+1}^{s_0,s_1,\cdots ,s_{\mathcal{l}+1}}\left[I_{\mathcal{l}}(f_0+f_1+\cdots +f_{\mathcal{l}})\right]\right]}^2={\left[p_{\mathcal{l}}^{s_0,s_1,\cdots ,s_{\mathcal{l}}}(f_0+f_1+\cdots +f_{\mathcal{l}})\right]}^2.$$

(47)

Needless to say, $\Phi ={\cup }_{\mathcal{l}=0}^{\infty }{\Phi }_{\mathcal{l}}$ is a linear space. Then, let us endow it with the strict inductive limit topology (This topology is the finest topology that makes all the identity mappings $J_{\mathcal{l}}:{\Phi }_{\mathcal{l}}⟼\Phi$, i.e., the so-called final topology in the language of Bourbaki [46]) produced on $\Phi$ by the family ${\left\{{\Phi }_{\mathcal{l}}\right\}}_{\mathcal{l}\in \left\{0\right\}\cup \mathbb{N}}$. One usually calls LF the spaces that are strict inductive limits of Frèchet spaces (from “Limit Frèchet”). Thus, $\Phi$ is an LF space.

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

$$f\left(x\right)=f_0+f_1+\cdots +f_k=\sum _{n=0}^{\infty }{a}_n^0{\psi }_n^0+\sum _{n=1}^{\infty }{a}_n^1{\psi }_n^1+\cdots +\sum _{n=k}^{\infty }{a}_n^k{\psi }_n^k.$$

(48)

Recalling diagram (41), let us define the linear operator $A^{+}$ on $\Phi$ as, for any $f\left(x\right)\in \Phi$, we have

$$\begin{array}{c}\hfill A^{+}f\left(x\right)=A_0^{+}{f}_0+A_1^{+}{f}_1+\cdots +A_k^{+}{f}_k=\sum _{n=1}^{\infty }{a}_{n-1}^0\sqrt{E_{n-1}+2\lambda }{\psi }_n^1\\ \hfill +\sum _{n=2}^{\infty }{a}_{n-1}^1\sqrt{E_{n-1}+2\lambda }{\psi }_n^2+\cdots +\sum _{n=k+1}^{\infty }{a}_{n-1}^k\sqrt{E_{n-1}+2\lambda }{\psi }_n^{k+1}.\end{array}$$

(49)

Obviously, $A^{+}{\Phi }_{\mathcal{l}}⟼{\Phi }_{\mathcal{l}+1}$, $\mathcal{l}=0,1,2,\cdots$, and it is a linear mapping.

In order to show the topological properties of A, as well as those of the operators included in the diagram (41) and also of Hamiltonians $H_{\mathcal{l}}$, $\mathcal{l}=0,1,2,\cdots$, we need to specify which model we 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 Pöschl–Teller one-dimensional Hamiltonian.

3.1. $H_0$ Is the One-Dimensional Harmonic Oscillator

SUSY partners of the one-dimensional harmonic oscillator have been extensively studied by several authors. Let us cite here [36,39,40,47,48,49]. In this case, we have the following equations ($n,\mathcal{l}=0,1,2,\cdots$):

$$\begin{array}{c}\hfill H_{\mathcal{l}}{\psi }_n^{\mathcal{l}}=\left(n+{\displaystyle \frac{1}{2}}\right){\psi }_n^{\mathcal{l}},{\psi }_n^{\mathcal{l}}={\displaystyle \frac{A_0^{+}{\psi }_{n-1}^{\mathcal{l}-1}}{\sqrt{n}}},A_{\mathcal{l}-1}{\psi }_n^{\mathcal{l}}=\sqrt{n}{\psi }_{n-1}^{\mathcal{l}-1}.\end{array}$$

(50)

Analogous to $A^{+}$, we may define A on each of the ${\Phi }_{\mathcal{l}}$ as ($f_i\in {\mathcal{S}}_i$)

$$\begin{array}{c}\hfill A(f_1+f_2+\cdots +f_{\mathcal{l}}):=A_1{f}_1+A_2{f}_2+\cdots +A_{\mathcal{l}}{f}_{\mathcal{l}}\\ \hfill =\sum _{n=1}^{\infty }{a}_n^1\sqrt{n}{\psi }_{n-1}^0+\sum _{n=1}^{\infty }{a}_n^2\sqrt{n}{\psi }_{n-1}^1+\cdots +\sum _{n=1}^{\infty }{a}_n^{\mathcal{l}}\sqrt{n}{\psi }_{n-1}^{\mathcal{l}-1}.\end{array}$$

(51)

Thus, $A:{\Phi }_{\mathcal{l}}⟼{\Phi }_{\mathcal{l}}$, for all $\mathcal{l}=1,2,\cdots$, is a linear mapping. As for the case of $A^{+}$, A can be extended to a linear mapping on $\Phi$.

Theorem 1. 

The mappings $A^{+}$ and A are continuous linear mappings on Φ.

Proof. 

Obviously, they are linear. Consider $A^{+}$. Let us show that it is continuous as a mapping $A^{+}:{\Phi }_{\mathcal{l}}⟼{\Phi }_{\mathcal{l}+1}$ for all $\mathcal{l}=0,1,2,\cdots$.

$$\begin{array}{c}\hfill {\left[p_{k+1}^{s_0,s_1,\cdots ,s_{k+1}}\left(A^{+}f\right)\right]}^2=\sum _{n=1}^{\infty }{|a_{n-1}^0|}^2{(n+1)}^{2s_0}(n+1)\\ \hfill +\sum _{n=2}^{\infty }|a_{n-1}^1{|}^2{(n+1)}^{2s_1}(n+1)+\cdots +\sum _{n=k+1}^{\infty }{|a_{n-1}^k|}^2{(n+1)}^{2s_k}(n+1)\\ \hfill \le \sum _{n=1}^{\infty }|a_{n-1}^0{|}^2{(n+1)}^{2(s_0+1)}+\sum _{n=2}^{\infty }|a_{n-1}^1{|}^2{(n+1)}^{2(s_1+1)}+\cdots +\sum _{n=k+1}^{\infty }{|a_{n-1}^k|}^2{(n+1)}^{2(s_k+1)}\\ \hfill ={\left[p_k^{s_0+1,s_1+1,\cdots ,s_k+1}\left(f\right)\right]}^2.\end{array}$$

(52)

with $k=0,1,2,\cdots$. This proves the continuity from ${\Phi }_{\mathcal{l}}$ to ${\Phi }_{\mathcal{l}+1}$ because of (20). Then, note that the canonical injection $J_{\mathcal{l}}:{\Phi }_{\mathcal{l}}⟼\Phi$ is continuous due to the definition of the strict inductive limit topology on $\Phi$. Then, for all $\mathcal{l}=0,1,2,\cdots$, we have that the mappings

$$J_{\mathcal{l}+1}\circ A^{+}:{\Phi }_{\mathcal{l}}⟼\Phi$$

(53)

are continuous. Then, following a well-known result ([50] page 58), $B^{+}$ is continuous as an operator on $\Phi$ (This result establishes that if Y is a locally convex space, X is a strict inductive limit of the spaces $X_n$ and $f:X⟼Y$ is a linear mapping, then f is continuous if and only if all mappings $f:X_n⟼Y$ are continuous, where each of the $f_n$ is the restriction of f to $X_n$). The proof of the continuity of B on $\Phi$ is similar. □

Now, take ${\mathcal{S}}_{\mathcal{l}}$ and $f\left(x\right)\in {\mathcal{S}}_{\mathcal{l}}$, $f_{\mathcal{l}}\left(x\right)={\sum }_{k=\mathcal{l}}^{\infty }{a}_k^{\mathcal{l}}{\psi }_k^{\mathcal{l}}\left(x\right)$. Define $B_{\mathcal{l}}^{+}$ on ${\mathcal{S}}_{\mathcal{l}}$ as

$$B_{\mathcal{l}}^{+}{f}_{\mathcal{l}}:=\sum _{k=\mathcal{l}}^{\infty }{a}_k^{\mathcal{l}}{B}_{\mathcal{l},k}^{+}{\psi }_k^{\mathcal{l}}\left(x\right)=\sum _{k=\mathcal{l}}^{\infty }{a}_k^{\mathcal{l}}\sqrt{k+1}{\psi }_{k+1}^{\mathcal{l}}.$$

(54)

Obviously, $B_{\mathcal{l}}^{+}{\psi }_k^{\mathcal{l}}\left(x\right)=\sqrt{k+1}{\psi }_{k+1}^{\mathcal{l}}$. Since ${\left[p_{\mathcal{l}}^s\left(B_{\mathcal{l}}^{+}{f}_{\mathcal{l}}\right)\right]}^2\le {\left[p_{\mathcal{l}}^{s+1}\left(f_{\mathcal{l}}\right)\right]}^2$, $s=0,1,2,\cdots$, $B_{\mathcal{l}}^{+}$ is continuous on ${\mathcal{S}}_{\mathcal{l}}$, $\mathcal{l}=0,1,2,\cdots$. Analogously,

$$B_{\mathcal{l}}{f}_{\mathcal{l}}:=\sum _{k=\mathcal{l}}{a}_k^{\mathcal{l}}{B}_{\mathcal{l},k}{\psi }_k^{\mathcal{l}}=\sum _{k=\mathcal{l}+1}^{\infty }\sqrt{k}{\psi }_{k-1}^{\mathcal{l}}.$$

(55)

Note that $B_0^{+}$ and $B_0$ are the creation and annihilation operators of the harmonic oscillator, respectively, $B_1^{+}$ and $B_1$ of the first SUSY partner of the oscillator, etc. The Hamiltonian $H_{\mathcal{l}}$ has the same eigenvalues as the harmonic oscillator except for the $\mathcal{l}-1$ first eigenvalues. Thus, for $f_{\mathcal{l}}\in {\mathcal{S}}_{\mathcal{l}}$

$$H_{\mathcal{l}}{f}_{\mathcal{l}}:=\sum _{k=\mathcal{l}}^{\infty }{a}_k^{\mathcal{l}}(k+1/2){\psi }_k^{\mathcal{l}}.$$

(56)

Considering the above discussion, it is quite straightforward to show that $B_{\mathcal{l}}^{+}$, $B_{\mathcal{l}}$ and $H_{\mathcal{l}}$ are continuous operators on ${\mathcal{S}}_{\mathcal{l}}$ and, therefore, continuously extendable to the dual ${\mathcal{S}}_{\mathcal{l}}^{\times }$, $\mathcal{l}=0,1,2,\cdots$. Their extension to all ${\Phi }_{\mathcal{l}}$ is immediate, as it is the proof of the continuity of these extensions to $\Phi$.

3.2. $H_0$ Is the One-Dimensional Trigonometric Pöschl–Teller Hamiltonian

Although we have not written the operators $B_{\mathcal{l},k}$ on the diagram (42), it is clear how they proceed. This construction goes for many cases. When $E_n$ is polynomially bounded by $P_k(n+1)$, all considerations of continuity go exactly the same as in the above example. In particular, if $H_0$ is either the one-dimensional Pöschl–Teller trigonometric potential or the one-dimensional infinite square well [51], they obey the above scheme.

Let us take the one-dimensional Pöschl–Teller Hamiltonian as in (11). Then, a SUSY transform takes (12) and transforms it into a similar Hamiltonian where ℓ has been replaced by $\mathcal{l}+1$ and so on. Thus, vectors in the first column in (41) should be denoted by $\left\{{\psi }_{\mathcal{l}}^n\right\}$, just replacing 0 by ℓ, $n=0,1,2\cdots$. Vectors in the second column are denoted as $\left\{{\psi }_{\mathcal{l}+1}^n\right\}$, $n=1,2,\cdots$ and so on. At the same time, we replace $A_k^{+}$ by $A_{\mathcal{l}+k}^{+}$, $k=0,1,2,\cdots$. Now [43],

$$A^{+}{\psi }_n^{\mathcal{l}}=\sqrt{(n+1)\left[\right(2(\mathcal{l}+1)+n+1]}{\psi }_{n+1}^{\mathcal{l}+1},\mathcal{l},n=0,1,2\cdots ,$$

(57)

and

$$A{\psi }_n^{\mathcal{l}}=\sqrt{n(2\mathcal{l}+n)}{\psi }_{n-1}^{\mathcal{l}-1},\mathcal{l},n=1,2\cdots .$$

(58)

Then, the whole construction proceeds essentially as in the above example. In particular, $A^{+}$ and A are linear continuous operators on $\Phi$.

4. Concluding Remarks

Gelfand triplets, also called Rigged Hilbert Spaces (RHSs), are the suitable framework for a rigorous mathematical formulation of the Dirac formalism of Quantum Mechanics, quantum systems (relativistic and non-relativistic and sometimes even classical) showing resonances or different types of singularities. In addition, RHS is the suitable arena to describe with the due mathematical rigor objects of common use in standard quantum mechanics, such as continuous and discrete bases, bases of special functions, and Lie algebras of continuous operators of the symmetries of a given quantum model. All of them exist inside a common framework, a property that does not have the standard formalism on Hilbert spaces.

In the present paper, we give an interesting example of the latter based on the one-dimensional Pöschl–Teller potential. This is a very interesting example showing all features, as described at the end of the last paragraph. We believe that this example is very illustrative and at the same time nontrivial.

The factorization method and the SUSY quantum mechanics provide an efficient method to obtain Hamiltonians with a similar spectrum to the one given. The use of ladder operators in this factorization, as well as in the process of relating eigenvectors of the different Hamiltonians, result from the SUSY transformations. These ladder operators are not bounded operators on Hilbert space, so their proper mathematical manipulation would require a cumbersome and non-trivial analysis, contrary to the formal analysis that is performed by physicists. The context of RHS solves this problem from a strict mathematical point of view. In the present article, we describe a particularly standard model in which the seed Hamiltonian, $H_0$, from which all others follow after reiterative SUSY transformations, is the Harmonic oscillator. We have chosen this seed Hamiltonian because its factorization and partners are well known. In addition, it has an infinite number of partners, a fact that adds some further mathematical interest to the problem. The same analysis can be performed when the seed Hamiltonian is the one-dimensional trigonometric Pöschl–Teller.

There are many examples and studies of this factorization method and SUSY transformations [40,41,47,48], which may be the point of departure of other similar studies in the future. This may be the case with respect to the spectrum-generating algebras [36,42,43,52].

Among the perspectives for future work, we may mention the analysis of pseudo-Hermitian systems [53,54,55], or more specifically PT-symmetric models [56] using Gelfand triplets. An interesting point of departure could be the non-Hermitian Hamiltonians associated with the complex Lie algebra $sl(2,\mathbb{C})$ [57].