Content-Type: multipart/mixed; boundary="-------------0203160616601" This is a multi-part message in MIME format. ---------------0203160616601 Content-Type: text/plain; name="02-126.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-126.comments" Journ. Ref.: Fortschr. Phys. 50 (2002) 2, 185-216 ---------------0203160616601 Content-Type: text/plain; name="02-126.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-126.keywords" Rigged Hilbert Space; Sturm-Liouville theory; continuous spectrum; Schrodinger equation; exactly soluble problems ---------------0203160616601 Content-Type: application/x-tex; name="02fp.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="02fp.tex" \documentstyle[preprint,aps,eqsecnum,amssymb,amsfonts,verbatim]{revtex} \tightenlines \begin{document} \title{\bf Rigged Hilbert Space Treatment of Continuous Spectrum} \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}} \author{A.~Bohm} \address{Center for Particle Physics, The University of Texas at Austin, Austin, Texas 78712-1081, USA} \author{M.~Gadella} \address{Departamento de F\'\i sica Te\'orica, Facultad de Ciencias, 47011 Valladolid, Spain} \date{March 16, 2002} \maketitle \begin{abstract} \noindent The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schr\"odinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave functions. They are also basis vectors that expand any physical wave function in a Dirac basis vector expansion. It is shown that an acceptable physical wave function must fulfill stronger conditions than just square integrability---the space of physical wave functions is not the whole Hilbert space but rather a dense subspace of the Hilbert space. We construct the position and energy representations of the Rigged Hilbert Space generated by the square barrier potential Hamiltonian. We shall also construct the unitary operator that transforms from the position into the energy representation. We shall see that in the energy representation the Dirac kets act as the antilinear Schwartz delta functional. In constructing the Rigged Hilbert Space of the square barrier potential, we will find a systematic procedure to construct the Rigged Hilbert Space of a large class of spherically symmetric potentials. The example of the square barrier potential will also make apparent that the natural framework for the solutions of a Schr\"odinger operator with continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert space. \end{abstract} \pacs{03.65.-w, 02.30.Hq} %PACS numbers: 03.65.-w, 02.30.Hq \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} In the late 1920s, Dirac introduced a new mathematical model of Quantum Mechanics based upon a uniquely smooth and elegant abstract algebra of linear operators defined on an infinite dimensional complex vector space equipped with an inner product norm~\cite{DIRAC}. Dirac's abstract algebraic model of {\it bras} and {\it kets} (from the bracket notation for inner product) proved to be of great heuristic value in the ensuing years, especially in dealing with Hamiltonians whose spectrum is continuous. The Hilbert space (HS) was the first mathematical idealization proposed for Quantum Mechanics~\cite{VON}. However, as von Neumann explains in the introduction to his book~\cite{VON}, the HS theory and Dirac's formalism are two different things. Although there were attempts to realize the Dirac formalism in Hilbert space, there were a number of serious problems resulting from the fact that the Hilbert space cannot allocate such things as bras, kets, the Dirac delta function or the Dirac basis vector expansion, all of which are essential in any physical formulation of Quantum Mechanics that deals with continuous spectrum. Indeed in his textual presentation~\cite{DIRAC} Dirac himself states that ``the bra and ket vectors that we now use form a more general space than a Hilbert Space'' (see~\cite{DIRAC}, page 40). In the late 1940s, L.~Schwartz gave a precise meaning to the Dirac delta function as a functional over a space of test functions (cf.~\cite{SCHWARTZ}). This led to the development of a new branch of functional analysis, the theory of distributions~\cite{SCHWARTZ}. About the same time, von Neumann published the theory of direct integral decompositions of a Hilbert space induced by a self-adjoint operator~\cite{VON2} (also valid for more general cases). This spectral theory was closer to classical Fourier analysis and represented an improvement over the earlier von Neumann's spectral theory~\cite{VON}. I.~Gelfand always thought that von Neumann's spectral theory was not the whole story of the theory of linear operators defined on infinite dimensional vector spaces. Prompted by the theory of distributions, he and his school introduced the Rigged Hilbert Space (RHS). Starting out with this RHS and von Neumann's direct integral decomposition, they were able to prove the Nuclear Spectral theorem (also called Gelfand-Maurin theorem)~\cite{GELFAND}. This theorem justifies Dirac basis vector expansion. One of the aspects of Dirac's formalism, the continuity of the elements of the algebra of observables, was discussed in the early 1960s in Refs.~\cite{MEJLBO,KRISTENSEN}. If two operators of the algebra of observables satisfy the canonical (Heisenberg) 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 (locally convex) topologies that make those operators continuous; the largest of those subdomains is the Schwartz space. In the mid 1960s, some physicists~\cite{B60,ROBERTS,ANTOINE} independently realized that the RHS provides a rigorous mathematical rephrasing of all the aspects of Dirac's formalism. In particular, the Nuclear Spectral theorem restates Dirac basis vector expansion along with the Dirac bras and kets within a mathematical theory. Later on, the RHS was used to accommodate resonance states (Gamow vectors) (cf.~\cite{BOHM81,BOHMGA,ANTONIOU93,BOLLINI96,BOHM97,ANTOINE98,DIS} and references therein). Applications of the RHS formalism to Quantum Mechanics can now be found in some textbooks~\cite{BOHM,GALINDO}. Although there are some explicit examples of RHS in the literature (see for instance~\cite{B70}), no example of the RHS generated by a Schr\"odinger Hamiltonian with continuous spectrum has been constructed yet. Here we try to fill in this gap~\cite{DIS}. The dynamical equation that governs the behavior of a quantum system at any time is the time-dependent Schr\"odinger equation, \begin{equation} i\hbar \frac{\partial}{\partial t}\varphi (t) =H \varphi (t) \, , \label{tdSequ} \end{equation} where $H$ denotes the Hamiltonian of the system and $\varphi (t)$ denotes the value of the wave function $\varphi$ at time $t$. In order to solve (\ref{tdSequ}), we associate to each energy $E$ in the spectrum ${\rm Sp}(H)$ of the Hamiltonian a ket $|E\rangle$ which is an eigenvector of $H$, \begin{equation} H|E\rangle =E |E\rangle \, , \quad E\in {\rm Sp}(H) \, . \label{tiSequ} \end{equation} These eigenkets form a complete basis system that expands any wave function $\varphi$ as \begin{equation} \varphi =\int dE\, |E\rangle \langle E|\varphi \rangle \equiv \int dE \, \varphi (E) |E\rangle \, . \label{DbveI} \end{equation} The time-dependent solution of Eq.~(\ref{tdSequ}) is obtained by Fourier-transforming the time-independent solution (\ref{DbveI}), \begin{equation} \varphi (t)=\int dE\, e^{-iEt/\hbar}\, \varphi (E) \, . \label{tiemdesodye} \end{equation} If the spectrum of the Hamiltonian has a continuous part, and if the energy $E$ belongs to this continuous part of the spectrum, then the corresponding eigenket $|E\rangle$ that solves Eq.~(\ref{tiSequ}) is not square integrable, i.e., $|E\rangle$ is not an element of the Hilbert space. Therefore, the eigenket $|E\rangle$ cannot represent an experimentally preparable physical state. We will show that the expansion (\ref{DbveI}) is only valid for those $\varphi$ that belong to a space of test functions ${\mathbf \Phi} \subset {\cal H}$. We will also show that the kets $|E\rangle$ can be understood mathematically as continuous antilinear functionals over the space of test functions $\mathbf \Phi$, i.e., $|E\rangle \in \mathbf \Phi ^{\times}$. According to the RHS mathematics, equation (\ref{tiSequ}) means that \begin{equation} \langle H\varphi |E\rangle =E\langle \varphi |E\rangle \, , \quad {\rm for \ every \ } \varphi \in \mathbf \Phi \, , \label{geing} \end{equation} where $H$ is (the restriction to $\mathbf \Phi$ of) the self-adjoint Hamiltonian operator, which is a continuous operator on the linear topological space $\mathbf \Phi$. For every such an operator, one defines the conjugate operator $H^{\times}$ on $\mathbf \Phi ^{\times}$ by \begin{equation} \langle H\varphi |F\rangle =\langle \varphi |H^{\times}F\rangle \, , \quad {\rm for \ all \ } \varphi \in {\mathbf \Phi} \, , \ F\in \mathbf \Phi ^{\times} \, . \label{def} \end{equation} The operator $H^{\times}$ is a uniquely defined extension of the Hilbert space adjoint operator $H^{\dagger}$ (which for the case of a essentially self-adjoint operator coincides with the closure of $H$). Using the definition (\ref{def}), we write (\ref{geing}) formally as \begin{equation} H^{\times}|E\rangle =E|E\rangle \, , \quad |E\rangle \in \mathbf \Phi ^{\times} \, , \label{timseqcors} \end{equation} which is understood as a functional equation over the space $\mathbf \Phi$. The quantities $E$ and $|E\rangle$ are called generalized eigenvalues and generalized eigenvectors, respectively. The general statement of the Nuclear Spectral theorem just assures the existence of the generalized eigenvectors $|E\rangle$, but it does not provide a prescription to construct them. In this paper, we construct the generalized eigenvectors $|E\rangle$ of the square barrier Hamiltonian along with the RHS. We shall use the Sturm-Liouville theory (Weyl theory)~\cite{DUNFORD} to find the RHS of the square barrier potential. By applying the Sturm-Liouville theory to the Schr\"odinger equation of the square barrier potential, we will obtain a domain ${\cal D}(H)$ on which the Hamiltonian is self-adjoint. The Green functions, the spectrum, and the unitary transformation that diagonalizes our Hamiltonian will be also computed. The diagonalization of the Hamiltonian will allow us to obtain the energy (spectral) representation and the direct integral decomposition of the Hilbert space induced by our Hamiltonian. We will see why this direct integral decomposition is not enough for the purposes of Quantum Mechanics and why the RHS is necessary. Next, we will construct the space $\mathbf \Phi$ and therewith the RHS of the square barrier potential, \begin{equation} \mathbf \Phi \subset {\cal H} \subset \mathbf \Phi ^{\times} \, . \label{GT} \end{equation} Dirac kets will be accommodated as elements of $\mathbf \Phi ^{\times}$, and the Schwartz delta function will appear in the energy (spectral) representation of the triplet (\ref{GT}). The Nuclear Spectral theorem will be proved, and it will be shown that this theorem is just a restatement of the heuristic Dirac basis vector expansion. \def\thesection{\arabic{section}} \section{Sturm-Liouville Theory Applied to the Square Barrier Potential} \def\thesection{\arabic{section}} \setcounter{equation}{0} \label{sec:SLT} \def\thesubsection{\thesection.\arabic{subsection}} \subsection{Schr\"odinger Equation in the Position Representation} \label{sec:SE} In order to calculate the set of real generalized eigenvalues of the square barrier Hamiltonian (the physical spectrum) and their corresponding generalized eigenvectors, we solve equation~(\ref{timseqcors}) in the position representation, \begin{equation} \langle \vec{x}|H^{\times}|E\rangle =E\langle \vec{x}|E\rangle \, . \label{Schrroenpso} \end{equation} The expression of the Hamiltonian in the position representation is \begin{equation} \langle \vec{x}|H^{\times}|E\rangle = \left(\frac{-\hbar ^2}{2m} \Delta +V(\vec{x})\right) \langle \vec{x}|E\rangle \, , \label{delta} \end{equation} where $\Delta$ is the three-dimensional Laplacian and \begin{equation} V(\vec{x})=V(r)=\left\{ \begin{array}{ll} 0 &0s \end{array} \right. \quad \Re (E)<0 \, , \ \Im (E)\neq 0 \, . \label{-} \end{equation} The eigenfunction $\widetilde{\chi}(r;E)$ satisfies the Schr\"odinger equation (\ref{info}) and the boundary conditions \begin{mathletters} \label{eigsbo0co} \begin{eqnarray} && \widetilde{\chi} (0;E)=0 \, , \label{bca01} \\ && \widetilde{\chi} (r;E)\in AC^2([0,\infty )) \, ,\\ && \widetilde{\chi}(r;E) {\rm \ is \ square \ integrable \ at \ } 0 \, . \label{sbca03} \end{eqnarray} \end{mathletters} The boundary conditions (\ref{eigsbo0co}) can be written as \begin{mathletters} \label{bctchiphis} \begin{eqnarray} \widetilde{\chi} (0;E)&=&0 \, , \\ \widetilde{\chi} (a-0;E)&=& \widetilde{\chi} (a+0;E) \, , \label{bocchiata1}\\ \widetilde{\chi}' (a-0;E)&=& \widetilde{\chi}' (a+0;E) \, ,\\ \widetilde{\chi} (b-0;E)&=& \widetilde{\chi} (b+0;E) \, ,\\ \widetilde{\chi}' (b-0;E)&=& \widetilde{\chi}' (b+0;E) \, , \label{bocchiata4}\\ \widetilde{\chi}(r;E) && \!\!\!\!\! \mbox{is square integrable at}\ 0 \, , \end{eqnarray} \end{mathletters} and lead to \begin{equation} \widetilde{\chi}(r;E) =\left\{ \begin{array}{lll} e^{\sqrt{- \frac{2m}{\hbar ^2}E}r}- e^{-\sqrt{- \frac{2m}{\hbar ^2}E}r} \quad &00$, $\Im (E)> 0$} \vskip0.2cm When $\Re (E)>0$, $\Im (E)>0$, the expression of the Green function is \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 \Re (E)>0, \ \Im (E)>0 \, . \label{++} \end{equation} The eigenfunction $\chi (r;E)$ satisfies the Schr\"odinger equation (\ref{info}) and the boundary conditions (\ref{eigsbo0co}), \begin{equation} \chi (r;E)=\left\{ \begin{array}{lll} \sin (\sqrt{\frac{2m}{\hbar ^2}E}r) \quad &00$, $\Im (E)< 0$} \vskip0.2cm In the region $\Re (E)>0$, $\Im (E)<0$ the Green function reads \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 \Re (E)>0, \ \Im (E)<0 \, . \label{+-} \end{equation} The eigenfunction $\chi (r;E)$ is given by (\ref{chi}). The eigenfunction $\Theta _-(r;E)$ satisfies the Schr\"odinger equation (\ref{info}) and the boundary conditions (\ref{bcainfty1}), \begin{equation} \Theta _-(r;E)=\left\{ \begin{array}{lll} {\cal A}_1^-(E)e^{i\sqrt{\frac{2m}{\hbar ^2}E}r} +{\cal A}_2^-(E) e^{-i\sqrt{\frac{2m}{\hbar ^2}E}r} \quad &00, \Im (E)>0\, , \, r>s \, . \label{G++tofpuon} \end{eqnarray} By substituting Eq.~(\ref{Thet-inosigm}) into Eq.~(\ref{+-}) 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 \Re (E)>0, \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}) leads to \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 \Re (E)>0, \Im (E)>0\, , \, r>s \, , \label{redaot++} \end{eqnarray} and Eq.~(\ref{G+-tofpuon}) leads 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)\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 \Re (E)>0, \Im (E)<0\, , \, r>s \, , \label{redaot+-} \end{eqnarray} The expression of the resolvent in terms of the basis $\sigma _1,\sigma _2$ can be written as (see Theorem~4 in Appendix~\ref{sec:A3}) \begin{equation} G(r,s;E)=\sum_{i,j=1}^{2} \theta _{ij}^+ (E)\sigma _i(r;E)\overline{\sigma _j(s;\overline{E})}\, , \quad r>s \, . \label{GF++} \end{equation} By comparing (\ref{GF++}) to (\ref{redaot++}) we get to \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{lc} \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 \Re (E)>0 \, , \ \Im (E)>0 \, , \label{theta++} \end{equation} By comparing (\ref{GF++}) to (\ref{redaot+-}) we get to \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{lc} -\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 \Re (E)>0 \, , \ \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~4 of Appendix~\ref{sec:A3} 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, ${\rm Sp}(H)=[0,\infty )$. Thus the resolvent set of $H$ is ${\rm Re}(H)={\mathbb C} -[0,\infty )$. \subsubsection{\bf Diagonalization and Eigenfunction Expansion} We are now in a position to diagonalize the Hamiltonian. By Theorem~2 of Appendix~\ref{sec:A3}, there is a unitary map $\widetilde U$ defined by \begin{eqnarray} \widetilde{U}:L^2([0,\infty ),dr) &\longmapsto & L^2( (0,\infty ),\rho (E)dE) \nonumber \\ f(r)& \longmapsto & \widetilde{f}(E)=(\widetilde{U}f )(E)=\int_0^{\infty}dr f(r) \overline{\chi (r;E)} \label{rhoU} \end{eqnarray} that brings ${\cal D}(H)$ onto the space \begin{equation} {\cal D}(\widetilde{E})=\{ \widetilde{f}(E) \in L^2( (0,\infty ),\rho (E)dE) \, | \ \int_0^{\infty}dE \, E^2|\widetilde{f}(E)|^2 \rho (E) <\infty \} \, . \label{rhospace} \end{equation} Eqs.~(\ref{rhoU}) and (\ref{rhospace}) provide a $\rho$-diagonalization of $H$. If we seek a $\delta$-diagonalization, i.e., if we seek eigenfunctions that are $\delta$-normalized, then the measure $\rho (E)$ must be absorbed by the eigenfunctions and by the wave functions.\footnote{The meaning of the $\delta$-normalization of the eigenfunctions will be explained in Section~2.9.} This is why we define \begin{equation} \sigma (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 \begin{equation} \widehat{f}(E):=\sqrt{\rho (E)}{\widetilde f}(E) \, , \quad \widetilde{f}(E) \in L^2( (0,\infty ),\rho (E)dE) \, , \end{equation} and construct the unitary operator \begin{eqnarray} \widehat{U}:L^2((0,\infty)),\rho (E)dE) &\longmapsto & L^2((0,\infty),dE) \nonumber \\ {\widetilde f} &\longmapsto & \widehat{f}(E)= \widehat{U}{\widetilde f}(E):= \sqrt{\rho (E)}{\widetilde f}(E) \, . \end{eqnarray} The operator that $\delta$-diagonalizes our Hamiltonian is $U:={\widehat U}{\widetilde U}$, \begin{eqnarray} U:L^2([0,\infty)),dr) &\longmapsto & L^2((0,\infty),dE) \nonumber \\ f &\longmapsto & Uf:={\widehat f} \, . \end{eqnarray} The action of $U$ can be written as an integral operator, \begin{equation} \widehat{f}(E)=(Uf)(E)= \int_0^{\infty}dr f(r) \overline{\sigma (r;E)} \, , \quad f(r)\in L^2([0,\infty ),dr) \, . \label{inteexpre} \end{equation} The image of ${\cal D}(H)$ under the action of $U$ is \begin{equation} {\cal D}(\widehat{E}):=U{\cal D}(H)= \{ {\widehat f}(E) \in L^2((0,\infty ),dE) \, | \ \int_0^{\infty} E^2|\widehat{f}(E)|^2 dE<\infty \} \, . \end{equation} Therefore, we have constructed a unitary operator \begin{eqnarray} U:{\cal D}(H) \subset L^2([0,\infty ),dr) &\longmapsto & {\cal D}(\widehat{E}) \subset L^2((0,\infty ),dE) \nonumber \\ f &\longmapsto & \widehat{f}=Uf \end{eqnarray} that transforms from the position representation into the energy representation. The operator $U$ diagonalizes our Hamiltonian in the sense that $\widehat{E}\equiv UHU^{-1}$ is the multiplication operator, \begin{eqnarray} \widehat{E}:{\cal D}(\widehat{E}) \subset L^2((0,\infty ),dE) &\longmapsto & L^2((0,\infty ),dE) \nonumber \\ \widehat{f} & \longmapsto & (\widehat{E} \widehat{f})(E):= E\widehat{f}(E) \, . \end{eqnarray} The inverse operator of $U$ is given by (see Theorem~3 of Appendix~\ref{sec:A3}) \begin{equation} f(r)=(U^{-1}\widehat{f})(r)= \int_0^{\infty}dE\, \widehat{f}(E)\sigma(r,E) \, , \quad \widehat{f}(E)\in L^2((0,\infty ),dE) \, . \label{invdiagonaliza} \end{equation} The operator $U^{-1}$ transforms from the energy representation into the position representation. The expressions (\ref{inteexpre}) and (\ref{invdiagonaliza}) provide the eigenfunction expansion of any square integrable function in terms of the eigensolutions $\sigma (r;E)$ of $h$. The unitary operator $U$ can be looked at as a sort of generalized Fourier transform: the Fourier transform connects the position and the momentum representations. $U$ connects the position and the energy representations. The role played by the plane waves $e^{-ipx}$ (which are generalized eigenfunctions of the operator $-id/dx$) is here played by the $\sigma (r;E)$ (which are generalized eigenfunctions of the differential operator $h$). Therefore $\sigma (r;E)\equiv \langle r|E\rangle$, which are the $\delta$-normalized eigensolutions of the Schr\"odinger equation, can be viewed as ``transition elements'' between the $r$- and the $E$-representations. The label $f$ of the functions in the position representation is different to the label $\widehat f$ of the functions in the energy representation, because they have different functional dependences. Similar considerations apply to the Hamiltonian, the domain, the resolvent, etc. This is not the standard practice in the physics literature, where different representations are usually identified and labeled by the same symbol (see, for instance, \cite{NEWTON,COHEN,TAYLOR,BOHM}). \def\thesubsection{\thesection.\arabic{subsection}} \subsection{The Need of the Rigged Hilbert Space} \label{sec:tneodRHS} The Sturm-Liouville theory only provides a domain ${\cal D}(H)$ on which the Hamiltonian $H$ is self-adjoint and a unitary operator $U$ that diagonalizes $H$. This unitary operator induces a direct integral decomposition of the Hilbert space (see \cite{VON2,GELFAND}), \begin{eqnarray} {\cal H} &\longmapsto & U{\cal H} \equiv \widehat{\cal H}=\oplus \int_{{\rm Sp}(H)}{\cal H}(E)dE \nonumber \\ f &\longmapsto & Uf\equiv \{ \widehat{f}(E) \}, \, \quad \widehat{f}(E) \in {\cal H}(E) \, , \label{dirintdec} \end{eqnarray} where ${\cal H}$ is realized by $L^2([0,\infty ),dr)$, and $\widehat{\cal H}$ is realized by $L^2([0,\infty ),dE)$. The Hilbert space ${\cal H}(E)$ associated to each energy eigenvalue of ${\rm Sp}(H)$ is 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 \{ E\widehat{f}(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}}= \int_{ {\rm Sp}(H)} \left( \widehat{f}(E), \widehat{g}(E) \right) _E dE \, , \end{equation} where the scalar product $(\, \cdot \, , \, \cdot \, )_E$ on ${\cal H}(E)$ is the usual scalar product on $\mathbb C$, \begin{equation} \left( \widehat{f}(E), \widehat{g}(E) \right) _E= \overline{\widehat{f}(E)} \; \widehat{g}(E) \, . \end{equation} As we shall explain below, the direct integral decomposition does not accommodate some of the basic requirements needed in Quantum Mechanics. These requirements can be accommodated by the RHS. One of the most important principles of Quantum Mechanics is that the quantity $(\varphi ,H\varphi)$ should fit the experimental expectation value of the observable $H$ in the state $\varphi$. However, $(\varphi ,H\varphi)$ is not defined for every element in $\cal H$, but only for those square normalizable wave functions that are also in ${\cal D}(H)$. Therefore, not every square normalizable function can represent a ``physical wave function'', but only those that are (at least) in ${\cal D}(H)$. Another fundamental assumption of quantum physics is that the quantity \begin{equation} {\rm disp}_{\varphi}H=(\varphi , H^2\varphi )-(\varphi ,H\varphi )^2 \label{dr} \end{equation} represents the dispersion of the observable $H$ in the state $\varphi$, and that \begin{equation} \Delta _{\varphi}H\equiv \sqrt{{\rm disp}_{\varphi}H} \label{uncer} \end{equation} represents the uncertainty of the observable $H$ in the state $\varphi$. The quantities (\ref{dr}) and (\ref{uncer}) are not defined for every element of the Hilbert space either. Therefore, we would like to find a subdomain $\mathbf \Phi$ included in ${\cal D}(H)$ on which the expectation values \begin{equation} (\varphi ,H^n\varphi )\, , \quad n=0,1,2,\ldots\, , \quad \varphi \in \mathbf \Phi \label{ev} \end{equation} are well-defined. Another important requirement of Quantum Mechanics is that algebraic operations such as the sum and multiplication of two operators are well-defined. In the HS formalism, these algebraic operations are not always well-defined because the domains on which these operators are self-adjoint do not remain stable under their actions in general. In fact, much of the trouble of the HS formalism comes from domain questions. In our case, the domain ${\cal D}(H)$ in (\ref{domain}) does not remain stable under $H$. We therefore would like to find a subdomain $\mathbf \Phi$ included in ${\cal D}(H)$ that remains stable under the action of $H$ and all of its powers, \begin{equation} H^n:{\mathbf \Phi} \longmapsto {\mathbf \Phi} \, , \quad n=0,1,2,\ldots \label{stabeilis} \end{equation} One can see that if Eq.~(\ref{stabeilis}) holds, then the expectation values (\ref{ev}) are well-defined for each $\varphi$ in $\mathbf \Phi$, i.e., if the domain $\mathbf \Phi$ remains stable under the action of $H$, then the expectation values of $H$ in any state $\varphi \in \mathbf \Phi$ are well-defined. In Quantum Mechanics, it is always assumed that for each $E\in {\rm Sp}(H)$ there is a Dirac ket $|E\rangle$ such that \begin{equation} H^{\times}|E\rangle =E|E\rangle \label{ke} \end{equation} and such that the Dirac basis vector expansion~(\ref{DbveI}) holds. Equation~(\ref{ke}) has no solution in the Hilbert space when $E$ belongs to the continuous part of the spectrum of the Hamiltonian. In fact, Eq.~(\ref{ke}) has to be related to the equation \begin{equation} \langle \vec{x}|H^{\times}|E\rangle =E\langle \vec{x}|E\rangle \, , \end{equation} which in the radial representation reads \begin{equation} h \sigma (r;E)=E\sigma (r;E) \, , \end{equation} where $h$ is the differential operator (\ref{doh}) and $\sigma (r;E)$ is the delta-normalized eigenfunction (\ref{dnes}). Since $\sigma (r;E)\equiv \langle r|E\rangle$ lies outside $L^2([0,\infty ),dr)$, i.e., \begin{equation} \int_0^{\infty}dr \, |\sigma (r;E)|^2 =\infty \, , \end{equation} the corresponding eigenket $|E\rangle$, which is defined by \begin{eqnarray} |E\rangle :\mathbf \Phi & \longmapsto & \mathbb C \nonumber \\ \varphi & \longmapsto & \langle \varphi |E\rangle := \int_0^{\infty}\overline{\varphi (r)}\sigma (r;E)dr\, , \label{needefinitionket} \end{eqnarray} should also lie outside the Hilbert space. We shall show that $|E\rangle$ is an element of $\mathbf \Phi ^{\times}$. In summary, what our mathematical framework should provide us with is: \begin{enumerate} \item a dense invariant domain $\mathbf \Phi$ on which all the powers of $H$ and all the expectation values (\ref{ev}) are well-defined, \item smooth enough wave functions so that Eq.~(\ref{ke}) holds in the sense \begin{equation} \langle \varphi |H^{\times}|E\rangle = E\langle \varphi |E\rangle \, , \end{equation} \item any wave function can be expanded by a Dirac basis vector expansion. \end{enumerate} In the direct integral decomposition formalism, there is not enough room for either of these three requirements. This is why we introduce the RHS. \def\thesubsection{\thesection.\arabic{subsection}} \subsection{Construction of the Rigged Hilbert Space} \label{sec:RotD} 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 $\cal D$ of the operator $H$, \begin{equation} {\cal D}:= \bigcap _{n=0}^{\infty}{\cal D}(H^n) \, . \label{misus} \end{equation} The space $\cal D$ is the largest subspace of ${\cal D}(H)$ that remains stable under the action of the Hamiltonian $H$ and all of its powers. It is easy to check that \begin{eqnarray} {\cal D}=\{ \varphi \in L^2([0,\infty ),dr) \, | && \ h^n\varphi (r)\in L^2([0,\infty ),dr),\ h^n\varphi (0)=0, \varphi ^{(n)}(a)=\varphi ^{(n)}(b)=0, \nonumber \\ && n=0,1,2,\ldots ; \ \varphi (r) \in C^{\infty}([0,\infty)) \} \, . \label{mainisexi} \end{eqnarray} The conditions $\varphi ^{(n)}(a)=\varphi ^{(n)}(b)=0$ in (\ref{mainisexi}) come from taking the discontinuities of the potential $V(r)$ at $r=a$ and at $r=b$ into consideration (cf.~\cite{ROBERTS}). The second step is to find a subspace $\mathbf \Phi$ on which the eigenkets $|E\rangle$ of $H$ are well-defined as antilinear functionals. For each $E\in {\rm Sp}(H)$, we associate a ket $|E\rangle$ to the generalized eigenfunction $\sigma (r;E)$ through \begin{eqnarray} |E\rangle :\mathbf \Phi & \longmapsto & \mathbb C \nonumber \\ \varphi & \longmapsto & \langle \varphi |E\rangle := \int_0^{\infty}\overline{\varphi (r)}\sigma (r;E)dr =\overline{(U\varphi )(E)} \, . \label{definitionket} \end{eqnarray} As actual computations show, the ket $|E\rangle$ in (\ref{definitionket}) is a generalized eigenfunctional of $H$ if $\mathbf \Phi$ is included in the maximal invariant subspace of $H$, \begin{equation} \mathbf \Phi \subset {\cal D} \, . \end{equation} Due to the non-square integrability of the eigenfunction $\sigma (r;E)$, we need to impose further restrictions on the elements of $\cal D$ in order to make the eigenfunctional $|E\rangle$ in Eq.~(\ref{definitionket}) continuous, \begin{equation} \int_0^{\infty}dr \, \left| (r+1)^n(h+1)^m\varphi (r)\right| ^2<\infty, \quad n,m=0,1,2,\ldots \label{condcodogis} \end{equation} The imposition of conditions (\ref{condcodogis}) upon the space $\cal D$ leads to the space of wave functions of the square barrier potential, \begin{equation} {\mathbf \Phi} =\{ \varphi \in {\cal D} \, | \ \int_0^{\infty}dr \, \left| (r+1)^n(h+1)^m\varphi (r)\right| ^2<\infty, \quad n,m=0,1,2,\ldots \} \, . \end{equation} On $\mathbf \Phi$, we define the family of norms \begin{equation} \| \varphi \| _{n,m} := \sqrt{\int_0^{\infty}dr \, \left| (r+1)^n(h+1)^m\varphi (r)\right| ^2} \, , \quad n,m=0,1,2,\ldots \label{nmnorms} \end{equation} The quantities (\ref{nmnorms}) fulfill the conditions to be a norm (cf.~Proposition~1 of Appendix~\ref{sec:A4}) and can be used to define a countably normed topology $\tau _{\mathbf \Phi}$ on $\mathbf \Phi$ (see~\cite{GELFAND}), \begin{equation} \varphi _{\alpha}\, \mapupdown{\tau_{\mathbf \Phi}}{\alpha \to \infty} \, \varphi \quad {\rm iff} \quad \| \varphi _{\alpha}-\varphi \| _{n,m} \, \mapupdown{}{\alpha \to \infty}\, 0 \, , \quad n,m=0,1,2, \ldots \end{equation} One can see that the space $\mathbf \Phi$ is stable under the action of $H$ and that $H$ is $\tau _{\mathbf \Phi}$-continuous (cf.~Proposition~2 of Appendix~\ref{sec:A4}). Once we have constructed the space $\mathbf \Phi$, we can construct its topological dual $\mathbf \Phi ^{\times}$ as the space of $\tau _{\mathbf \Phi}$-continuous antilinear functionals on $\mathbf \Phi$ (see~\cite{GELFAND}) and therewith the RHS of the square barrier potential (for $l=0$) \begin{equation} {\mathbf \Phi} \subset L^2([0,\infty ),dr) \subset {\mathbf \Phi}^{\times} \, . \end{equation} The ket $|E\rangle$ in Eq.~(\ref{definitionket}) is a well-defined antilinear functional on $\mathbf \Phi$, i.e., $|E\rangle$ belongs to $\mathbf \Phi ^{\times}$ (cf.~Proposition~3 of Appendix~\ref{sec:A4}). The ket $|E\rangle$ is a generalized eigenvector of the Hamiltonian $H$ (cf.~Proposition~3 of Appendix~\ref{sec:A4}), \begin{equation} H^{\times}|E\rangle=E|E\rangle \, , \end{equation} i.e., \begin{equation} \langle \varphi |H^{\times}|E\rangle= \langle H\varphi |E\rangle = E\langle \varphi|E\rangle \, , \quad \forall \varphi \in \mathbf \Phi \, . \label{afegenphis} \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 generalized eigenvalue equation (\ref{afegenphis}) holds. As we shall see in the next section, the functions $\varphi$ of $\mathbf \Phi$ can be expanded by a Dirac basis vector expansion. \def\thesubsection{\thesection.\arabic{subsection}} \subsection{Dirac Basis Vector Expansion} \label{sec:DBVE} We are now in a position to derive the Dirac basis vector expansion. This derivation consists of the restriction of the Weyl-Kodaira expansions (\ref{inteexpre}) and (\ref{invdiagonaliza}) to the space $\mathbf \Phi$. If we denote $\langle r|\varphi \rangle \equiv \varphi (r)$ and $\langle E|r\rangle \equiv \overline{\sigma (r;E)}$, and if we define the action of the left ket $\langle E|$ on $\varphi \in \mathbf \Phi$ as $\langle E| \varphi \rangle := \widehat{\varphi}(E)$, then Eq.~(\ref{inteexpre}) becomes \begin{equation} \langle E|\varphi \rangle =\int_0^{\infty}dr \, \langle E |r \rangle \langle r|\varphi \rangle \, , \quad \varphi \in \mathbf \Phi \, . \label{DFeps} \end{equation} If we denote $\langle r|E \rangle \equiv \sigma (r;E)$, then Eq.~(\ref{invdiagonaliza}) becomes \begin{equation} \langle r|\varphi \rangle =\int_0^{\infty}dE \, \langle r|E \rangle \langle E|\varphi \rangle \, , \quad \varphi \in \mathbf \Phi \, . \label{inveqDva} \end{equation} This equation is the Dirac basis vector expansion of the square barrier potential. In fact, when we formally write (\ref{DbveI}) in the position representation, we get to (\ref{inveqDva}). In Eq.~(\ref{inveqDva}), the wave function $\langle r|\varphi \rangle$ is spanned in a ``Fourier-type'' expansion by the eigenfunctions $\langle r|E \rangle$. In this expansion, each eigenfunction $\langle r|E \rangle$ is weighted by $\langle E|\varphi \rangle =\widehat{\varphi}(E)$, which is the value of the wave function in the energy representation at the point $E$. Thus any function $\varphi (r)=\langle r|\varphi \rangle$ of $\mathbf \Phi$ can be written as a linear superposition of the monoenergetic eigenfunctions $\sigma (r;E)=\langle r|E \rangle$. Although the Weyl-Kodaira expansions (\ref{inteexpre}) and (\ref{invdiagonaliza}) are valid for every element of the Hilbert space, the Dirac basis vector expansions (\ref{DFeps}) and (\ref{inveqDva}) are only valid for functions $\varphi \in \mathbf \Phi$ because only those functions fulfill both \begin{equation} \overline{\widehat{\varphi}(E)}=\langle \varphi |E\rangle \end{equation} and \begin{equation} \langle \varphi |H^{\times}|E\rangle = \langle H\varphi |E\rangle = E \langle \varphi|E\rangle \, . \end{equation} Another way to rephrase the Dirac basis vector expansion is the Nuclear Spectral (Gelfand-Maurin) theorem. Instead of using the general statement of \cite{GELFAND}, we prove this theorem using the Sturm-Liouville theory (see Proposition~4 of Appendix~\ref{sec:A5}). The Nuclear Spectral Theorem allows us to write the scalar product of any two functions $\varphi ,\psi$ of $\mathbf \Phi$ in terms of the action of the kets $|E\rangle$ on $\varphi ,\psi$: \begin{equation} (\varphi ,\psi )=\int_0^{\infty} dE \, \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in \mathbf \Phi \, . \label{GMT1a} \end{equation} It also allows us to write the matrix elements of the Hamiltonian and all of its powers between two elements $\varphi ,\psi$ of $\mathbf \Phi$ in terms of the action of the kets $|E\rangle$ on $\varphi ,\psi$: \begin{equation} (\varphi ,H^n \psi )=\int_0^{\infty}dE \, E^n \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in {\mathbf \Phi} \, , n=1,2,\ldots \label{GMT2a} \end{equation} \def\thesubsection{\thesection.\arabic{subsection}} \subsection{Energy Representation of the RHS} \label{ESrepr} In this section, we construct the energy representation of the RHS. Since the unitary operator $U$ transforms from the position representation into the energy representation, the action of $U$ on the RHS provides the energy representation of the RHS. We have already shown that in the energy representation the Hamiltonian $H$ acts as the multiplication operator $\widehat{E}$. The energy representation of the space $\mathbf \Phi$ is defined as \begin{equation} \widehat{\mathbf \Phi}:= U\mathbf \Phi \, . \end{equation} It is very easy to see that $\widehat{\mathbf \Phi}$ is a linear subspace of $L^2([0,\infty ),dE)$. In order to endow $\widehat{\mathbf \Phi}$ with a topology $\tau _{\widehat{\mathbf \Phi}}$, we carry the topology on $\mathbf \Phi$ into $\widehat{\mathbf \Phi}$, \begin{equation} \tau _{\widehat{\mathbf \Phi}}:=U\tau _{\mathbf \Phi} \, . \end{equation} With this topology, the space $\widehat{\mathbf \Phi}$ is a linear topological space. If we denote the dual space of $\widehat{\mathbf \Phi}$ by $\widehat{\mathbf \Phi}^{\times}$, then we have \begin{equation} U^{\times}{\mathbf \Phi}^{\times}= (U{\mathbf \Phi})^{\times}= \widehat{\mathbf \Phi}^{\times} \, . \end{equation} If we denote $|\widehat{E}\rangle \equiv U^{\times}|E\rangle$, then we can prove that $|\widehat{E}\rangle$ is the antilinear Schwartz delta functional, i.e., $|\widehat{E}\rangle$ is the antilinear functional that associates to each function $\widehat{\varphi}$ the complex conjugate of its value at the point $E$ (see Proposition~5 of Appendix~\ref{sec:A6}), \begin{eqnarray} |\widehat{E}\rangle: \widehat{\mathbf \Phi} &\longmapsto & \mathbb C \nonumber \\ \widehat{\varphi} &\longmapsto & \langle \widehat{\varphi}|\widehat{E}\rangle := \overline{ \widehat{\varphi}(E)} \, . \end{eqnarray} Therefore, the Schwartz delta functional appears in the (spectral) energy representation of the RHS associated to the Hamiltonian. If we write the action of the Schwartz delta functional as an integral operator, then the Dirac $\delta$-function appears as the kernel of that integral operator. It is very helpful to show the different realizations of the RHS through the following diagram: \begin{equation} \begin{array}{ccclccclccll} H; & \varphi (r) & \ & \mathbf \Phi & \subset & L^2([0,\infty),dr) & \subset & \mathbf \Phi ^{\times} & \ & |E\rangle & \ & {\rm position \ repr.} \nonumber \\ [2ex] & & \ & \downarrow U & &\downarrow U & & \downarrow U^{\times} & \ & & & \nonumber \\ [2ex] \widehat{E}; & \widehat{\varphi}(E) & \ & \widehat{\mathbf \Phi} & \subset & L^2([0,\infty),dE) & \subset & \widehat{\mathbf \Phi} ^{\times} & \ & |\widehat{E}\rangle &\ & {\rm energy \ repr.} \\ \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. \def\thesubsection{\thesection.\arabic{subsection}} \subsection{Meaning of the $\delta$-Normalization of the Eigenfunctions} \label{sec:MFM} In this section, we show that the $\delta$-normalization of the eigenfunctions is related to the measure $d\mu (E)$ that is used to compute the scalar product of the wave functions in the energy representation, \begin{equation} (\varphi , \psi )=\int_0^{\infty}\overline{\varphi (E)} \psi (E) d\mu (E) \, . \label{sprusduyms} \end{equation} We will see that if the measure in (\ref{sprusduyms}) is the Lebesgue measure $dE$, then the eigenfunctions are $\delta$-normalized, and that if the measure is $\rho (E)dE$, then the eigenfunctions are $\rho$-normalized. For the sake of simplicity, in this section we label the wave functions in the position and in the energy representation with the same symbol. With this notation, Eq.~(\ref{inveqDva}) reads \begin{mathletters} \label{expnsofisn} \begin{eqnarray} &&\varphi (r)=\int_0^{\infty}dE\, \varphi (E) \sigma (r;E) \, , \\ && \psi (r)= \int_0^{\infty}dE\, \psi (E) \sigma (r;E) \, . \end{eqnarray} \end{mathletters} Since $\varphi (r), \psi (r)\in L^2( [0,\infty),dr)$, their scalar product is well-defined, \begin{equation} (\varphi ,\psi )=\int_0^{\infty}dr \, \overline{\varphi (r)}\psi (r)\, . \label{scparlpodof} \end{equation} Plugging (\ref{expnsofisn}) into (\ref{scparlpodof}), we obtain \begin{equation} (\varphi ,\psi )=\int_0^{\infty}dE\int_0^{\infty}dE'\, \overline{\varphi (E)} \psi (E')\int_0^{\infty}dr \, \overline{\sigma (r;E)}\sigma (r;E') \, . \label{fe1} \end{equation} If we use the Lebesgue measure $dE$, then the scalar product (\ref{sprusduyms}) can be written as \begin{equation} (\varphi ,\psi )=\int_0^{\infty}dE\, \overline{\varphi (E)}\psi (E) \, . \label{form2} \end{equation} Comparison of (\ref{fe1}) and (\ref{form2}) leads to \begin{equation} \int_0^{\infty}dr \, \overline{\sigma (r;E)}\sigma (r;E')= \delta (E-E') \, , \label{formdeno} \end{equation} i.e., the eigenfunctions $\sigma (r;E)$ are $\delta$-normalized. We now consider the case in which the eigenfunctions are $\rho$-normalized. If we use the measure $d\mu (E)=\rho (E) dE$, then the scalar product of $\varphi$ and $\psi$ is given by \begin{equation} (\varphi ,\psi )=\int_0^{\infty} \overline{\varphi _{\rho}(E)}\psi _{\rho}(E) \rho (E)\, dE \, , \label{rhospsiner} \end{equation} where $\varphi _{\rho}(E):=\varphi (E)/\sqrt{\rho (E)}$ and $\psi _{\rho}(E):= \psi (E)/\sqrt{\rho (E)}$. If we define $\sigma _{\rho}(r;E):=\sigma (r;E)/\sqrt{\rho (E)}$, then Eq.~(\ref{expnsofisn}) reads \begin{mathletters} \label{rhoexpnsofisn} \begin{eqnarray} &&\varphi (r)=\int_0^{\infty}\varphi _{\rho}(E) \sigma _{\rho}(r;E) \rho (E) \, dE \, , \\ && \psi (r)= \int_0^{\infty}\psi _{\rho}(E) \sigma _{\rho}(r;E) \rho (E) \, dE \, . \end{eqnarray} \end{mathletters} Plugging Eq.~(\ref{rhoexpnsofisn}) into (\ref{scparlpodof}), we obtain \begin{equation} (\varphi ,\psi )=\int_0^{\infty}dE\int_0^{\infty}dE'\, \overline{\varphi _{\rho}(E)} \psi _{\rho}(E')\rho (E) \rho (E') \int_0^{\infty}dr \, \overline{\sigma _{\rho}(r;E)}\sigma _{\rho}(r;E') \, . \label{compsexps} \end{equation} Comparison of (\ref{compsexps}) and (\ref{rhospsiner}) leads to \begin{equation} \int_0^{\infty}dr \, \overline{\sigma _{\rho}(r;E)}\sigma _{\rho}(r;E')= \frac{1}{\rho (E)}\delta (E-E') \, , \end{equation} i.e., the eigenfunctions $\sigma _{\rho}(r;E)$ are $\rho$-normalized. \def\thesection{\arabic{section}} \section{Conclusion} \def\thesection{\arabic{section}} \setcounter{equation}{0} \label{sec:C} In this paper, we have constructed the Rigged Hilbert Space \begin{equation} {\mathbf \Phi} \subset L^2([0,\infty),dr) \subset {\mathbf \Phi}^{\times} \label{ctriplet} \end{equation} of the square barrier Hamiltonian and its energy representation \begin{equation} \widehat{\mathbf \Phi} \subset L^2([0,\infty),dE) \subset \widehat{\mathbf \Phi} ^{\times} \, . \end{equation} The spectrum of the Hamiltonian $H$ is the positive real semiaxis. For each value $E$ of the spectrum of $H$, we have constructed a Dirac ket $|E\rangle$ that is a generalized eigenfunctional of $H$ whose corresponding generalized eigenvalue is $E$. In the energy representation, $|E\rangle$ acts as the antilinear Schwartz delta functional. On the space $\mathbf \Phi$, all algebraic operations involving the Hamiltonian $H$ are well-defined. In particular, the expectation values of the Hamiltonian in any element of $\mathbf \Phi$ are well-defined. Any element of $\mathbf \Phi$ can be expanded in terms of the eigenkets $|E\rangle$ by a Dirac basis vector expansion. The elements of $\mathbf \Phi$ are represented by well-behaved functions, in contrast to the elements of the Hilbert space, which are represented by sets of equivalent functions that can vary arbitrarily on any set of zero Lebesgue measure. Therefore, it seems natural to conclude that a physically acceptable wave function is not any element of the Hilbert space but rather an element of the subspace $\mathbf \Phi$. The monoenergetic eigensolutions $\sigma (r;E)$ of the time-independent Schr\"odinger equation are not square integrable, and therefore they cannot represent an acceptable wave function. Those eigensolutions has been used to define the eigenkets $|E\rangle$ that expand the physical wave functions $\varphi \in \mathbf \Phi$ in a Dirac basis vector expansion. The eigenkets $|E\rangle$ belong to the space $\mathbf \Phi ^{\times}$. The eigensolutions $\sigma (r;E)=\langle r|E\rangle$ have been also used to construct the unitary operator $U$ that transforms from the position representation into the energy representation. In our quest for the RHS of the square barrier potential, we have found a systematic method to construct the RHS of a large class of spherically symmetric potentials: \begin{enumerate} \item Expression of the formal differential operator. \item Hilbert space $\cal H$ of square integrable functions on which the formal differential operator acts. \item A domain ${\cal D}(H)$ of the Hilbert space on which the formal differential operator is self-adjoint. \item Green functions (resolvent) of that self-adjoint operator. \item Diagonalization of the self-adjoint operator, eigenfunction expansion of the elements of $\cal H$ in terms of the eigensolutions of the formal differential operator, and direct integral decomposition of $\cal H$ induced by the self-adjoint operator. \item Subspace $\mathbf \Phi$ of ${\cal D}(H)$ on which all the expectation values of $H$ are well-defined and on which the Dirac kets act as antilinear functionals. \item Rigged Hilbert space $\mathbf \Phi \subset {\cal H}\subset \mathbf \Phi ^{\times}$. \end{enumerate} \def\thesection{\arabic{section}} \section*{Acknowledgments} \setcounter{equation}{0} One of the authors (R.~de la Madrid) wishes to thank Prof.~J.-P.~Antoine and T.~Kuna for introducing him to the Sturm-Liouville theory and Profs.~R.~De la Llave, A.~Galindo, L.~Caffarelli and J.~Bona for illuminating discussions. Many thanks are due to N.~Harshman and C.~Koeninger for invaluable advise on English style. Financial support from the U.E.~TMR Contract number ERBFMRX-CT96-0087 ``The Physics of Quantum Information'', from the Welch Foundation, from La Junta de Castilla y Le\'on Project PC02 1999, from DGCYT PB98-0370, and from DGICYT PB98-0370 is gratefully acknowledged. \appendix \def\thesection{\Alph{section}} \section{Self-Adjoint Extension} \setcounter{equation}{0} \label{sec:A1} In this appendix, we list the possible self-adjoint extensions associated to the differential operator $h$. We first need some definitions (cf.~\cite{DUNFORD}). \vskip0.5cm {\bf Definition~1}\quad By $AC^2 ([0,\infty ) )$ we denote the space of all functions $f$ which have a continuous derivative in $[0,\infty )$, and for which $f'$ is not only continuous but also absolutely continuous over each compact subinterval of $[0,\infty )$. Thus $f ^{(2)}$ exists almost everywhere, and is integrable over any compact subinterval of $[0,\infty )$. At $0$ $f'$ is continuous from the right. \vskip0.5cm The space $AC ^2 ([0,\infty ) )$ is the largest space of functions on which the differential operator $h$ can be defined. \vskip0.5cm {\bf Definition~2}\quad We define the spaces \begin{eqnarray} &&{\cal H}^2 _h ([0,\infty ) ):= \{ f\in AC ^2 ([0,\infty )) \, | \ f, hf \in L^2 ( [0,\infty ),dr) \} \\ &&{\cal H}^2 ([0,\infty ) ):= \{ f\in AC ^2 ([0,\infty ) \, | \ f, f^{(2)} \in L^2 ( [0,\infty ),dr) \} \\ &&{\cal H}^2 _0 ([0,\infty ) ):= \{ f\in {\cal H}^2([0,\infty )) \, | \ f \ \mbox{vanishes outside some compact subset of} \ (0,\infty ) \} \, . \end{eqnarray} Using these spaces, we can define the necessary operators to calculate the self-adjoint extensions associated to $h$. \vskip1cm {\bf Definition~3}\quad If $h$ is the formal differential operator (\ref{doh}), we define the operators $H_0$ and $H_1$ on $L^2 ([0,\infty ),dr )$ by the formulas \begin{eqnarray} &&{\cal D}(H_0)={\cal H}^2 _0 ([0,\infty ) ), \quad H_0f:=hf, \quad f \in {\cal D}(H_0) \, . \\ &&{\cal D}(H_1)={\cal H}^2 _h ([0,\infty ) ), \quad H_1f:=hf, \quad f\in {\cal D}(H_1) \, . \end{eqnarray} \vskip0.5cm The operators $H_0$ and $H_1$ are sometimes called the {\it minimal} and the {\it maximal} operators associated to the differential operator $h$, respectively. The domain ${\cal D}(H_1)$ is the largest domain of the Hilbert space $L^2 ([0,\infty ),dr )$ on which the action of the differential operator $h$ can be defined and remains inside $L^2 ([0,\infty ),dr )$. Further, $H_0^{\dagger}=H_1$. The self-adjoint extensions of $H_0$ are given by the restrictions of the operator $H_1$ to domains determined by the conditions (see \cite{DUNFORD}, page 1306) \begin{equation} f(0)+\alpha \, f'(0)=0 \, , \quad -\infty < \alpha \leq \infty \, . \end{equation} These boundary conditions lead to the domains \begin{equation} {\cal D}_{\alpha}(H)=\{ f \in {\cal D}(H_1) \, | \ f(0)+\alpha \, f'(0)=0 \} \, , \quad -\infty < \alpha \leq \infty \, . \end{equation} On these domains, the formal differential operator $h$ is self-adjoint. The boundary condition that fits spherically symmetric potentials is $f(0)=0$, i.e., $\alpha =0$. This condition selects our domain (\ref{domain}), \begin{equation} {\cal D}(H)={\cal D}_{\alpha =0}(H) = \{ f \in {\cal D}(H_1) \, | \ f(0)=0 \} \, . \end{equation} \def\thesection{\Alph{section}} \section{Resolvent and Green Function} \setcounter{equation}{0} \label{sec:A2} The following theorem provides the procedure to compute the Green function of the Hamiltonian $H$ (cf.~Theorem~XIII.3.16 of Ref.~\cite{DUNFORD}): \vskip0.5cm {\bf Theorem~1}\quad Let $H$ be the self-adjoint operator (\ref{operator}) derived from the real formal differential operator (\ref{doh}) by the imposition of the boundary condition (\ref{sac}). Let $\Im E \neq 0$. Then there is exactly one solution $\chi (r;E)$ of $(h-E)\sigma =0$ square-integrable at $0$ and satisfying the boundary condition (\ref{sac}), and exactly one solution $\Theta (r;E)$ of $(h-E)\sigma =0$ square-integrable at infinity. The resolvent $(E-H)^{-1}$ is an integral operator whose kernel $G(r,s;E)$ is given by \begin{equation} G(r,s;E)=\left\{ \begin{array}{ll} \frac{2m}{\hbar ^2} \, \frac{\chi (r;E) \, \Theta (s;E)}{W(\chi ,\Theta )} &rs \, , \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 If we define \begin{mathletters} \begin{eqnarray} &&\widetilde{k} :=\sqrt{-\frac{2m}{\hbar ^2}\, E} \, , \\ && \widetilde{Q}:=\sqrt{-\frac{2m}{\hbar ^2}\, (E-V_0)} \, , \end{eqnarray} \end{mathletters} then the functions $\widetilde{{\cal J}}(E)$ of Eq.~(\ref{tildechifunction}) are given by \begin{mathletters} \label{tildejfunc} \begin{eqnarray} &&\widetilde{{\cal J}}_1(E)= \frac{1}{2}e^{-\widetilde{Q}a} \left[ \left(1+\frac{\widetilde{k}} {\widetilde{Q}}\right) e^{\widetilde{k}a} + \left(-1+\frac{\widetilde{k}} {\widetilde{Q}} \right) e^{-\widetilde{k}a} \right] , \\ && \widetilde{{\cal J}}_2(E)= \frac{1}{2}e^{\widetilde{Q}a} \left[ \left(1-\frac{\widetilde{k}} {\widetilde{Q}} \right) e^{\widetilde{k}a} + \left(-1-\frac{\widetilde{k}} {\widetilde{Q}} \right) e^{-\widetilde{k}a} \right] , \\ && \widetilde{{\cal J}}_3(E)=\frac{1}{2}e^{-\widetilde{k}b} \left[ \left(1+\frac{\widetilde{Q}} {\widetilde{k}}\right) e^{\widetilde{Q}b}\widetilde{{\cal J}}_1(E)+ \left(1-\frac{\widetilde{Q}} {\widetilde{k}} \right) e^{-\widetilde{Q}b}\widetilde{{\cal J}}_2(E) \right] , \\ &&\widetilde{{\cal J}}_4(E)=\frac{1}{2}e^{\widetilde{k}b} \left[ \left(1-\frac{\widetilde{Q}} {\widetilde{k}} \right) e^{\widetilde{Q}b}\widetilde{{\cal J}}_1(E)+ \left(1+\frac{\widetilde{Q}} {\widetilde{k}} \right) e^{-\widetilde{Q}b}\widetilde{{\cal J}}_2(E) \right] , \end{eqnarray} \end{mathletters} and the functions $\widetilde{\cal A}(E)$ of Eq.~(\ref{tildethetfunc}) by \begin{mathletters} \label{tildeAfunc} \begin{eqnarray} && \widetilde{{\cal A}}_3(E)= \frac{1}{2}e^{-\widetilde{Q}b} \left(1-\frac{\widetilde{k}} {\widetilde{Q}}\right) e^{-\widetilde{k}b} , \\ && \widetilde{{\cal A}}_4(E)= \frac{1}{2}e^{\widetilde{Q}b} \left(1+\frac{\widetilde{k}} {\widetilde{Q}} \right) e^{-\widetilde{k}b} , \\ && \widetilde{{\cal A}}_1(E)=\frac{1}{2}e^{-\widetilde{k}a} \left[ \left(1+\frac{\widetilde{Q}} {\widetilde{k}}\right) e^{\widetilde{Q}a} \widetilde{{\cal A}}_3(E)+ \left(1-\frac{\widetilde{Q}} {\widetilde{k}} \right) e^{-\widetilde{Q}a} \widetilde{{\cal A}}_4(E) \right] , \\ && \widetilde{ {\cal A}}_2(E)= \frac{1}{2}e^{\widetilde{k}a} \left[ \left(1-\frac{\widetilde{Q}} {\widetilde{k}} \right) e^{\widetilde{Q}a} \widetilde{{\cal A}}_3(E)+ \left(1+\frac{\widetilde{Q}} {\widetilde{k}} \right) e^{-\sqrt{- \widetilde{Q}}a} \widetilde{{\cal A}}_4(E) \right] . \end{eqnarray} \end{mathletters} The expression for the Wronskian of $\widetilde{\chi}$ and $\widetilde{\Theta}_-$ is \begin{equation} W(\widetilde{\chi} ,\widetilde{\Theta}_- )= -2\widetilde{k} \widetilde{\cal{J}}_3(E) \, . \end{equation} If we define \begin{mathletters} \begin{eqnarray} &&k:=\sqrt{\frac{2m}{\hbar ^2} \, E} \, , \\ &&Q:=\sqrt{\frac{2m}{\hbar ^2} \, (E-V_0)} \, , \end{eqnarray} \end{mathletters} then the functions ${\cal J}(E)$ of Eq.~(\ref{chi}) are given by \begin{mathletters} \label{Jfunction} \begin{eqnarray} &&{\cal J}_1(E)=\frac{1}{2}e^{-iQa} \left( \sin (ka)+ \frac{k}{iQ} \cos (ka) \right) , \\ &&{\cal J} _2 (E)=\frac{1}{2}e^{iQa} \left( \sin (ka)- \frac{k}{iQ} \cos (ka) \right) , \\ &&{\cal J}_3(E)=\frac{1}{2}e^{-ikb} \left[ \left(1+ \frac{Q}{k} \right) e^{iQb}{\cal J}_1(E)+ \left(1-\frac{Q}{k} \right) e^{-iQb}{\cal J}_2(E) \right] , \\ &&{\cal J}_4(E)=\frac{1}{2}e^{ikb} \left[ \left(1-\frac{Q}{k} \right) e^{iQb}{\cal J} _1(E)+ \left(1+\frac{Q}{k}\right) e^{-iQb}{\cal J}_2(E) \right] , \end{eqnarray} \end{mathletters} and the functions ${\cal A}^+(E)$ of Eq.~(\ref{theta+fun}) by \begin{mathletters} \label{A+functions} \begin{eqnarray} &&{\cal A}_3^+(E)=\frac{1}{2}e^{-iQb} \left(1+\frac{k}{Q}\right) e^{ikb} , \\ &&{\cal A}_4^+(E)=\frac{1}{2}e^{iQb} \left(1-\frac{k}{Q} \right) e^{ikb} , \\ &&{\cal A}_1^+(E)=\frac{1}{2}e^{-ika} \left[ \left(1+\frac{Q}{k}\right) e^{iQa}{\cal A}_3^+(E)+ \left(1-\frac{Q}{k} \right) e^{-iQa} {\cal A}_4^+(E) \right] , \\ &&{\cal A}_2^+(E)=\frac{1}{2}e^{ika} \left[ \left(1-\frac{Q}{k} \right) e^{iQa}{\cal A}_3^+(E)+ \left(1+\frac{Q}{k} \right) e^{-iQa}{\cal A}_4^+(E) \right] . \end{eqnarray} \end{mathletters} The Wronskian of $\chi$ and $\Theta _+$ is \begin{equation} W(\chi , \Theta _+)=2ik {\cal J}_4(E) \, . \end{equation} The functions ${\cal A}^-(E)$ of Eq.~(\ref{thetafun-}) are given by \begin{mathletters} \label{A-functions} \begin{eqnarray} &&{\cal A}_3^-(E)=\frac{1}{2}e^{-iQb} \left(1-\frac{k}{Q}\right) e^{-ikb} , \\ &&{\cal A}_4^-(E)=\frac{1}{2}e^{iQb} \left(1+\frac{k}{Q} \right) e^{-ikb} , \\ &&{\cal A}_1^-(E)=\frac{1}{2}e^{-ika} \left[ \left(1+\frac{Q}{k}\right) e^{iQa}{\cal A}_3^-(E)+ \left(1-\frac{Q}{k} \right) e^{-iQa}{\cal A}_4^-(E) \right] , \\ &&{\cal A}_2^-(E)=\frac{1}{2}e^{ika} \left[ \left(1-\frac{Q}{k} \right)e^{iQa}{\cal A}_3^-(E) + \left(1+\frac{Q}{k} \right) e^{-iQa}{\cal A}_4^-(E) \right] . \end{eqnarray} \end{mathletters} The Wronskian of $\chi$ and $\Theta _-$ is \begin{equation} W(\chi , \Theta _-)=-2ik {\cal J}_3(E) \, . \end{equation} \def\thesection{\Alph{section}} \section{Diagonalization and Eigenfunction \newline Expansion} \setcounter{equation}{0} \label{sec:A3} The theorem that provides the operator $U$ that diagonalizes $H$ is (cf.~Theorem XIII.5.13 of Ref.~\cite{DUNFORD}) \vskip0.5cm {\bf Theorem~2} (Weyl-Kodaira)\quad Let $h$ be the formally self-adjoint differential operator (\ref{doh}) defined on the interval $[0,\infty )$. Let $H$ be the self-adjoint operator (\ref{operator}). Let $\Lambda$ be an open interval of the real axis, and suppose that there is given a set $\{ \sigma _1(r;E),\, \sigma _2(r;E)\}$ of functions, defined and continuous on $(0,\infty )\times \Lambda$, such that for each fixed $E$ in $\Lambda$, $\{ \sigma _1(r;E),\, \sigma _2(r;E)\}$ forms a basis for the space of solutions of $h\sigma =E\sigma$. Then there exists a positive $2\times 2$ matrix measure $\{ \rho _{ij} \}$ defined on $\Lambda$, such that \begin{enumerate} \item the limit \begin{equation} (U f)_i(E)=\lim_{c\to 0}\lim_{d\to \infty} \left[ \int_c^d f(r) \overline{\sigma _i(r;E)}dr \right] \end{equation} exists in the topology of $L^2(\Lambda ,\{ \rho _{ij}\})$ for each $f$ in $L^2([0,\infty ),dr)$ and defines an isometric isomorphism $U$ of $E(\Lambda )L^2([0,\infty ),dr)$ onto $L^2(\Lambda ,\{ \rho _{ij}\})$; \item for each Borel function $G$ defined on the real line and vanishing outside $\Lambda$, \begin{equation} U{\cal D}(G(H))=\{ [f_i]\in L^2(\Lambda ,\{ \rho _{ij}\}) \, | \ [Gf_i]\in L^2(\Lambda ,\{ \rho _{ij}\}) \} \end{equation} and \begin{equation} (UG(H)f)_i(E)=G(E)(Uf)_i(E), \quad i=1,2, \, E\in \Lambda , \, f\in {\cal D}(G(H)) \, . \end{equation} \end{enumerate} \vskip0.5cm The theorem that provides the inverse of the operator $U$ is (cf.~Theorem XIII.5.14 of Ref.~\cite{DUNFORD}) \vskip0.5cm {\bf Theorem~3} (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 \begin{enumerate} \item the inverse of the isometric isomorphism $U$ of $E(\Lambda )L^2([0,\infty ),dr)$ 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 $L^2([0, \infty ),dr)$; \item if $G$ is a bounded Borel function vanishing outside a Borel set $e$ whose closure is compact and contained in $\Lambda$, then $G(H)$ has the representation \begin{equation} G(H)f(r)=\int _0^{\infty}f(s)K(H,r,s)ds \, , \end{equation} where \begin{equation} K(H,r,s)=\sum_{i,j=1}^2 \int_e G(E)\overline{\sigma _i(s;E)}\sigma _j(r;E)\rho _{ij}(dE) \, . \end{equation} \end{enumerate} \vskip0.5cm The spectral measures are provided by the following theorem (cf.~Theorem XIII.5.18 of Ref.~\cite{DUNFORD}): \vskip0.5cm {\bf Theorem~4} (Titchmarsh-Kodaira)\quad Let $\Lambda$ be an open interval of the real axis and $O$ be an open set in the complex plane containing $\Lambda$. Let $\{ \sigma _1(r;E),\, \sigma _2(r;E)\}$ be a set of functions which form a basis for the solutions of the equation $h\sigma =E\sigma$, $E\in O$, and which are continuous on $(0,\infty )\times O$ and analytically dependent on $E$ for $E$ in $O$. Suppose that the kernel $G(r,s;E)$ for the resolvent $(E-H)^{-1}$ has a representation \begin{equation} G(r,s;E)=\left\{ \begin{array}{lll} \sum_{i,j=1}^2 \theta _{ij}^-(E)\sigma _i(r;E) \overline{\sigma _j(s;\overline{E})}\, , & \qquad & rs \, , \end{array} \right. \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 \\ \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 The functions $\widetilde{\cal B}(E)$ of Eq.~(\ref{tildesigma1}) are given by \begin{mathletters} \label{tildeBfunctions} \begin{eqnarray} && \widetilde{{\cal B}}_3(E)= \frac{1}{2}e^{-\widetilde{Q}b} \left(1+\frac{\widetilde{k}}{\widetilde{Q}}\right) e^{\widetilde{k}b} , \\ && \widetilde{{\cal B}}_4(E)= \frac{1}{2}e^{\widetilde{Q}b} \left(1-\frac{\widetilde{k}}{\widetilde{Q}} \right) e^{\widetilde{k}b} , \\ && \widetilde{{\cal B}}_1(E)=\frac{1}{2}e^{-\widetilde{k}a} \left[ \left(1+\frac{\widetilde{Q}}{\widetilde{k}}\right) e^{\widetilde{Q}a} \widetilde{{\cal B}}_3(E)+ \left(1-\frac{\widetilde{Q}}{\widetilde{k}} \right) e^{-\widetilde{Q}a} \widetilde{{\cal B}}_4(E) \right] , \\ && \widetilde{ {\cal B}}_2(E)= \frac{1}{2}e^{\widetilde{k}a} \left[ \left(1-\frac{\widetilde{Q}}{\widetilde{k}} \right) e^{\widetilde{Q}a} \widetilde{{\cal B}}_3(E)+ \left(1+\frac{\widetilde{Q}}{\widetilde{k}} \right) e^{-\widetilde{Q}a} \widetilde{{\cal B}}_4(E) \right] . \end{eqnarray} \end{mathletters} The functions ${\cal C}(E)$ of Eq.~(\ref{sigam2cos}) are given by \begin{mathletters} \label{Cfunctions} \begin{eqnarray} &&{\cal C}_1(E)= \frac{1}{2}e^{-iQa} \left( \cos (ka)-\frac{k}{iQ} \sin (ka) \right), \\ && {\cal C}_2(E)= \frac{1}{2}e^{iQa} \left( \cos (ka)+\frac{k}{iQ} \sin (ka) \right), \\ && {\cal C}_3(E)=\frac{1}{2}e^{-ikb} \left[ \left(1+\frac{Q}{k}\right) e^{iQb}{\cal C}_1(E)+ \left(1-\frac{Q}{k} \right) e^{-iQb}{\cal C}_2(E) \right] , \\ &&{\cal C}_4(E)=\frac{1}{2}e^{ikb} \left[ \left(1-\frac{Q}{k} \right) e^{iQb}{\cal C}_1(E)+ \left(1+\frac{Q}{k} \right) e^{-iQb}{\cal C}_2(E) \right] . \end{eqnarray} \end{mathletters} \def\thesection{\Alph{section}} \section{Construction of the RHS} \setcounter{equation}{0} \label{sec:A4} {\bf Proposition~1} \quad The quantities \begin{equation} \| \varphi \| _{n,m} := \sqrt{\int_0^{\infty}dr \, \left| (r+1)^n(h+1)^m\varphi (r)\right| ^2}, \quad \varphi \in {\mathbf \Phi} \, , \, n,m=0,1,2,\ldots, \label{anmnorms} \end{equation} are norms. \vskip0.2cm {\it Proof}\quad It is very easy to show that the quantities (\ref{anmnorms}) fulfill the conditions to be a norm, \begin{mathletters} \begin{eqnarray} &&\| \varphi +\psi \| _{n,m} \leq \| \varphi \| _{n,m} + \| \psi \| _{n,m} \, , \\ && \| \alpha \varphi \| _{n,m}=|\alpha |\, \| \varphi \| _{n,m} \, , \\ && \| \varphi \| _{n,m} \geq 0 \, , \\ && {\rm If }\ \| \varphi \| _{n,m} =0, \ {\rm then} \ \varphi =0 \, . \label{homiensi} \end{eqnarray} \end{mathletters} The only condition that is somewhat difficult to prove is (\ref{homiensi}): if $\| \varphi \| _{n,m}=0$, then \begin{equation} (1+r)^n(h+1)^m\varphi (r)=0 \, , \end{equation} which yields \begin{equation} (h+1)^m\varphi (r)=0 \, . \label{homiodhiis} \end{equation} If $m=0$, then Eq.~(\ref{homiodhiis}) implies $\varphi (r)=0$. If $m=1$, then Eq.~(\ref{homiodhiis}) implies that $-1$ is an eigenvalue of $H$ whose corresponding eigenvector is $\varphi$. Since $-1$ is not an eigenvalue of $H$, $\varphi$ must be the zero vector. If $m>1$, the proof is similar. \vskip0.5cm {\bf Proposition~2}\quad The space $\mathbf \Phi$ is stable under the action of $H$, and $H$ is $\tau _{\mathbf \Phi}$-continuous. \vskip0.2cm {\it Proof}\quad In order to see that $H$ is $\tau _{\mathbf \Phi}$-continuous, we just have to realize that \begin{eqnarray} \| H\varphi \| _{n,m}&=&\| (H+I)\varphi -\varphi \| _{n,m} \nonumber \\ &\leq & \| (H+I)\varphi \| _{n,m}+ \| \varphi \| _{n,m} \nonumber \\ &=&\| \varphi \| _{n,m+1}+\| \varphi \| _{n,m} \, . \label{tauphiscont} \end{eqnarray} We now prove that $\mathbf \Phi$ is stable under the action of $H$. Let $\varphi \in \mathbf \Phi$. To say that $\varphi \in \mathbf \Phi$ is equivalent to say that $\varphi \in {\cal D}$ and that the norms $\| \varphi \| _{n,m}$ are finite for every $n,m=0,1,2,\ldots \ $ Since $H\varphi$ is also in ${\cal D}$, and since the norms $\| H\varphi \| _{n,m}$ are also finite (see Eq.~(\ref{tauphiscont})), the vector $H\varphi$ is also in $\mathbf \Phi$. \vskip0.5cm {\bf Proposition~3}\quad The function \begin{eqnarray} |E\rangle :\mathbf \Phi & \longmapsto & \mathbb C \nonumber \\ \varphi & \longmapsto & \langle \varphi |E\rangle := \int_0^{\infty}\overline{\varphi (r)}\sigma (r;E)dr =\overline{(U\varphi )(E)} \, . \label{adefinitionket} \end{eqnarray} is an antilinear functional on $\mathbf \Phi$ that is a generalized eigenvector of (the restriction to $\mathbf \Phi$ of) $H$. \vskip0.2cm {\it Proof}\quad From the definition (\ref{adefinitionket}), 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| \sigma (r;E) \right| \, . \end{equation} Since \begin{eqnarray} |\langle \varphi |E\rangle | &=& |\overline{U\varphi (E)}| \nonumber \\ &=&\left| \int_0^{\infty}dr \, \overline{\varphi (r)}\sigma(r;E)\right| \nonumber \\ &\leq & \int_0^{\infty}dr \, |\overline{\varphi (r)}| |\sigma(r;E)| \nonumber \\ &\leq & {\cal M}(E) \int _0^{\infty}dr \, |\varphi (r)| \nonumber \\ &=& {\cal M}(E) \int_0^{\infty}dr \, \frac{1}{1+r} (1+r) |\varphi (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 (r) \right| ^2 \right) ^{1/2} \nonumber \\ &=&{\cal M}(E) \left( \int_0^{\infty}dr \, \frac{1}{(1+r)^2} \right) ^{1/2} \| \varphi \| _{1,0} \nonumber \\ &=&{\cal M}(E) \| \varphi \| _{1,0} \, , \end{eqnarray} the functional $|E\rangle$ is continuous when $\mathbf \Phi$ is endowed with the $\tau _{\mathbf \Phi}$ topology. In order to prove that $|E\rangle$ is a generalized eigenvector of $H$, we make use of the conditions (\ref{mainisexi}) and (\ref{condcodogis}) satisfied by the elements of $\mathbf \Phi$, \begin{eqnarray} \langle \varphi |H^{\times}|E\rangle &=& \langle H\varphi |E\rangle \nonumber \\ &=& \int_0^{\infty}dr \, \left( -\frac{\hbar ^2}{2m}\frac{d^2}{dr^2}+V(r) \right) \overline{\varphi(r)} \sigma (r;E) \nonumber \\ &=&-\frac{\hbar ^2}{2m} \left[ \frac{d\overline{\varphi (r)}}{dr} \sigma(r;E) \right] _0^{\infty} +\frac{\hbar ^2}{2m} \left[ \overline{\varphi (r)} \frac{d\sigma(r;E)}{dr} \right] _0^{\infty} \nonumber \\ &&+ \int_0^{\infty}dr \, \overline{\varphi(r)} \left( -\frac{\hbar ^2}{2m}\frac{d^2}{dr^2}+V(r) \right)\sigma (r;E) \nonumber \\ &=&E\langle \varphi |E\rangle \, . \end{eqnarray} Similarly, one can also prove that \begin{equation} \langle \varphi | (H^{\times})^n|E\rangle = E^n \langle \varphi |E\rangle \, . \end{equation} \def\thesection{\Alph{section}} \section{Dirac Basis Vector Expansion} \setcounter{equation}{0} \label{sec:A5} {\bf Proposition~4} (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 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 $E$ in the spectrum of $H$ there is a generalized eigenvector $|E\rangle$ such that \begin{equation} H^{\times}|E\rangle =E|E\rangle \end{equation} and such that \begin{equation} (\varphi ,\psi )=\int_{ {\rm Sp}(H)}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^n \psi )=\int_{ {\rm Sp}(H)}dE \, E^n \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in {\mathbf \Phi} \, , n=1,2,\ldots \label{GMT2} \end{equation} \vskip0.2cm {\it Proof} \quad Let $\varphi$ and $\psi$ be in $\mathbf \Phi$. Since $U$ is unitary, \begin{equation} (\varphi ,\psi )=(U\varphi ,U\psi )=(\widehat{\varphi} ,\widehat{\psi}) \, . \label{Usiuni} \end{equation} The wave functions $\widehat{\varphi}$ and $\widehat{\psi}$ are in particular elements of $L^2([0,\infty ),dE)$. Therefore their scalar product is well-defined, \begin{equation} (\widehat{\varphi} ,\widehat{\psi} )= \int_{{\rm Sp}(H)}dE \, \overline{ \widehat{\varphi}(E)} \widehat{\psi}(E) \, . \label{sphatvhaps} \end{equation} Since $\varphi$ and $\psi$ belong to $\mathbf \Phi$, the action of each eigenket $|E\rangle$ on them is well-defined, \begin{mathletters} \label{actionofEpsi} \begin{eqnarray} \langle \varphi |E\rangle =\overline{ \widehat{\varphi}(E)} \, ,\\ \langle E|\psi \rangle =\widehat{\psi}(E) \, . \end{eqnarray} \end{mathletters} Plugging Eq.~(\ref{actionofEpsi}) into Eq.~(\ref{sphatvhaps}) and Eq.~(\ref{sphatvhaps}) into Eq.~(\ref{Usiuni}), we get to Eq.~(\ref{GMT1}). The proof of (\ref{GMT2}) is similar: \begin{eqnarray} (\varphi ,H^n\psi )&=&(U\varphi , UH^nU^{-1}U\psi ) \nonumber \\ &=& (\widehat{\phi} ,\widehat{E}^n\widehat{\psi} ) \nonumber \\ &=&\int_{{\rm Sp}(H)}dE \,\overline{ \widehat{\varphi}(E)} (\widehat{E}^n\widehat{\psi})(E) \nonumber \\ &=&\int_{{\rm Sp}(H)}dE\, E^n \overline{ \widehat{\varphi}(E)} \widehat{\psi}(E) \nonumber \\ &=& \int_{{\rm Sp}(H)}dE \, E^n \langle \varphi |E\rangle \langle E|\psi \rangle \, . \end{eqnarray} \def\thesection{\Alph{section}} \section{Energy Representation of the RHS} \setcounter{equation}{0} \label{sec:A6} {\bf Proposition~5} \quad The energy representation of the eigenket $|E\rangle$ is the antilinear Schwartz delta functional $|\widehat{E}\rangle$. \vskip0.2cm {\it Proof} \quad Since \begin{eqnarray} \langle \widehat{\varphi }|U^{\times}|E\rangle &=& \langle U^{-1}\widehat{\varphi }|E\rangle \nonumber \\ &=& \langle \varphi |E\rangle \nonumber \\ &=& \int_0^{\infty}\overline{\varphi (r)}\sigma (r;E)dr \nonumber \\ &=& \overline{ \widehat{\varphi}(E)} \, , \end{eqnarray} the functional $U^{\times}|E\rangle =|\widehat{E}\rangle$ is the antilinear Schwartz delta functional. \begin{thebibliography}{99} \bibitem{DIRAC} P.~A.~M.~Dirac, {\it The principles of Quantum Mechanics}, 3rd ed., Clarendon Press, Oxford (1947). \bibitem{VON} J.~von Neumann, {\it Mathematische Grundlagen der Quantentheorie}, Springer, Berlin (1931); English translation by R.~T.~Beyer, Princeton University Press, Princeton (1955). \bibitem{SCHWARTZ} L.~Schwartz, {\it Th\'eory de Distributions}, Hermann, Paris (1950). \bibitem{VON2} J.~von Neumann, Ann.~Math.~(N.~Y.) {\bf 50}, 401 (1949). \bibitem{GELFAND} I.~M.~Gel'fand, N.~Y.~Vilenkin, {\it Generalized Functions, Vol.~4} Academic Press, New York, (1964); K.~Maurin, {\it Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups}, Polish Scientific Publishers, Warsaw (1968). \bibitem{MEJLBO} L.~C.~Mejlbo, Math.~Scand.~{\bf 13}, 129 (1963). \bibitem{KRISTENSEN} P.~Kristensen, L.~Mejlbo, E.~Thue Poulsen, Commun.~Math.~Phys.~{\bf 1} 175 (1965); Math.~Scand.~{\bf 14} 129 (1964). \bibitem{B60} A.~Bohm, {\it Rigged Hilbert Spaces}, International Center for Theoretical Physics Lecture Notes, Publication ICTP 64/9, Trieste (1964); {\it Boulder Lectures in Theoretical Physics, 1966}, Volume 9A, Gordon and Breach, New York (1967). \bibitem{ROBERTS} J.~E.~Roberts, J.~Math.~Phys.~{\bf 7}, 1097 (1966); Commun.~Math.~Phys., {\bf 3}, 98 (1966). \bibitem{ANTOINE} J.~P.~Antoine, J.~Math.~Phys.~{\bf 10}, 53 (1969); J.~Math.~Phys.~{\bf 10}, 2276 (1969). \bibitem{BOHM81} A.~Bohm, J.~Math.~Phys.~{\bf 22}, 2813 (1981). \bibitem{BOHMGA} A.~Bohm, M.~Gadella, {\it Dirac Kets, Gamow Vectors and Gelfand Triplets}, Springer Lecture Notes in Physics, {\bf 348}, Berlin (1989). \bibitem{ANTONIOU93} I.~Antoniou, S.~Tasaki, Int.~J.~Quantum Chemistry {\bf 46}, 425 (1993). \bibitem{BOLLINI96} C.~G.~Bollini, O.~Civitarese, A.~L.~De Paoli, M.~C.~Rocca, J.~Math.~Phys.~{\bf 37}, 4235 (1996). \bibitem{BOHM97} A.~Bohm, M.~Loewe, S.~Maxson, P.~Patuleanu, C.~Puntmann and M.~Gadella, J.~Math.~Phys., {\bf 38}, 6072 (1997). \bibitem{ANTOINE98} J.-P.~Antoine, in {\it Irreversibility and Causality}, edited by A.~Bohm, H.-D.~Doebner, and P.~Kielanowski, Springer-Verlag (1998), page~3. \bibitem{DIS} R.~de la Madrid, {\it Quantum Mechanics in Rigged Hilbert Space Language}, PhD Thesis, Universidad de Valladolid (2001). 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