Content-Type: multipart/mixed; boundary="-------------0205010458504" This is a multi-part message in MIME format. ---------------0205010458504 Content-Type: text/plain; name="02-209.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-209.comments" Journal Reference: J. Phys. A: Math. Gen. 35 (2002) 319-342 ---------------0205010458504 Content-Type: text/plain; name="02-209.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-209.keywords" Rigged Hilbert Spaces, Sturm-Liouville theory, continuous spectrum, Dirac's formalism, Schrodinger equation. ---------------0205010458504 Content-Type: application/x-tex; name="02jpa.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="02jpa.tex" \documentstyle[preprint,aps,eqsecnum,amssymb,amsfonts,verbatim]{revtex} \tightenlines \begin{document} \title{\bf Rigged Hilbert Space Approach to the Schr\"odinger Equation} \author{R.~de la Madrid} \address{Institute for Scientific Interchange (ISI), Villa Gualino, Viale Settimio Severo 65, I-10133, Torino, Italy \\ and \\ Departamento de F\'\i sica Te\'orica, Facultad de Ciencias, 47011 Valladolid, Spain \\ E-mail: \texttt{rafa@isiosf.isi.it}} \date{October 1, 2001} \maketitle \begin{abstract} \noindent It is shown that the natural framework for the solutions of any Schr\"odinger equation whose spectrum has a continuous part is the Rigged Hilbert Space rather than just the Hilbert space. The difficulties of using only the Hilbert space to handle unbounded Schr\"odinger Hamiltonians whose spectrum has a continuous part are disclosed. Those difficulties are overcome by using an appropriate Rigged Hilbert Space (RHS). The RHS is able to associate an eigenket to each energy in the spectrum of the Hamiltonian, regardless of whether the energy belongs to the discrete or to the continuous part of the spectrum. The collection of eigenkets corresponding to both discrete and continuous spectra forms a basis system that can be used to expand any physical wave function. Thus the RHS treats discrete energies (discrete spectrum) and scattering energies (continuous spectrum) on the same footing. \end{abstract} \pacs{03.65.-w, 02.30.-f} %PACS numbers: 03.65.-w, 02.30.-f \def\llra{\relbar\joinrel\longrightarrow} %THIS IS LONG \def\mapright#1{\smash{\mathop{\llra}\limits_{#1}}} %ARROW ON LINE \def\mapup#1{\smash{\mathop{\llra}\limits^{#1}}} %CAN PUT SOMETHING OVER IT \def\mapupdown#1#2{\smash{\mathop{\llra}\limits^{#1}_{#2}}} %over&under it% \newpage \def\thesection{\arabic{section}} \section{Introduction} \def\thesection{\arabic{section}} \setcounter{equation}{0} \label{sec:introduction} Several authors have realized that the Hilbert space is not sufficient for the purposes of Quantum Mechanics, and that an extension of the Hilbert space to the Rigged Hilbert Space (RHS) is needed. The RHS was introduced in physics for the first time in the 1960s independently by J.~P.~Antoine~\cite{ANTOINE}, A.~Bohm~\cite{B60}, and J.~E.~Roberts~\cite{ROBERTS}. These authors realized that the RHS provides a rigorous mathematical rephrasing of Dirac's formalism. In essence, the Nuclear Spectral Theorem~\cite{GELFAND} (also known as Gelfand-Maurin Theorem) restates Dirac basis vector expansion along with the Dirac bras and kets within a mathematical theory. Later on, other authors such as A.~Galindo and P.~Pascual~\cite{GALINDO}, and O.~Melsheimer~\cite{MELSH} came to the same conclusion. Earlier attempts to go beyond the Hilbert space framework are reported in Refs.~\cite{MEJLBO,KRISTENSEN}. If two operators of the algebra of observables satisfy Heisenberg's commutation relation, at least one of them cannot be continuous (bounded) with respect to the Hilbert space topology. In Refs.~\cite{MEJLBO,KRISTENSEN}, it is shown that there are subdomains of the Hilbert space that can be endowed with topologies that make those operators continuous. Since the early 1990s, the RHS has become a standard tool in many areas of theoretical physics, especially in those that deal with continuous and resonance spectra. For instance, the RHS has been used to treat the Lippmann-Schwinger equation~\cite{DIS,BKR}, the Gamow vectors (cf.~\cite{DIS,BOLLINI96,BOHM97,ANTOINE98,KUKULIN} and references therein), and certain generalized spectral decompositions of chaotic maps~\cite{AT93,SUCHANECKI}. The dynamical equation of Quantum Mechanics is the Schr\"odinger equation. Thus any attempt to show that the RHS contains the mathematical methods needed by Quantum Mechanics should show that the natural framework for the solutions of the Schr\"odinger equation is the RHS. The objective of this paper is to show that the solutions of the Schr\"odinger equation fall in a RHS. To illustrate this, we shall use the example of the square well-barrier potential Hamiltonian. We recall that none of Refs.~\cite{BKR,BOLLINI96,BOHM97,ANTOINE98,KUKULIN,AT93,SUCHANECKI} took the Schr\"odinger equation as the dynamical equation. A RHS is a triplet of spaces \begin{equation} \mathbf \Phi \subset {\cal H} \subset \mathbf \Phi ^{\times} \, , \label{IRHS} \end{equation} where $\cal H$ is a Hilbert space, $\mathbf \Phi$ is a dense subspace of the Hilbert space, and $\mathbf \Phi ^{\times}$ is the dual space of $\mathbf \Phi$, i.e., $\mathbf \Phi ^{\times}$ is the set of antilinear functionals over the space $\mathbf \Phi$ . The space $\mathbf \Phi$ has a topology that is finer than the topology inherited from $\cal H$. The domain ${\cal D}(H)$ of the Hamiltonian lies between $\mathbf \Phi$ and $\cal H$, \begin{equation} {\mathbf \Phi} \subset {\cal D}(H) \subset {\cal H} \, . \end{equation} If a Schr\"odinger Hamiltonian is defined on the whole Hilbert space and has only discrete spectrum, then the Hilbert space is sufficient for the purposes of Quantum Mechanics. However, if the Hamiltonian is not defined on the whole Hilbert space (e.g., it is unbounded) and its spectrum has a continuous part, then the mathematical methods of the Hilbert space are not sufficient, and an extension of those methods is needed. The RHS arises as the natural extension when dealing with unbounded operators that have continuous spectrum. The reason why the RHS is the extension of the Hilbert space that we need is twofold. On the one hand, one of the key assumptions of Quantum Mechanics is that the quantity \begin{equation} (\varphi ,H\varphi ) \label{IexintrodispP} \end{equation} accounts for the expectation value of the measurement of the observable $H$ in the state $\varphi$, and that \begin{equation} \Delta _{\varphi}H= \sqrt{ (\varphi ,H^2\varphi )-(\varphi ,H\varphi )^2} \label{IintrodispA} \end{equation} accounts for the uncertainty of the measurement of the observable $H$ in the state $\varphi$. The expectation value (\ref{IexintrodispP}) cannot be computed for every element of the Hilbert space $\cal H$, but only for those $\varphi \in {\mathcal H}$ that also belong to ${\cal D}(H)$. Similarly, the uncertainty (\ref{IintrodispA}) cannot be computed for every element of $\cal H$ either. If we take as physical states those square integrable functions for which physical quantities such as the expectation value (\ref{IexintrodispP}) and the uncertainty (\ref{IintrodispA}) can be computed, then not every square integrable function (i.e., every element of $\cal H$) can represent a physical state. As we shall see, the natural space of physical wave functions is the space $\mathbf \Phi$ in (\ref{IRHS}), because all physical quantities such as expectation values and uncertainties can be computed for its elements. On the other hand, in Quantum Mechanics it is assumed that for each energy in the spectrum of the Hamiltonian $H$, there corresponds a ket that is an eigenvector of $H$. If we denote the ket corresponding to an energy $E_n$ in the discrete spectrum by $|E_n)$ and denote the ket corresponding to an energy $E$ in the continuous spectrum by $|E\rangle$, then we should have \begin{mathletters} \label{Ieigenkets} \begin{eqnarray} \def\theequation{\thesection.\arabic{equation}} &H|E_n)=E_n|E_n) \, , \quad & \label{Ibeigenkets} \\ &H|E\rangle=E|E\rangle \, . \quad & \label{Iceigenkets} \end{eqnarray} \end{mathletters} These eigenkets are normalized according to the following rule: \begin{equation} (E_n|E_m)=\delta _{nm} \, , \quad \langle E |E ^{\prime} \rangle = \delta (E -E ^{\prime}) \, , \quad \langle E|E_n)=0 \, . \label{Ideltanorintro} \end{equation} It is also assumed that those kets form a complete basis that can be used to expand any wave function $\varphi$, \begin{equation} \varphi = \sum_{n}|E_n)(E_n|\varphi ) +\int dE \, |E\rangle \langle E|\varphi \rangle \, . \label{IDiracbsis} \end{equation} The solutions of Eq.~(\ref{Ibeigenkets}) are square normalizable, i.e., they lie in the Hilbert space. However, the solutions of Eq.~(\ref{Iceigenkets}) are not square normalizable, i.e., they lie outside the Hilbert space. Therefore, the Hilbert space is not large enough to contain the non-normalizable eigenkets that are associated to the energies in the continuous spectrum---a larger space than the Hilbert space is needed. This larger space, which contains the eigenkets $|E\rangle$, is the space $\mathbf \Phi ^{\times}$ in (\ref{IRHS}). The action of the Hamiltonian, which is in principle defined only on the elements of $\mathbf \Phi$ (or on the elements of ${\cal D}(H)$), can be extended to the elements $|F\rangle$ of $\mathbf \Phi ^{\times}$ by defining the following conjugate operator $H^{\times}$: \begin{equation} \langle \varphi |H^{\times}|F\rangle := \langle H^{\dagger}\varphi |F\rangle \, , \quad \forall \varphi \in {\mathbf \Phi} \, , \ |F\rangle \in \mathbf \Phi ^{\times} \, . \label{idef} \end{equation} The operator $H^{\times}$ is a uniquely defined extension of $H$. Using definition (\ref{idef}), the RHS formalism restates Eq.~(\ref{Iceigenkets}) as \begin{equation} \langle \varphi |H^{\times}|E\rangle = \langle H^{\dagger}\varphi |E\rangle =E\langle \varphi |E\rangle \, , \quad \forall \varphi \in \mathbf \Phi \, . \label{igeing} \end{equation} When the arbitrary $\varphi \in \mathbf \Phi$ is omitted in this equation, we recover the formal equation (\ref{Iceigenkets}). In this way, the Gelfand triplet (\ref{IRHS}) arises in a natural way. The Hilbert space $\mathcal H$ appears because the wave functions must be square normalizable. The subspace $\mathbf \Phi$ is the set of physical wave functions, i.e., the set of square integrable functions for which any expectation value and any uncertainty can be computed. The dual space $\mathbf \Phi ^{\times}$ contains the eigenkets associated to energies in the continuous part of the spectrum of $H$. These eigenkets are defined as functionals over the space $\mathbf \Phi$, and they can be used to expand any $\varphi \in \mathbf \Phi$ as in Eq.~(\ref{IDiracbsis}). The eigenequation (\ref{Iceigenkets}) holds within the RHS in the sense of Eq.~(\ref{igeing}). Thus the Rigged Hilbert Space is the extension of the Hilbert space that is needed when the solutions of the Schr\"odinger equation are not normalizable. These non-normalizable solutions are treated as distributions in the dual space $\mathbf \Phi ^{\times}$. Unlike the Hilbert space, the RHS treats the discrete and continuous spectra on the same footing: for each energy of the spectrum of the Hamiltonian (either in the discrete or in the continuous part), there is an eigenket of the Hamiltonian (given by Eq.~(\ref{Ibeigenkets}) in the discrete case, and by Eq.~(\ref{Iceigenkets}) in the continuous case) that belongs to a basis that expands any physical wave function as in Eq.~(\ref{IDiracbsis}). The main shortcoming of the RHS theory is that it does not provide a prescription to construct the space $\mathbf \Phi$ and the eigenvectors $|E\rangle$. The general statement of the Nuclear Spectral Theorem~\cite{GELFAND} just assures the existence of the generalized eigenvectors $|E\rangle$. In that theorem, the space $\mathbf \Phi$ is assumed to be given beforehand. In this paper, the RHS of the square well-barrier will be constructed explicitly along with the eigenkets $|E\rangle$ by applying the method proposed in Ref.~\cite{DIS}. This method has been successfully applied to a Hamiltonian with purely continuous spectrum: the square {\it barrier} Hamiltonian~\cite{FP}. Here we apply it to a Hamiltonian with both discrete and continuous spectrum: the square {\it well-barrier} Hamiltonian. The steps of our method can be summarized as follows: \vskip0.2cm (i) Construction of the self-adjoint Hamiltonian from the formal Schr\"odinger differential operator. \vskip0.2cm (ii) Construction of the Green function and therewith of the resolvent. \vskip0.2cm (iii) Computation of the spectrum of the Hamiltonian. \vskip0.2cm (iv) Construction of the direct integral decomposition. \vskip0.2cm (v) Construction of the Rigged Hilbert Space. \vskip0.2cm \noindent Steps (i)-(iv) will be addressed by using the Sturm-Liouville theory~\cite{DUNFORD}. Step (v) will be handled by the methods proposed in Ref.~\cite{DIS}. In Section~\ref{sec:fdosehami}, we construct the self-adjoint Hamiltonian from the formal Schr\"odinger differential operator and from a domain of the Hilbert space. In Section~\ref{sec:greenfunctions}, we construct the Green function and the resolvent of $H$. In Section~\ref{sec:spectrum}, we compute the spectrum of the Hamiltonian. In Section~\ref{sec:directindeco}, we construct the direct integral decomposition of the Hilbert space generated by the Hamiltonian. In Section~\ref{sec:constRHS}, the RHS of the square well-barrier Hamiltonian is constructed. \def\thesection{\arabic{section}} \section{Formal Differential Operator and Self-Adjoint Hamiltonian} \def\thesection{\arabic{section}} \setcounter{equation}{0} \label{sec:fdosehami} The first step toward the construction of a RHS associated to a Hamiltonian is to define that Hamiltonian. In non-relativistic quantum mechanics, the Hamiltonian is defined by the formal differential operator of the time-independent Schr\"odinger equation and by a Hilbert space domain on which the Hamiltonian acts. The time-independent Schr\"odinger equation reads in the position representation as \begin{equation} \left(\frac{-\hbar ^2}{2m} \Delta +V(\vec{x})\right) \langle \vec{x}|E\rangle =E\langle \vec{x}|E\rangle \, , \label{delta} \end{equation} where $\Delta$ is the three-dimensional Laplacian and \begin{equation} V(\vec{x})\equiv V(r)=\left\{ \begin{array}{cl} -V_1 &0s \, , \end{array} \right. \label{exofGFA} \end{equation} where $W(\chi ,\Theta )$ is the Wronskian of $\chi$ and $\Theta$, \begin{equation} W(\chi ,\Theta )=\chi \Theta '-\chi ' \Theta \, . \end{equation} \vskip0.5cm In order to compute $G(r,s;E)$, we shall divide the complex energy plane in three different regions and obtain $G(r,s;E)$ for each region separately. In our calculations, we shall use the following branch of the square root function: \begin{equation} \sqrt{\cdot}:\{ E\in {\mathbb C} \, | \ -\pi <{\rm arg}(E)\leq \pi \} \longmapsto \{E\in {\mathbb C} \, | \ -\pi/2 <{\rm arg}(E)\leq \pi/2 \} \, . \label{branch} \end{equation} \def\thesubsection{\thesection.\arabic{subsection}} \subsection{Region ${\rm Re}(E)<0$, ${\rm Im}(E)\neq 0$} We first apply Theorem~1 to the region of the complex energy plane where ${\rm Re}(E)<0$ and ${\rm Im}(E)\neq 0$. According to that theorem, the Green function reads \begin{equation} G(r,s;E)=\left\{ \begin{array}{ll} -\frac{2m/\hbar ^2}{\sqrt{-2m/\hbar ^2 \, E}} \, \frac{\widetilde{\chi}(r;E) \, \widetilde{\Theta}(s;E)} {2\widetilde{\mathcal{J}}_3(E)} &rs \end{array} \right. \quad \mbox{Re}(E)<0 \, , \ \mbox{Im}(E)\neq 0 \, . \label{green-} \end{equation} In this equation, the eigenfunction $\widetilde{\chi}(r;E)$ satisfies the time-independent Schr\"odinger equation (\ref{rSe0}) subject to the following boundary conditions: \begin{mathletters} \begin{eqnarray} && \widetilde{\chi} (0;E)=0 \, , \label{bca01} \\ && \widetilde{\chi} (r;E)\in AC^2([0,\infty )) \, , \label{bcchiac} \\ && \widetilde{\chi}(r;E) {\rm \ is \ square \ integrable \ at \ } 0 \, , \label{sbca03} \end{eqnarray} \label{eigsbo0co} \end{mathletters} which yield \begin{equation} \widetilde{\chi}(r;E)= \left\{ \begin{array}{lll} \frac{i}{2}\, (e^{\sqrt{-\frac{2m}{\hbar ^2}(E+V_1)}\, r}- e^{-\sqrt{-\frac{2m}{\hbar ^2}(E+V_1)}\, r}) \quad &00$, ${\rm Im}(E)> 0$} \label{sec:region++} In this region of the complex plane, Theorem~1 leads to the following Green function: \begin{equation} G(r,s;E)=\left\{ \begin{array}{ll} \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \frac{\chi (r;E) \, \Theta _+(s;E)}{2i{\cal J}_4(E)} &rs \end{array} \right. \quad {\rm Re}(E)>0 \, , \ {\rm Im}(E)> 0 \, . \label{green++} \end{equation} In this equation, the eigenfunction $\chi (r;E)$ satisfies the differential equation (\ref{rSe0}) subject to the boundary conditions (\ref{eigsbo0co}), \begin{equation} \chi (r;E)=\left\{ \begin{array}{lll} \sin (\sqrt{\frac{2m}{\hbar ^2}(E+V_1)}\, r) \quad &00$, ${\rm Im}(E)< 0$} If ${\rm Re}(E)>0$, ${\rm Im}(E)< 0$ and we use the branch of the square root function~(\ref{branch}), then the Green function has the form \begin{equation} G(r,s;E)=\left\{ \begin{array}{ll} -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2\, E}} \, \frac{\chi (r;E) \, \Theta _-(s;E)}{2i{\cal J}_3(E)} &rs \end{array} \right. \quad {\rm Re}(E)>0 \, , \ {\rm Im}(E)< 0 \, . \label{green+-} \end{equation} The eigenfunction $\chi (r;E)$ satisfies the time-independent Schr\"odinger equation (\ref{rSe0}) subject to the boundary conditions (\ref{eigsbo0co}). Thus $\chi (r;E)$ is given by Eq.~(\ref{chi})---hence the same symbol as in the previous subsection---although now $E$ lies in the fourth quadrant of the complex energy plane. The eigenfunction $\Theta _-(r;E)$ satisfies the differential equation (\ref{rSe0}) subject to the boundary conditions (\ref{thetcoabejej}), \begin{equation} \Theta _-(r;E)=\left\{ \begin{array}{lll} {\cal A}^-_1(E)e^{i\sqrt{\frac{2m}{\hbar ^2}(E+V_1)}\, r} +{\cal A}^-_2(E) e^{-i\sqrt{\frac{2m}{\hbar ^2}(E+V_1)}\, r} \quad &0s \, , \end{array} \right. \label{greenfunitthes} \end{equation} for all $E$ in ${\rm Re}(H)\cap O$, and that $\{ \rho _{ij} \}$ is a positive matrix measure on $\Lambda$ associated with $H$ as in Theorem 2. Then the functions $\theta _{ij}^{\pm}$ are analytic in ${\rm Re}(H)\cap O$, and given any bounded open interval $(E_1,E_2)\subset \Lambda$, we have for $1\leq i,j\leq 2$, \begin{equation} \begin{array}{lll} \rho _{ij}((E_1,E_2))&=& \lim_{\delta \to 0}\lim_{\epsilon \to 0+} \frac{1}{2\pi i}\int_{E_1+\delta}^{E_2-\delta} [ \theta _{ij}^-(E-i\epsilon )-\theta _{ij}^-(E+i\epsilon ) ]dE \\ [2ex] \quad &=& \lim_{\delta \to 0}\lim_{\epsilon \to 0+} \frac{1}{2\pi i}\int_{E_1+\delta}^{E_2-\delta} [ \theta _{ij}^+(E-i\epsilon )-\theta _{ij}^+(E+i\epsilon ) ]dE \, . \end{array} \end{equation} \vskip0.5cm Using these spectral measures we can compute the spectrum of $H$. This spectrum is the subset of the real line on which the Green function fails to be analytic. This non-analyticity of $G(r,s;E)$ will be built into the functions $\theta _{ij}^{\pm}(E)$ that appear in Theorem~3. From the expression of the Green function computed in Section~\ref{sec:greenfunctions}, it is clear that the subsets $(-\infty ,0)$ and $(0,\infty )$ should be studied separately. We will denote either of these subsets by $\Lambda$. \subsection{Negative Energy Real Line: $\Lambda =(-\infty ,0)$} \label{sec:NeERlin} We first take $\Lambda$ from Theorem~3 to be $(-\infty ,0)$. We choose a basis for the space of solutions of the equation $h\sigma =E\sigma$ that is continuous on $(0,\infty )\times \Lambda$ and analytically dependent on $E$ as \begin{mathletters} \begin{eqnarray} &&\sigma _1(r;E)=\left\{ \begin{array}{lll} \widetilde{\cal B}_1(E)e^{\sqrt{-\frac{2m}{\hbar ^2}(E+V_1)}r} +\widetilde{\cal B}_2(E) e^{-\sqrt{-\frac{2m}{\hbar ^2}(E+V_1)}r} \quad &00, \mbox{Im}(E)>0\, , \, r>s \, . \label{G++tofpuon} \end{eqnarray} By substituting Eq.~(\ref{Thet-inosigm}) into Eq.~(\ref{green+-}) we get to \begin{eqnarray} &&G(r,s;E)= -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \frac{1}{2i{\cal J}_3(E)}\, \left[ \frac{ {\cal C}_3(E)}{W(E)} \sigma _1(r;E)-\frac{{\cal J}_3(E)}{W(E)}\sigma _2(r;E)\right] \sigma _1 (s;E) \, , \nonumber \\ &&\qquad \hskip7cm \mbox{Re}(E)>0, \mbox{Im}(E)<0\, , \, r>s \, . \label{G+-tofpuon} \end{eqnarray} Since \begin{equation} \overline{\sigma _1(s;\overline{E})}=\sigma _1(s;E) \, , \end{equation} Eq.~(\ref{G++tofpuon}) can be written as \begin{eqnarray} &&G(r,s;E)= \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \frac{1}{2i{\cal J}_4(E)}\, \left[ -\frac{ {\cal C}_4(E)}{W(E)} \sigma _1(r;E)\overline{\sigma _1(s;\overline{E})} +\frac{ {\cal J}_4(E)}{W(E)}\sigma _2(r;E) \overline{\sigma _1(s;\overline{E})}\right] \nonumber \\ &&\qquad \hskip8cm \mbox{Re}(E)>0, \mbox{Im}(E)>0\, , \, r>s \, , \label{redaot++} \end{eqnarray} whereas Eq.~(\ref{G+-tofpuon}) can be written as \begin{eqnarray} &&G(r,s;E)= -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \frac{1}{2i{\cal J}_3(E)}\, \left[ \frac{ {\cal C}_3(E)}{W(E)} \sigma _1(r;E)\overline{\sigma _1(s;\overline{E})} -\frac{{\cal J}_3(E)}{W(E)}\sigma _2(r;E) \overline{\sigma _1(s;\overline{E})}\right] \nonumber \\ &&\qquad \hskip8cm \mbox{Re}(E)>0, \mbox{Im}(E)<0\, , \, r>s \, . \label{redaot+-} \end{eqnarray} By comparing (\ref{greenfunitthes}) to (\ref{redaot++}) we get to \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \frac{1}{2i} \frac{- {\cal C}_4(E)}{ {\cal J}_4(E)W(E)} & 0 \\ \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \frac{1}{2i} \frac{1}{W(E)} & 0 \end{array} \right) \, , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)>0 \, . \label{theta++} \end{equation} By comparing (\ref{greenfunitthes}) to (\ref{redaot+-}) we get to \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \frac{1}{2i} \frac{{\cal C}_3(E)}{ {\cal J}_3(E)W(E)} & 0 \\ \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \frac{1}{2i} \frac{1}{W(E)} & 0 \end{array} \right) \, , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)<0 \, . \label{theta+-} \end{equation} From Eqs.~(\ref{theta++}) and (\ref{theta+-}) we can see that the measures $\rho _{12}$, $\rho _{21}$ and $\rho _{22}$ in Theorem~3 are zero and that the measure $\rho _{11}$ is given by \begin{eqnarray} \rho _{11}((E_1,E_2))&=&\lim _{\delta \to 0} \lim _{\epsilon \to 0+} \frac{1}{2\pi i} \int_{E_1+\delta}^{E_2-\delta} \left[ \theta _{11}^+ (E-i\epsilon ) -\theta _{11}^+ (E+i\epsilon ) \right] dE \nonumber \\ &=&\int_{E_1}^{E_2} \frac{1}{4\pi}\, \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \frac{1}{ {\cal J}_3(E) {\cal J}_4(E)}\, dE \, , \end{eqnarray} which leads to \begin{equation} \rho (E)\equiv \rho _{11}(E)= \frac{1}{4\pi}\, \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \frac{1}{ |{\cal J}_4(E)|^2}\, , \quad E\in (0,\infty ) \, . \end{equation} The function $\theta _{11}^+(E)$ has a branch cut along $(0,\infty)$, and therefore $(0,\infty )$ is included in ${\rm Sp}(H)$. Since ${\rm Sp}(H)$ is a closed set, we have that \begin{equation} \mbox{Sp}(H)=\{ E_1,\ldots ,E_N \} \cup [0,\infty )\, . \end{equation} Therefore, the spectrum of $H$ has a discrete part $\Lambda _{\rm b}=\{ E_1,\ldots ,E_N \}$ and a continuous part $\Lambda _{\rm c}=[0,\infty )$. \def\thesection{\arabic{section}} \section{Direct Integral Decomposition} \def\thesection{\arabic{section}} \setcounter{equation}{0} \label{sec:directindeco} The fourth step toward the construction of the RHS is to compute the unitary operator that diagonalizes the Hamiltonian and the direct integral decomposition induced by that unitary operator. In order to compute them, we apply Theorem~2 of Section~\ref{sec:spectrum} to the discrete and continuous spectrum separately, since they can be treated independently. In the end, the independence of the discrete and continuous spectra will lead to the splitting of the Hilbert space into a direct sum of a Hilbert space associated to the discrete spectrum and another Hilbert space associated to the continuous spectrum. We first apply Theorem~2 to the discrete part of the spectrum. By that theorem, there is a unitary operator $U_{\rm b}$ from ${\mathcal H}_{\rm b} :={\sf E}(\Lambda _{\rm b})L^2([0,\infty ),dr)$ onto $\widehat{\mathcal{H}}_{\rm b}:={\mathbb C}^N$ defined by \begin{eqnarray} U_{\rm b}: \mathcal{H}_{\rm b} &\longmapsto & \widehat{\mathcal{H}}_{\rm b} \nonumber \\ f_{\rm b}(r)& \longmapsto & U_{\rm b}f_{\rm b}=\{ (\phi _1,f_{\rm b}), \ldots ,(\phi _N,f_{\rm b}) \} \, . \label{Ubound} \end{eqnarray} In this equation, $\mathcal{H}_{\rm b}$ is the Hilbert space spanned by the bound states $\phi _1,\ldots ,\phi _N$, $\widehat{\mathcal{H}}_{\rm b}$ is the space of $N$-tuples of the form $\{ (\phi _n,f_{\rm b})\}_{n=1}^N$ (hence $\widehat{\mathcal{H}}_{\rm b}$ is isomorphic to ${\mathbb C}^N$), $f_{\rm b}={\sf E}(\Lambda _{\rm b})f$, i.e., $f_{\rm b}$ is the component of $f$ along the space $\mathcal{H}_{\rm b}$, \begin{equation} f_{\rm b}(r)=\sum_{n=1}^N (\phi _n,f)\, \phi _n(r) \, , \end{equation} and $(\phi _n,f_{\rm b})$ is the scalar product of the $n$-th bound state $\phi _n(r)$ of Eq.~(\ref{normaiegin}) with $f_{\rm b}$, \begin{equation} (\phi _n, f_{\rm b})=\int_0^{\infty}dr\, \overline{\phi _n(r)}f_{\rm b}(r) \, , \quad n=1, \ldots , N \, . \end{equation} On the space $\widehat{\mathcal{H}}_{\rm b}$, the Hamiltonian acts as the following $N\times N$ diagonal matrix: \begin{equation} \widehat{H}_{\rm b}:=U_{\rm b}H_{\rm b}U_{\rm b}^{-1}= \left( \begin{array}{cccc} E_1 & 0 & .\, . & 0 \\ 0 &E_2 & . \, . & 0 \\ . \, . & . \, . & . \, . & . \, . \\ 0 & 0 & . \, . & E_N \end{array} \right) \, , \label{diagonaHb} \end{equation} where $H_{\rm b}$ is the restriction of $H$ to $\mathcal{H}_{\rm b}$. Next, we apply Theorem~2 to the continuous part of the spectrum. By that theorem, there is a unitary map $\widetilde{U}_{\rm c}$ from ${\mathcal H}_{\rm c} :={\sf E}(\Lambda _{\rm c})L^2([0,\infty ),dr)$ onto $L^2( (0,\infty ),\rho (E)dE)$ defined by \begin{eqnarray} \widetilde{U}_{\rm c}:\mathcal{H}_{\rm c} &\longmapsto & L^2( (0,\infty ),\rho (E)dE) \nonumber \\ f_{\rm c}(r)& \longmapsto & \widetilde{f}_{\rm c}(E)=(\widetilde{U}_{\rm c}f_{\rm c})(E)= \int_0^{\infty}dr f_{\rm c}(r) \overline{\chi (r;E)} \, , \label{rhoU} \end{eqnarray} where $f_{\rm c}={\sf E}(\Lambda _{\rm c})f$, i.e., $f_{\rm c}$ is the component of $f$ along the Hilbert space $\mathcal{H}_{\rm c}$. The space $\mathcal{H}_{\rm c}$ is the Hilbert space that corresponds to the continuous part of the spectrum---hence the subscript ${\rm c}$. If we write \begin{equation} {\cal D}(H)=\mathcal{H}_{\rm b} \oplus {\cal D}(H_{\rm c}) \end{equation} and denote the restriction of $H$ to ${\cal D}(H_{\rm c})$ by $H_{\rm c}$, then the operator $\widetilde{U}_{\rm c}$ of Eq.~(\ref{rhoU}) provides a $\rho$-diagonalization of $H_{\rm c}$. If we seek a $\delta$-normalization~\cite{DIS}, we just have to define the following eigenfunction: \begin{equation} \phi (r;E):=\sqrt{\rho (E)} \, \chi (r;E) \, , \label{dnes} \end{equation} which is the eigensolution of the differential operator $h$ that is $\delta$-normalized, and the following unitary operator: \begin{eqnarray} U_{\rm c}:\mathcal{H}_{\rm c} &\longmapsto & \widehat{\mathcal{H}}_{\rm c} \nonumber \\ f_{\rm c} &\longmapsto & \widehat{f}_{\rm c}(E)= (U_{\rm c}f_{\rm c})(E)=\int_0^{\infty}dr \, f_{\rm c}(r) \overline{\phi (r;E)} \, , \end{eqnarray} where $\widehat{\mathcal{H}}_{\rm c}=L^2([0,\infty ),dE)$. We note that the normalization of the bound states (\ref{normaiegin}) is different to the $\delta$-normalization of (\ref{dnes}). The inverses of the operators $U_{\rm b}$ and $U_{\rm c}$ are provided by the following theorem (cf.~Theorem XIII.5.14 of Ref.~\cite{DUNFORD}): \vskip0.5cm {\bf Theorem~4} (Weyl-Kodaira)\quad Let $H$, $\Lambda$, $\{ \rho _{ij} \}$, etc., be as in Theorem~2. Let $E_0$ and $E_1$ be the end points of $\Lambda$. Then the inverse of the isometric isomorphism $U$ of ${\sf E}(\Lambda )L^2([0,\infty ),dr)$ onto $L^2(\Lambda ,\{ \rho _{ij}\})$ is given by the formula \begin{equation} (U^{-1}F)(r)=\lim_{\mu _0 \to E_0}\lim_{\mu _1 \to E_1} \int_{\mu _0}^{\mu _1} \left( \sum_{i,j=1}^{2} F_i(E)\sigma _j(r;E)\rho _{ij}(dE) \right) \, , \end{equation} where $F=[F_1,F_2]\in L^2(\Lambda ,\{ \rho _{ij}\})$, the limit existing in the topology of $L^2([0, \infty ),dr)$. \vskip0.5cm According to Theorem~4, the inverse of $U_{\rm b}$ is given by \begin{equation} (U_{\rm b}^{-1}\widehat{f}_{\rm b})(r)= \sum_{n=1}^N (\phi _n,f_{\rm b}) \phi _n(r) \, , \end{equation} where $\widehat{f}_{\rm b} =\{ (\phi _1,f_{\rm b}), \ldots , (\phi _N,f_{\rm b}) \}$ is an $N$-tuple of complex numbers. The operator $U_{\rm b}^{-1}$ transforms from $\widehat{\mathcal{H}}_{\rm b}$ onto $\mathcal{H}_{\rm b}$. According to Theorem~4, the inverse of $U_{\rm c}$ is given by \begin{equation} f_{\rm c}(r)=(U_{\rm c}^{-1}\widehat{f}_{\rm c})(r)= \int_0^{\infty}dE\, \widehat{f}_{\rm c}(E)\phi (r;E) \, , \quad \widehat{f}_{\rm c}(E)\in L^2([0,\infty ),dE) \, . \label{invdiagonaliza} \end{equation} The operator $U_{\rm c}^{-1}$ transforms from $\widehat{\mathcal{H}}_{\rm c}$ onto $\mathcal{H}_{\rm c}$. Using the operators $U_{\rm b}$ and $U_{\rm c}$ we can construct an operator $U$ defined as \begin{eqnarray} U:{\cal H} &\longmapsto & \widehat{\cal H}= \oplus _{n=1}^N \widehat{\mathcal H}(E_n) \oplus \int_{0}^{\infty} \widehat{\cal H}(E)dE \nonumber \\ f &\longmapsto & Uf:= [U_{\rm b}f,U_{\rm c}f]= \left[ \widehat{f}_{\rm b},\{ \widehat{f}_{\rm c}(E) \} \right] \, , \label{dirintdec} \end{eqnarray} where ${\cal H}$ is realized by $\mathcal{H}_{\rm b} \oplus \mathcal{H}_{\rm c}$, and $\widehat{\cal H}$ is realized by $\widehat{\mathcal{H}}_{\rm b} \oplus \widehat{\mathcal{H}}_{\rm c}$. The Hilbert space $\widehat{\mathcal H}(E_n)$, which is associated to each energy $E_n$ in the discrete spectrum, and the Hilbert space $\widehat{\cal H}(E)$, which is associated to each energy $E$ in the continuous spectrum, are realized by the Hilbert space of complex numbers $\mathbb C$. On $\widehat{\cal H}$, the operator $H$ acts as the multiplication operator, \begin{equation} Hf \longmapsto UHf\equiv \sum_{n=1}^N E_n(\phi _n,f_{\rm b}) + \{ E\widehat{f}_{\rm c}(E) \} \, , \quad f\in {\cal D}(H) \, . \end{equation} The scalar product on $\widehat{\cal H}$ can be written as \begin{equation} \left( \widehat{f},\widehat{g} \right) _{\widehat{\cal H}}= \sum_{n=1}^N (f_{\rm b},\phi _n)(\phi _n,g_{\rm b})+ \int_0^{\infty}dE\, \overline{\widehat{f}_{\rm c}(E)} \, \widehat{g}_{\rm c}(E) \, , \end{equation} where $\widehat{f}_{\rm b}=\{(\phi _1,f_{\rm b}), \ldots , (\phi _N,f_{\rm b})\}$ and $\widehat{g}_{\rm b}=\{(\phi _1,g_{\rm b}), \ldots , (\phi _N,g_{\rm b})\}$. Therefore, $U$ induces a direct integral decomposition of the Hilbert space associated to the Hamiltonian $H$ (see~\cite{VON2,GELFAND}). In particular, we can write the eigenfunction expansions of any element of the Hilbert spaces $L^2([0,\infty ),dr )$ and $L^2([0,\infty ),dE)$ in terms of the eigensolutions $\phi _n(r)$ and $\phi (r;E)$ of $h$, \begin{equation} f(r)=f_{\rm b}(r)+f_{\rm c}(r)= \sum_{n=1}^N(\phi _n,f)\phi_n(r) + \int_0^{\infty}dE\, \widehat{f}(E) \phi (r;E) \, , \label{eigenfunex1} \end{equation} \begin{equation} \widehat{f}\equiv \widehat{f}_{\rm b}+\widehat{f}_{\rm c}(E)= \{ (\phi _n,f)\}_{n=1}^N + \int_0^{\infty}dr\, f(r) \overline{\phi (r;E)} \, . \label{eigenfunex2} \end{equation} \def\thesection{\arabic{section}} \section{Rigged Hilbert Space} \def\thesection{\arabic{section}} \setcounter{equation}{0} \label{sec:constRHS} As explained in the Introduction, the Hilbert space framework (in particular, the direct integral decomposition) is not sufficient for the purposes of Quantum Mechanics: an extension of the Hilbert space to the RHS is needed. To construct the RHS, we shall consider the discrete and the continuous spectra separately. \subsection{Construction of the Rigged Hilbert Space} \label{sec:RotD} First, we construct the RHS associated to the discrete spectrum. In this case, the Hilbert space ${\cal H}_{\rm b}$ is finite dimensional. Therefore, all the difficulties exposed in the Introduction do not arise. The ``Gelfand triplet'' that corresponds to the discrete spectrum consists of three copies of the Hilbert space ${\cal H}_{\rm b}$, \begin{equation} \mathbf \Phi _{\rm b}=\mathcal{H}_{\rm b}=\mathbf \Phi _{\rm b}^{\times}\, . \label{RHSdisc} \end{equation} The eigenket $|E_n)$ associated to any energy $E_n$ in the discrete spectrum is defined by \begin{eqnarray} |E_n) :\mathbf \Phi _{\rm b}& \longmapsto & {\mathbb C} \nonumber \\ \varphi _{\rm b}& \longmapsto & (\varphi _{\rm b}|E_n) :=(\varphi _{\rm b},\phi _n)= \int_0^{\infty}dr \, \overline{\varphi _{\rm b}(r)}\phi _n(r) \, . \label{bneedefinitionket} \end{eqnarray} Clearly, the function $|E_n)$ is an antilinear functional over $\mathbf \Phi _{\rm b}$, i.e., $|E_n)\in \mathbf \Phi _{\rm b}^{\times}$. Besides being an eigenvector of the Hamiltonian in the usual sense, $|E_n)$ is also an eigenvector of $H_{\rm b}$ in the RHS sense of Eq.~(\ref{idef}), \begin{equation} (\varphi _{\rm b}|H_{\rm b}^{\times}|E_n)=E_n(\varphi _{\rm b}|E_n) \, , \quad \forall \varphi _{\rm b} \in \mathbf \Phi _{\rm b} \, . \end{equation} Although it is not necessary to use the RHS formalism when dealing with the discrete spectrum, we have constructed the ``RHS'' (\ref{RHSdisc}) and the eigenket (\ref{bneedefinitionket}) in order to make a parallel to the continuous case. Now, we construct the RHS associated to the continuous spectrum. The first step is to make all the powers of the Hamiltonian well-defined. In order to do so, we construct the maximal invariant subspace ${\cal D}_{\rm c}$ of the operator $H_{\rm c}$, \begin{equation} {\cal D}_{\rm c}:= \bigcap _{n=0}^{\infty}{\cal D}(H_{\rm c}^n) \, . \label{misus} \end{equation} The space ${\cal D}_{\rm c}$ is the largest subspace of ${\cal D}(H_{\rm c})$ that remains stable under the action of the Hamiltonian $H_{\rm c}$ and of its powers. Recalling that $h$ denotes the formal differential operator (\ref{doh}), it is easy to check that \begin{eqnarray} {\cal D}_{\rm c}=\{ \varphi _{\rm c}\in {\cal H}_{\rm c} \, | && \ h^n\varphi _{\rm c}(r)\in {\cal H}_{\rm c},\ h^n\varphi _{\rm c}(0)=0, \ \varphi _{\rm c}^{(n)}(a)=\varphi _{\rm c}^{(n)}(b)=0, \ n=0,1,2,\ldots ; \nonumber \\ && \varphi _{\rm c}(r) \in C^{\infty}([0,\infty)) \} \, . \label{mainisexi} \end{eqnarray} The second step is to find the subspace $\mathbf \Phi _{\rm c}\subset {\cal D}_{\rm c}$ on which the eigenkets $|E\rangle$ of Eq.~(\ref{Iceigenkets}) act as antilinear functionals. Those eigenkets are to be defined as integral operators whose kernel is the eigenfunction $\phi (r;E)$, \begin{eqnarray} |E\rangle :\mathbf \Phi _{\rm c}& \longmapsto & {\mathbb C} \nonumber \\ \varphi _{\rm c}& \longmapsto & \langle \varphi _{\rm c}|E\rangle := \int_0^{\infty}dr\, \overline{\varphi _{\rm c}(r)}\phi (r;E) =\overline{(U_{\rm c}\varphi _{\rm c})(E)} \, . \label{definitionket} \end{eqnarray} A necessary condition for $|E\rangle$ to be well-defined is that $\mathbf \Phi _{\rm c}$ be included in ${\cal D}_{\rm c}$. This condition, however, is not sufficient to obtain a continuous functional. In order to make the eigenfunctional $|E\rangle$ continuous, we have to impose further restrictions on the behavior at infinity of the elements of ${\cal D}_{\rm c}$, \begin{equation} \int_0^{\infty}dr \, \left| (r+1)^n(h+1)^m\varphi _{\rm c}(r)\right| ^2<\infty \, , \quad n,m=0,1,2,\ldots \label{condcodogis} \end{equation} The imposition of (\ref{condcodogis}) upon the space ${\cal D}_{\rm c}$ yields \begin{equation} {\mathbf \Phi}_{\rm c}=\{ \varphi _{\rm c}\in {\cal D}_{\rm c} \, | \ \int_0^{\infty}dr \, \left| (r+1)^n(h+1)^m\varphi _{\rm c}(r)\right| ^2<\infty, \quad n,m=0,1,2,\ldots \} \, . \end{equation} On $\mathbf \Phi _{\rm c}$, we define the family of norms \begin{equation} \| \varphi _{\rm c}\| _{n,m} := \sqrt{\int_0^{\infty}dr \, \left| (r+1)^n(h+1)^m\varphi _{\rm c}(r)\right| ^2} \, , \quad n,m=0,1,2,\ldots \label{nmnorms} \end{equation} These norms can be used to define a countably normed topology $\tau _{\mathbf \Phi _{\rm c}}$ on $\mathbf \Phi _{\rm c}$ (see~\cite{GELFAND}), \begin{equation} \varphi _{{\rm c},\alpha}\, \mapupdown{\tau_{\mathbf \Phi _{\rm c}}}{\alpha \to \infty} \, \varphi _{\rm c}\quad {\rm iff} \quad \| \varphi _{{\rm c},\alpha }-\varphi _{\rm c}\| _{n,m} \, \mapupdown{}{\alpha \to \infty}\, 0 \, , \quad n,m=0,1,2, \ldots \end{equation} Once we have constructed the space $\mathbf \Phi _{\rm c}$, we can construct its topological dual $\mathbf \Phi _{\rm c}^{\times}$ as the space of $\tau _{\mathbf \Phi _{\rm c}}$-continuous antilinear functionals on $\mathbf \Phi _{\rm c}$ (see~\cite{GELFAND}) and therewith the RHS corresponding to the continuous part of the spectrum, \begin{equation} \mathbf \Phi _{\rm c}\subset {\cal H} _{\rm c}\subset \mathbf \Phi _{\rm c}^{\times} \, . \label{RHSCONT} \end{equation} In order to show that the RHS (\ref{RHSCONT}) is what we are looking for, we need to prove the following proposition (the proof can be found in Appendix~\ref{sec:appendixB}): \vskip0.5cm {\bf Proposition~1} \quad The triplet of spaces (\ref{RHSCONT}) is a Rigged Hilbert Space, and it satisfies all the requirements demanded in the Introduction. More specifically, \vskip0.3cm (i) The quantities (\ref{nmnorms}) fulfill the conditions to be a norm. \vskip0.3cm (ii) The space $\mathbf \Phi _{\rm c}$ is stable under the action of $H_{\rm c}$, and $H_{\rm c}$ is $\tau _{\mathbf \Phi _{\rm c}}$-continuous. \vskip0.3cm (iii) The ket $|E\rangle$ of Eq.~(\ref{definitionket}) is a well-defined antilinear functional on $\mathbf \Phi _{\rm c}$, i.e., $|E\rangle$ belongs to $\mathbf \Phi _{\rm c}^{\times}$. \vskip0.3cm (iv) The ket $|E\rangle$ is a generalized eigenvector of $H_{\rm c}$, \begin{equation} H_{\rm c}^{\times}|E\rangle=E|E\rangle \, , \end{equation} i.e., \begin{equation} \langle \varphi _{\rm c}|H_{\rm c}^{\times}|E\rangle= \langle H_{\rm c}^{\dagger}\varphi _{\rm c} |E\rangle = E\langle \varphi _{\rm c}|E\rangle \, , \quad \forall \varphi _{\rm c}\in \mathbf \Phi _{\rm c} \, . \label{nafegenphis} \end{equation} \vskip0.5cm (This proposition is a generalization to infinite dimensional spaces of the results that hold in finite dimensional spaces.) We now combine the RHSs (\ref{RHSdisc}) and (\ref{RHSCONT}) corresponding to the discrete and continuous spectrum into the RHS of the square well-barrier potential, \begin{equation} \mathbf \Phi \subset {\cal H} \subset \mathbf \Phi ^{\times} \, , \label{ABRHSCONT} \end{equation} where \begin{equation} \mathbf \Phi :=\mathbf \Phi _{\rm b}\oplus \mathbf \Phi _{\rm c} \, ; \qquad {\cal H}:={\cal H}_{\rm b}\oplus {\cal H}_{\rm c} \, ; \qquad \mathbf \Phi ^{\times} :=\mathbf \Phi _{\rm b}^{\times} \oplus \mathbf \Phi _{\rm c}^{\times} \, . \end{equation} On the space $\mathbf \Phi$, all the expectation values of the Hamiltonian and all the algebraic operations involving $H$ are well-defined, and the eigenvalue equations (\ref{Ieigenkets}) hold. The kets $|E_n)$ and $|E\rangle$ satisfy the normalization (\ref{Ideltanorintro}). Since the spaces $\mathbf \Phi _{\rm b}$ and $\mathbf \Phi _{\rm c}$ are orthogonal to each other, we also have that \begin{equation} (\varphi|E_n)=(\varphi _{\rm b}|E_n) \, ; \quad \langle \varphi|E\rangle =\langle \varphi _{\rm c}|E\rangle \, . \end{equation} As we shall see in the next section, the kets $|E_n)$ and $|E\rangle$ form a complete basis system. Before finishing this section, we would like to remark that $\phi (r;E)$ is \emph{not} the same object as the ket $|E\rangle$ of Eq.~(\ref{definitionket}). The function $\phi (r;E)$ is an eigenfunction of the formal differential operator $h$, whereas the ket $|E\rangle$ is a generalized eigenvector of the Hamiltonian. The eigenket $|E\rangle$ is defined by the eigenfunction $\phi (r;E)$ \emph{and} by the space of test functions $\mathbf \Phi _{\rm c}$ on which it acts as an antilinear functional. \subsection{Dirac Basis Vector Expansion} In the Introduction, we mentioned that in Quantum Mechanics it is assumed that the eigenkets of the Hamiltonian form a complete basis system that can be used to expand any physical wave function $\varphi$ as in Eq.~(\ref{IDiracbsis}). This expansion is derived in the present section. That derivation consists of the restriction of the eigenfunction expansion (\ref{eigenfunex1}) to the space $\mathbf \Phi$. If we denote $\langle r|\varphi \rangle \equiv \varphi (r)$, $\langle r|E_n)\equiv \phi _n(r)$, and $\langle r|E\rangle \equiv \phi (r;E)$, and if we define the action of the bra $\langle E|$ on the wave function $\varphi$ by $\langle E| \varphi \rangle := \overline{\langle \varphi|E\rangle}$, then Eq.~(\ref{eigenfunex1}) can be written as \begin{equation} \langle r|\varphi \rangle =\sum_{n=0}^{\infty} \langle r|E_n) (E_n|\varphi )+ \int_0^{\infty}dE \, \langle r|E \rangle \langle E|\varphi \rangle \, , \quad \varphi \in \mathbf \Phi \, . \label{inveqDva} \end{equation} This is the Dirac basis vector expansion of the square well-barrier potential. Although the eigenfunction expansion (\ref{eigenfunex1}) is valid for every element of the Hilbert space, Dirac basis vector expansion (\ref{inveqDva}) is only valid for functions $\varphi \in \mathbf \Phi$, because only those functions fulfill both \begin{equation} \overline{\widehat{\varphi}_{\rm c}(E)}=\langle \varphi |E\rangle \end{equation} and \begin{equation} \langle \varphi |H^{\times}|E\rangle = \langle H^{\dagger}\varphi |E\rangle = E \langle \varphi|E\rangle \, . \end{equation} For the sake of completeness, we include the Nuclear Spectral Theorem~\cite{GELFAND}, which is usually referred to as the mathematical justification of the heuristic Dirac basis vector expansion~\cite{ANTOINE,B60,ROBERTS,GELFAND,GALINDO,MELSH}. Instead of using the general proof of Ref.~\cite{GELFAND}, we prove that theorem using the machinery of the Sturm-Liouville theory (our proof can be found in Appendix~\ref{sec:appendixB}). \vskip0.5cm {\bf Proposition~2} (Nuclear Spectral Theorem) \quad Let \begin{equation} {\mathbf \Phi} \subset L^2([0,\infty ),dr)\subset {\mathbf \Phi}^{\times} \end{equation} be the RHS of the square well-barrier Hamiltonian $H$ such that $\mathbf \Phi$ remains invariant under $H$ and $H$ is a $\tau _{\mathbf \Phi}$-continuous operator on $\mathbf \Phi$. Then, for each energy in the spectrum of $H$ there is a generalized eigenvector such that \begin{mathletters} \begin{eqnarray} &H|E_n) =E_n|E_n) \, , \quad &E_n \in \{ E_1, \ldots ,E_N \} \, , \\ &H^{\times}|E\rangle =E|E\rangle \, , \quad &E\in [0,\infty ) \, , \end{eqnarray} \end{mathletters} and such that \begin{equation} (\varphi ,\psi )=\sum_{n=1}^N(\varphi |E_n)(E_n|\psi )+ \int_0^{\infty}dE\, \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in \mathbf \Phi \, , \label{GMT1} \end{equation} and \begin{equation} (\varphi ,H^m \psi )=\sum_{n=1}^NE_n^m(\varphi |E_n)(E_n|\psi )+ \int_0^{\infty} dE \, E^m \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in {\mathbf \Phi} \, , m=1,2,\ldots \label{GMT2} \end{equation} \vskip0.5cm Thus this theorem allows us to write the scalar product (\ref{GMT1}) of any two functions $\varphi ,\psi$ of $\mathbf \Phi$ and the matrix elements (\ref{GMT2}) in terms of the action of the kets $|E_n)$ and $|E\rangle$ on $\varphi ,\psi$. \subsection{Energy and Momentum Representations of the Rigged Hilbert Space} In this section, we construct the energy representation of the RHS (\ref{ABRHSCONT}) by applying to it the unitary operator $U$ of Eq.~(\ref{dirintdec}). As done throughout this paper, we study the discrete and the continuous case separately. In the discrete case (\ref{RHSdisc}), we have already seen that the energy representation of ${\cal H}_{\rm b}$ is ${\mathbb C}^N$. Hence the energy representation of the RHS (\ref{RHSdisc}) consists of three copies of the space ${\mathbb C}^N$. The image of the eigenket $|E_n)$ under $U_{\rm b}^{\times}$ is the $N$-tuple that has zeros everywhere but at the $n$-th position, $\{ \delta _{in} \}_{i=1}^N$. In the energy representation, the Hamiltonian $H_{\rm b}$ acts as the diagonal matrix (\ref{diagonaHb}). In the continuous case (\ref{RHSCONT}), we have already shown that in the energy representation the Hamiltonian $H_{\rm c}$ acts as the multiplication operator. The energy representation of the space $\mathbf \Phi _{\rm c}$ is defined as \begin{equation} \widehat{\mathbf \Phi}_{\rm c}:= U_{\rm c}\mathbf \Phi _{\rm c}\, . \end{equation} The space $\widehat{\mathbf \Phi}_{\rm c}$ is a linear subspace of $L^2([0,\infty ),dE)$. In order to endow $\widehat{\mathbf \Phi}_{\rm c}$ with a topology $\tau _{\widehat{\mathbf \Phi}_{\rm c}}$, we carry the topology on $\mathbf \Phi _{\rm c}$ into $\widehat{\mathbf \Phi}_{\rm c}$, \begin{equation} \tau _{\widehat{\mathbf \Phi}_{\rm c}}:= U_{\rm c}\tau _{\mathbf \Phi _{\rm c}} \, . \end{equation} With this topology, the space $\widehat{\mathbf \Phi}_{\rm c}$ is a linear topological space. If we denote the dual space of $\widehat{\mathbf \Phi}_{\rm c}$ by $\widehat{\mathbf \Phi}_{\rm c}^{\times}$, then we have \begin{equation} U_{\rm c}^{\times}{\mathbf \Phi}_{\rm c}^{\times}= (U_{\rm c}{\mathbf \Phi}_{\rm c})^{\times}= \widehat{\mathbf \Phi}_{\rm c}^{\times} \, . \end{equation} From the following chain of equalities: \begin{eqnarray} \langle \widehat{\varphi }_{\rm c}|U_{\rm c}^{\times}|E\rangle &=& \langle U_{\rm c}^{-1}\widehat{\varphi}_{\rm c}|E\rangle \nonumber \\ &=& \int_0^{\infty}dr\, \overline{\varphi _{\rm c}(r)}\phi (r;E) \nonumber \\ &=& \overline{\widehat{\varphi}_{\rm c}(E)} \, , \end{eqnarray} it follows that the energy representation of the Dirac ket $|E\rangle$, that we denote by $|\widehat{E}\rangle$, is the antilinear Schwartz delta functional, i.e., $|\widehat{E}\rangle$ is the antilinear functional that associates to each function $\widehat{\varphi}_{\rm c}$ the complex conjugate of its value at the point $E$. In order to summarize all these results, it is very helpful to show the different realizations of the RHS through the following diagram: \begin{equation} \begin{array}{cccccccccc} H; & \varphi & \ & \mathbf \Phi_{\rm b}\oplus \mathbf \Phi_{\rm c} & \subset & {\cal H}_{\rm b}\oplus {\cal H}_{\rm c} & \subset &\mathbf \Phi_{\rm b}^{\times}\oplus \mathbf \Phi_{\rm c}^{\times} & \ & |E_n)\oplus |E\rangle \nonumber \\ & & \ & U_{\rm b} \downarrow \oplus \downarrow U_{\rm c} & & U_{\rm b}\downarrow \oplus \downarrow U_{\rm c} & &U_{\rm b}^{\times} \downarrow \oplus \downarrow U_{\rm c}^{\times} & \ & \nonumber \\ \widehat{H}; & \widehat{\varphi} & \ & \widehat{\mathbf \Phi}_{\rm b}\oplus \widehat{\mathbf \Phi}_{\rm c}& \subset & \widehat{{\cal H}}_{\rm b}\oplus \widehat{{\cal H}}_{\rm c} & \subset & \widehat{\mathbf \Phi}_{\rm b}^{\times}\oplus \widehat{\mathbf \Phi}_{\rm c}^{\times} & \ & |\widehat{E}_n)\oplus |\widehat{E}\rangle \\ \end{array} \label{diagramsavp} \end{equation} On the top line of the diagram (\ref{diagramsavp}), we have the RHS, the Hamiltonian, the wave functions and the Dirac kets in the position representation. On the bottom line, we have their energy representation counterparts. Since every function we have computed (eigenfunctions of the Schr\"odinger differential operator, Green function, etc.) depends on the square root of the energy rather than on the energy itself, the momentum $k$ is a more convenient variable, especially for the continuous case. The momentum $k$ is defined by \begin{equation} k:=\sqrt{\frac{2m}{\hbar ^2} \, E} \, . \label{momenuks} \end{equation} In terms of $k$, the $\delta$-normalized eigensolution of the differential operator (\ref{doh}) reads \begin{equation} \langle r|k\rangle = \left[2\pi {\cal J}_3(k){\cal J}_4(k) \right]^{-1/2} \chi (r;k) \, . \label{continuoseign} \end{equation} This eigensolution can be used to obtain the unitary operator $V_{\rm c}$ that transforms from the position into the momentum representation, \begin{equation} \widehat{f}_{\rm c}(k)=(V_{\rm c}f_{\rm c})(k)= \int_0^{\infty}dr\, f_{\rm c}(r) \overline{\langle r|k\rangle}\, , \quad f_{\rm c}\in {\cal H}_{\rm c} \, . \label{Vcontinuoseign} \end{equation} (The operator $V_{\rm b}$ corresponding to the discrete case is $U_{\rm b}$.) On the $k$-representation, the (continuous part of the) Hamiltonian acts as multiplication by $k^2$. To each $k\in [0,\infty )$, there corresponds an eigenket $|k\rangle$ that acts on $\mathbf \Phi _{\rm c}$ as the following integral operator: \begin{equation} \langle \varphi _{\rm c}|k\rangle :=\int_0^{\infty}dr\, \langle \varphi _{\rm c}|r\rangle \langle r|k\rangle= \overline{(V_{\rm c}\varphi _{\rm c})(k)}\, , \quad \varphi _{\rm c} \in \mathbf \Phi _{\rm c} \, . \end{equation} This eigenket is a generalized eigenvector of the Hamiltonian corresponding to the eigenvalue $k^2$. In this way, we can construct the $k$-representation of all the objects of the diagram (\ref{diagramsavp}). If we express our functions in terms of $k$, there is no need to distinguish different regions in the $E$-plane, because in the $k$-plane the resolvent and the $S$-matrix do not have a cut. There is no need to work with a particular branch of the square root either, because Eq.~(\ref{momenuks}) provides a Riemann surface in a natural way. For instance, we can define the Jost function \begin{equation} {\cal J}(k):={\cal J}_+(k) \, , \quad k\in {\mathbb C} \, , \label{junfcion} \end{equation} the eigensolution \begin{equation} f(r;k):=\Theta _+(r;k) \, , \quad k\in {\mathbb C} \, , \end{equation} and the Green function \begin{equation} G(r,r';k):=-\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \frac{\chi (r_<;k)f(r_>;k)}{{\cal J}(k)} \, , \quad k\in {\mathbb C} \, , \label{grenkfunc} \end{equation} where $r_<$, $r_>$ refer to the smaller and to the bigger of $r$ and $r'$, respectively. Then, as $k$ approaches the real positive $k$ axis, or as $E$ approaches the right-hand cut from above, ${\cal J}(k)$, $f(r;k)$ and $G(r,r';k)$ become ${\cal J}_+(k)$, $\Theta _+(r;k)$ and $G^+(r,r';k)$, respectively; as $k$ approaches the negative real axis (from above), or $E$ the right-hand cut from below, ${\cal J}(k)$, $f(r;k)$ and $G(r,r';k)$ become ${\cal J}_-(k)$, $\Theta _-(r;k)$ and $G^-(r,r';k)$, respectively~\cite{NEWTON}. Also, the values that the functions of Eqs.~(\ref{junfcion})-(\ref{grenkfunc}) take on the second sheet of the Riemann surface are already specified (the Sturm-Liouville theory deals only with energies in the first sheet). This is particularly useful when studying resonance energies, which lie on the second sheet~\cite{MONDRAGON1,MONDRAGON2,MONDRAGON3}. \def\thesection{\arabic{section}} \section{Conclusions} \def\thesection{\arabic{section}} \setcounter{equation}{0} \label{sec:conclusions} We have shown that the solutions of the Schr\"odinger equation corresponding to the square well-barrier potential fall in the Rigged Hilbert Space \begin{equation} \mathbf \Phi _{\rm b} \oplus \mathbf \Phi _{\rm c} \subset {\cal H}_{\rm b} \oplus {\cal H}_{\rm c} \subset \mathbf \Phi _{\rm b}^{\times} \oplus \mathbf \Phi _{\rm c}^{\times} \, . \label{rihscon} \end{equation} The spectrum of the Hamiltonian has a discrete part $\{ E_1, \ldots ,E_N \}$ and a continuous part $[0,\infty)$. For each energy in the spectrum of $H$, we have constructed an eigenket that is an eigenvector of $H$. If the energy $E_n$ belongs to the discrete spectrum, its corresponding eigenket $|E_n)$, which is given by Eq.~(\ref{bneedefinitionket}), is an element of $\mathbf \Phi _{\rm b}^{\times}$. If the energy $E$ belongs to the continuous spectrum, its corresponding eigenket $|E\rangle$, which is given by Eq~(\ref{definitionket}), is an element of $\mathbf \Phi _{\rm c}^{\times}$. Thus the RHS treats the discrete and the continuous spectra independently and on the same footing. Each element $\varphi$ of $\mathbf \Phi _{\rm b} \oplus \mathbf \Phi _{\rm c}$ can be expanded by the eigenkets of the Hamiltonian as in Eq.~(\ref{inveqDva}). The elements of $\mathbf \Phi _{\rm b} \oplus \mathbf \Phi _{\rm c}$ are the only square normalizable functions for which such an expansion is possible. The expectation values and the uncertainties of the Hamiltonian are well defined quantities in each element $\varphi$ of $\mathbf \Phi _{\rm b} \oplus \mathbf \Phi _{\rm c}$. Therefore, the RHS (\ref{rihscon}) contains all the physically meaningful solutions of the Schr\"odinger equation---the physical wave functions are included in $\mathbf \Phi _{\rm b} \oplus \mathbf \Phi _{\rm c}$, whereas the monoenergetic solutions of the time independent Schr\"odinger equation are included in $\mathbf \Phi _{\rm b}^{\times} \oplus \mathbf \Phi _{\rm c}^{\times}$. Although illustrated within the example of the square well-barrier potential, these conclusions hold in general: the solutions of the Schr\"odinger equation fall in a RHS rather than just in a Hilbert space. \section*{Acknowledgments} The author thanks C.~Koeninger, N.~L.~Harshman, and Profs.~A.~Bohm, M.~Gadella and A.~Galindo for invaluable comments. Financial support from the U.E.~TMR Contract number ERBFMRX-CT96-0087 ``The Physics of Quantum Information'' is gratefully acknowledged. \appendix \def\thesection{\Alph{section}} \section{List of Auxiliary Functions} \setcounter{equation}{0} \label{sec:appendix} If we define \begin{equation} \widetilde{k} :=\sqrt{-\frac{2m}{\hbar ^2}\, E} \, ; \quad \widetilde{Q}_1:=\sqrt{-\frac{2m}{\hbar ^2}\, (E+V_1)} \, ; \quad \widetilde{Q}_2:=\sqrt{-\frac{2m}{\hbar ^2}\, (E-V_2)} \, . \end{equation} then the coefficients in Eq.~(\ref{tildechifunction}) are given by \begin{mathletters} \begin{eqnarray} &&\widetilde{{\cal J}}_1(E)\equiv \widetilde{{\cal J}}_1(\widetilde{k})= \frac{i}{4}e^{-\widetilde{Q}_2a} \left[ \left(1+\frac{\widetilde{Q}_1}{\widetilde{Q}_2}\right) e^{\widetilde{Q}_1a} - \left(1-\frac{\widetilde{Q}_1}{\widetilde{Q}_2} \right) e^{-\widetilde{Q}_1a} \right] , \hskip1cm \\ && \widetilde{{\cal J}}_2(E)\equiv\widetilde{{\cal J}}_2(\widetilde{k})= \frac{i}{4}e^{\widetilde{Q}_2a} \left[ \left(1-\frac{\widetilde{Q}_1}{\widetilde{Q}_2}\right) e^{\widetilde{Q}_1a} - \left(1+\frac{\widetilde{Q}_1}{\widetilde{Q}_2} \right) e^{-\widetilde{Q}_1a} \right] , \hskip1cm \\ && \widetilde{{\cal J}}_3(E)\equiv \widetilde{{\cal J}}_3(\widetilde{k})= \frac{1}{2}e^{-\widetilde{k}b} \left[ \left(1+\frac{\widetilde{Q}_2}{\widetilde{k}}\right) e^{\widetilde{Q}_2b}\widetilde{{\cal J}}_1(\widetilde{k})+ \left(1-\frac{\widetilde{Q}_2}{\widetilde{k}} \right) e^{-\widetilde{Q}_2b}\widetilde{{\cal J}}_2(\widetilde{k}) \right] , \hskip1cm \\ &&\widetilde{{\cal J}}_4(E)\equiv \widetilde{{\cal J}}_4(\widetilde{k})= \frac{1}{2}e^{\widetilde{k}b} \left[ \left(1-\frac{\widetilde{Q}_2}{\widetilde{k}} \right) e^{\widetilde{Q}_2b}\widetilde{{\cal J}}_1(\widetilde{k})+ \left(1+\frac{\widetilde{Q}_2}{\widetilde{k}} \right) e^{-\widetilde{Q}_2b}\widetilde{{\cal J}}_2(\widetilde{k}) \right] , \hskip1cm \end{eqnarray} \label{tildejfunc} \end{mathletters} and the coefficients in Eq.~(\ref{tildethetfunc}) by \begin{mathletters} \begin{eqnarray} && \widetilde{{\cal A}}_3(E)\equiv \widetilde{{\cal A}}_3(\widetilde{k})= \frac{1}{2}e^{-\widetilde{Q}_2b} \left(1-\frac{\widetilde{k}}{\widetilde{Q}_2}\right) e^{-\widetilde{k}b} , \hskip1.5cm \\ && \widetilde{{\cal A}}_4(E)\equiv \widetilde{{\cal A}}_4(\widetilde{k})= \frac{1}{2}e^{\widetilde{Q}_2b} \left(1+\frac{\widetilde{k}} {\widetilde{Q}_2} \right) e^{-\widetilde{k}b} , \hskip1.5cm \\ && \widetilde{{\cal A}}_1(E)\equiv \widetilde{{\cal A}}_1(\widetilde{k})= \frac{1}{2}e^{-\widetilde{Q}_1a} \left[ \left(1+\frac{\widetilde{Q}_2}{\widetilde{Q}_1}\right) e^{\widetilde{Q}_2a} \widetilde{{\cal A}}_3(\widetilde{k})+ \left(1-\frac{\widetilde{Q}_2}{\widetilde{Q}_1} \right) e^{-\widetilde{Q}_2a} \widetilde{{\cal A}}_4(\widetilde{k}) \right] , \hskip1.5cm \\ && \widetilde{ {\cal A}}_2(E)\equiv \widetilde{{\cal A}}_2(\widetilde{k})= \frac{1}{2}e^{\widetilde{Q}_1a} \left[ \left(1-\frac{\widetilde{Q}_2}{\widetilde{Q}_1}\right) e^{\widetilde{Q}_2a} \widetilde{{\cal A}}_3(\widetilde{k})+ \left(1+\frac{\widetilde{Q}_2}{\widetilde{Q}_1} \right) e^{-\widetilde{Q}_2a} \widetilde{{\cal A}}_4(\widetilde{k}) \right] . \hskip1.5cm \end{eqnarray} \label{tildeAfunc} \end{mathletters} If we define \begin{equation} k:=\sqrt{\frac{2m}{\hbar ^2} \, E} \, ; \quad Q_1:=\sqrt{\frac{2m}{\hbar ^2} \, (E+V_1)} \, ; \quad Q_2:=\sqrt{\frac{2m}{\hbar ^2} \, (E-V_2)} \, . \end{equation} then the functions ${\cal J}(E)$ of Eq.~(\ref{chi}) are given by \begin{mathletters} \begin{eqnarray} &&{\cal J}_1(E)\equiv {\cal J}_1(k)= \frac{1}{2}e^{-iQ_2a} \left( \sin (Q_1a)+\frac{Q_1}{iQ_2}\cos (Q_1a) \right) , \hskip1.5cm \\ &&{\cal J} _2 (E)\equiv {\cal J}_2(k)= \frac{1}{2}e^{iQ_2a} \left( \sin (Q_1a)-\frac{Q_1}{iQ_2} \cos (Q_1a) \right) , \hskip1.5cm \\ &&{\cal J}_3(E)\equiv {\cal J}_3(k)= \frac{1}{2}e^{-ikb} \left[ \left(1+ \frac{Q_2}{k} \right) e^{iQ_2b}{\cal J}_1(k)+ \left(1-\frac{Q_2}{k} \right) e^{-iQ_2b}{\cal J}_2(k) \right] , \hskip1.5cm \\ &&{\cal J}_4(E)\equiv {\cal J}_4(k)= \frac{1}{2}e^{ikb} \left[ \left(1-\frac{Q_2}{k} \right) e^{iQ_2b}{\cal J} _1(k)+ \left(1+\frac{Q_2}{k}\right) e^{-iQ_2b}{\cal J}_2(k) \right] , \hskip1.5cm \end{eqnarray} \label{Jfunction} \end{mathletters} and the functions ${\cal A}^+(E)$ of Eq.~(\ref{theta+fun}) by \begin{mathletters} \begin{eqnarray} &&{\cal A}_3^+(E)\equiv {\cal A}_3^+(k)= \frac{1}{2}e^{-iQ_2b} \left(1+\frac{k}{Q_2}\right) e^{ikb} , \hskip1.3cm \\ &&{\cal A}_4^+(E)\equiv {\cal A}_4^+(k)= \frac{1}{2}e^{iQ_2b} \left(1-\frac{k}{Q_2} \right) e^{ikb} , \hskip1.3cm \\ &&{\cal A}_1^+(E)\equiv {\cal A}_1^+(k)= \frac{1}{2}e^{-iQ_1a} \left[ \left(1+\frac{Q_2}{Q_1}\right) e^{iQ_2a}{\cal A}_3^+(k)+ \left(1-\frac{Q_2}{Q_1} \right) e^{-iQ_2a} {\cal A}_4^+(k) \right] , \hskip1.3cm \\ &&{\cal A}_2^+(E)\equiv {\cal A}_2^+(k)= \frac{1}{2}e^{iQ_1a} \left[ \left(1-\frac{Q_2}{Q_1}\right) e^{iQ_2a}{\cal A}_3^+(k)+ \left(1+\frac{Q_2}{Q_1} \right) e^{-iQ_2a} {\cal A}_4^+(k) \right] . \hskip1.3cm \end{eqnarray} \label{A+functions} \end{mathletters} The coefficients in Eq.~(\ref{thetafun-}) are given by \begin{mathletters} \begin{eqnarray} &&{\cal A}_3^-(E)\equiv {\cal A}_3^-(k)= \frac{1}{2}e^{-iQ_2b} \left(1-\frac{k}{Q_2}\right) e^{-ikb} , \hskip1.2cm \\ &&{\cal A}_4^-(E)\equiv {\cal A}_4^-(k)= \frac{1}{2}e^{iQ_2b} \left(1+\frac{k}{Q_2} \right) e^{-ikb} , \hskip1.2cm \\ &&{\cal A}_1^-(E)\equiv {\cal A}_1^-(k)= \frac{1}{2}e^{-iQ_1a} \left[ \left(1+\frac{Q_2}{Q_1}\right) e^{iQ_2a}{\cal A}_3^-(k)+ \left(1-\frac{Q_2}{Q_1} \right) e^{-iQ_2a}{\cal A}_4^-(k) \right] , \hskip1.2cm \\ &&{\cal A}_2^-(E)\equiv {\cal A}_2^-(k)= \frac{1}{2}e^{iQ_1a} \left[ \left(1-\frac{Q_2}{Q_1}\right) e^{iQ_2a}{\cal A}_3^-(k)+ \left(1+\frac{Q_2}{Q_1} \right) e^{-iQ_2a}{\cal A}_4^-(k) \right] . \hskip1.2cm \end{eqnarray} \label{A-functions} \end{mathletters} The coefficients in Eq.~(\ref{tildesigma1}) are given by \begin{mathletters} \begin{eqnarray} && \widetilde{{\cal B}}_3(E)\equiv \widetilde{{\cal B}}_3(\widetilde{k})= \frac{1}{2}e^{-\widetilde{Q}_2b} \left(1+\frac{\widetilde{k}}{\widetilde{Q}_2}\right) e^{\widetilde{k}b} , \\ && \widetilde{{\cal B}}_4(E)\equiv \widetilde{{\cal B}}_4(\widetilde{k}) = \frac{1}{2}e^{\widetilde{Q}_2b} \left(1-\frac{\widetilde{k}}{\widetilde{Q}_2} \right) e^{\widetilde{k}b} , \\ && \widetilde{{\cal B}}_1(E)\equiv \widetilde{{\cal B}}_1(\widetilde{k})= \frac{1}{2}e^{-\widetilde{Q}_1a} \left[ \left(1+\frac{\widetilde{Q}_2}{\widetilde{Q}_1}\right) e^{\widetilde{Q}_2a} \widetilde{{\cal B}}_3(\widetilde{k})+ \left(1-\frac{\widetilde{Q}_2}{\widetilde{Q}_1} \right) e^{-\widetilde{Q}_2a} \widetilde{{\cal B}}_4(\widetilde{k}) \right] , \\ && \widetilde{ {\cal B}}_2(E)\equiv \widetilde{{\cal B}}_2(\widetilde{k})= \frac{1}{2}e^{\widetilde{Q}_1a} \left[ \left(1-\frac{\widetilde{Q}_2}{\widetilde{Q}_1} \right) e^{\widetilde{Q}_2a} \widetilde{{\cal B}}_3(\widetilde{k})+ \left(1+\frac{\widetilde{Q}_2}{\widetilde{Q}_1} \right) e^{-\widetilde{Q}_2a} \widetilde{{\cal B}}_4(\widetilde{k}) \right] . \end{eqnarray} \label{tildeBfunctions} \end{mathletters} The functions ${\cal C}(E)$ of Eq.~(\ref{sigam2cos}) are given by \begin{mathletters} \begin{eqnarray} &&{\cal C}_1(E)\equiv {\cal C}_1(k)= \frac{1}{2}e^{-iQ_2a} \left( \cos (Q_1a)-\frac{Q_1}{iQ_2} \sin (Q_1a) \right), \\ && {\cal C}_2(E)\equiv {\cal C}_2(k)= \frac{1}{2}e^{iQ_2a} \left( \cos (Q_1a)+\frac{Q_1}{iQ_2} \sin (Q_1a) \right), \\ && {\cal C}_3(E)\equiv {\cal C}_3(k)=\frac{1}{2}e^{-ikb} \left[ \left(1+\frac{Q_2}{k}\right) e^{iQ_2b}{\cal C}_1(k)+ \left(1-\frac{Q_2}{k} \right) e^{-iQ_2b}{\cal C}_2(k) \right] , \\ &&{\cal C}_4(E)\equiv {\cal C}_4(k)=\frac{1}{2}e^{ikb} \left[ \left(1-\frac{Q_2}{k} \right) e^{iQb}{\cal C}_1(k)+ \left(1+\frac{Q_2}{k} \right) e^{-iQb}{\cal C}_2(k) \right] . \end{eqnarray} \label{Cfunctions} \end{mathletters} \def\thesection{\Alph{section}} \section{Proofs of Propositions 1 and 2} \setcounter{equation}{0} \label{sec:appendixB} {\it Proof of Proposition~1.} \vskip0.3cm (i) It is very easy to show that the quantities (\ref{nmnorms}) fulfill the conditions to be a norm, \begin{mathletters} \begin{eqnarray} &&\| \varphi _{\rm c}+\psi _{\rm c} \| _{n,m} \leq \| \varphi _{\rm c} \| _{n,m} + \| \psi _{\rm c} \| _{n,m} \, , \\ && \| \alpha \varphi _{\rm c} \| _{n,m}= |\alpha |\, \| \varphi _{\rm c}\| _{n,m} \, , \\ && \| \varphi _{\rm c} \| _{n,m} \geq 0 \, , \\ && {\rm If }\ \| \varphi _{\rm c} \| _{n,m} =0, \ {\rm then} \ \varphi _{\rm c} =0 \, . \label{homiensi} \end{eqnarray} \end{mathletters} The only condition that is somewhat difficult to prove is (\ref{homiensi}): if $\| \varphi _{\rm c}\| _{n,m}=0$, then \begin{equation} (1+r)^n(h+1)^m\varphi _{\rm c}(r)=0 \, , \end{equation} which yields \begin{equation} (h+1)^m\varphi _{\rm c}(r)=0 \, . \label{homiodhiis} \end{equation} If $m=0$, then Eq.~(\ref{homiodhiis}) implies $\varphi _{\rm c}(r)=0$. If $m=1$, then Eq.~(\ref{homiodhiis}) implies that $-1$ is an eigenvalue of $H_{\rm c}$ whose corresponding eigenvector is $\varphi _{\rm c}$. Since there is no discrete eigenvalue in the continuous part of the spectrum, $\varphi _{\rm c}$ must be the zero vector. The proof for $m>1$ is similar. \vskip0.3cm (ii) In order to see that $H_{\rm c}$ is $\tau _{\mathbf \Phi _{\rm c}}$-continuous, we just have to realize that \begin{eqnarray} \| H_{\rm c}\varphi _{\rm c}\| _{n,m}&=& \| (H_{\rm c}+I)\varphi _{\rm c}-\varphi _{\rm c}\| _{n,m} \nonumber \\ &\leq & \| (H_{\rm c}+I)\varphi _{\rm c}\| _{n,m}+ \| \varphi _{\rm c}\| _{n,m} \nonumber \\ &=&\| \varphi _{\rm c}\| _{n,m+1}+\| \varphi _{\rm c}\| _{n,m} \, . \label{tauphiscont} \end{eqnarray} The stability of $\mathbf \Phi _{\rm c}$ under the action of $H_{\rm c}$ also follows from Eq.~(\ref{tauphiscont}). \vskip0.3cm (iii) From the definition (\ref{definitionket}), it is pretty easy to see that $|E\rangle$ is an antilinear functional. In order to show that $|E\rangle$ is continuous, we define \begin{equation} {\cal M}(E):= \sup _{r\in [0,\infty )} \left| \phi (r;E) \right| \, . \end{equation} Since \begin{eqnarray} |\langle \varphi _{\rm c}|E\rangle | &=&\left| \int_0^{\infty}dr \, \overline{\varphi _{\rm c}(r)}\phi(r;E)\right| \nonumber \\ &\leq & {\cal M}(E) \int _0^{\infty}dr \, |\varphi _{\rm c}(r)| \nonumber \\ &=& {\cal M}(E) \int_0^{\infty}dr \, \frac{1}{1+r} (1+r) |\varphi _{\rm c}(r)| \nonumber \\ &\leq & {\cal M}(E) \left( \int_0^{\infty}dr \, \frac{1}{(1+r)^2} \right) ^{1/2} \left( \int_0^{\infty}dr \, \left| (1+r) \varphi _{\rm c}(r) \right| ^2 \right) ^{1/2} \nonumber \\ &=&{\cal M}(E) \| \varphi _{\rm c} \| _{1,0} \, , \end{eqnarray} the functional $|E\rangle$ is continuous when $\mathbf \Phi _{\rm c}$ is endowed with the $\tau _{\mathbf \Phi _{\rm c}}$ topology. \vskip0.3cm (iv) In order to prove that $|E\rangle$ is a generalized eigenvector of $H_{\rm c}$, we make use of the conditions (\ref{mainisexi}) and (\ref{condcodogis}) satisfied by the elements of $\mathbf \Phi _{\rm c}$, \begin{eqnarray} \langle \varphi _{\rm c}|H_{\rm c}^{\times}|E\rangle &=& \langle H_{\rm c}^{\dagger}\varphi _{\rm c}|E\rangle \nonumber \\ &=& \int_0^{\infty}dr \, \left( -\frac{\hbar ^2}{2m}\frac{d^2}{dr^2}+V(r) \right) \overline{\varphi _{\rm c}(r)} \phi (r;E) \nonumber \\ &=&-\frac{\hbar ^2}{2m} \left[ \frac{d\overline{\varphi _{\rm c}(r)}}{dr} \phi (r;E) \right] _0^{\infty} +\frac{\hbar ^2}{2m} \left[ \overline{\varphi _{\rm c}(r)} \frac{d\phi(r;E)}{dr} \right] _0^{\infty} \nonumber \\ &&+ \int_0^{\infty}dr \, \overline{\varphi _{\rm c}(r)} \left( -\frac{\hbar ^2}{2m}\frac{d^2}{dr^2}+V(r) \right)\phi (r;E) \nonumber \\ &=&E\langle \varphi _{\rm c}|E\rangle \, . \end{eqnarray} \vskip0.5cm {\it Proof of Proposition~2}. Let $\varphi$ and $\psi$ be in $\mathbf \Phi$. Since $U$ of Eq.~(\ref{dirintdec}) is unitary, \begin{equation} (\varphi ,\psi )=(U\varphi ,U\psi )= (\widehat{\varphi}_{\rm b} ,\widehat{\psi}_{\rm b})_{{\cal H}_{\rm b}}+ (\widehat{\varphi}_{\rm c} ,\widehat{\psi}_{\rm c})_{{\cal H}_{\rm c}} \, . \label{Usiuni} \end{equation} The scalar product of $\widehat{\varphi}_{\rm b}$ and $\widehat{\psi}_{\rm b}$ can be written as \begin{equation} (\widehat{\varphi}_{\rm b} ,\widehat{\psi}_{\rm b})_{{\cal H}_{\rm b}}= \sum_{n=1}^N(\varphi|E_n)(E_n|\psi ) \, . \label{discreslsl} \end{equation} The wave functions $\widehat{\varphi}_{\rm c}$ and $\widehat{\psi}_{\rm c}$ are in particular elements of $L^2([0,\infty ),dE)$. Hence their scalar product is well-defined, \begin{equation} (\widehat{\varphi}_{\rm c} ,\widehat{\psi}_{\rm c})_{{\cal H}_{\rm c}}= \int_0^{\infty}dE \, \overline{ \widehat{\varphi}_{\rm c}(E)}\, \widehat{\psi}_{\rm c}(E) \, . \label{sphatvhaps} \end{equation} Since $\varphi _{\rm c}$ and $\psi _{\rm c}$ belong to $\mathbf \Phi _{\rm c}$, the action of each eigenket $|E\rangle$ on them is well-defined, \begin{mathletters} \begin{eqnarray} \langle \varphi _{\rm c}|E\rangle = \overline{ \widehat{\varphi}_{\rm c}(E)} \, ,\\ \langle E|\psi _{\rm c}\rangle =\widehat{\psi}_{\rm c}(E) \, . \end{eqnarray} \label{actionofEpsi} \end{mathletters} By plugging Eq.~(\ref{actionofEpsi}) into Eq.~(\ref{sphatvhaps}) and Eqs.~(\ref{sphatvhaps}) and (\ref{discreslsl}) into Eq.~(\ref{Usiuni}), we get to Eq.~(\ref{GMT1}). 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