Content-Type: multipart/mixed; boundary="-------------0405310446565" This is a multi-part message in MIME format. ---------------0405310446565 Content-Type: text/plain; name="04-169.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-169.keywords" Rigged Hilbert space; exactly solvable system ---------------0405310446565 Content-Type: application/x-tex; name="03ijtp.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="03ijtp.tex" \documentclass[11pt]{article} \usepackage{amsmath,amsfonts,latexsym,graphicx,amssymb} \textheight21.5cm \textwidth15cm \topmargin-1cm \oddsidemargin0.1cm %\textwidth15.5cm %\topmargin=-.25in \oddsidemargin=1mm \evensidemargin=10mm %\textheight=216mm \textwidth=161mm %\baselineskip=18pt %\usepackage{subeqn} %\font\bbbss=cmr12 scaled\magstep 0 \begin{document} \def\theequation{\thesection.\arabic{equation}} \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% \title{\bf The Rigged Hilbert Space of the Free Hamiltonian} \author{Rafael de la Madrid \\ [1ex] \small{\it Institute for Scientific Interchange (ISI), Villa Gualino, Viale Settimio Severo 65} \\ [-0.5ex] \small{\it I-10133, Torino, Italy} \\ [-0.5ex] \small{\it E-mail: \texttt{rafa@isiosf.isi.it}}} \date{\small{(October 23, 2003)}} \maketitle \begin{abstract} \noindent We explicitly construct the Rigged Hilbert Space (RHS) of the free Hamiltonian $H_0$. The construction of the RHS of $H_0$ provides yet another opportunity to see that when continuous spectrum is present, the solutions of the Schr\"odinger equation lie in a RHS rather than just in a Hilbert space. \end{abstract} \vskip0.3cm PACS numbers: 03.65.-w, 02.30.Hq %\pacs{03.65.-w, 02.30.Hq} \newpage \section{Introduction} \setcounter{equation}{0} \label{sec:introduction} There is a growing realization that the Rigged Hilbert Space (RHS) provides the methods needed to handle Dirac's bra-ket formalism and continuous spectrum. Moreover, there is an increasing number of Quantum Mechanics textbooks that include the RHS as part of their contents~\cite{ATKINSON}--\cite{GALINDO} (this list is non-exhaustive). However, there is still a lack of simple examples for which the RHS is explicitly constructed (one exception is Ref.~\cite{B70}). Especially important is to construct the RHS generated by the Schr\"odinger equation, because claiming that Quantum Mechanics needs the RHS is tantamount to claiming that the solutions of the Schr\"odinger equation lie in a RHS (when continuous spectrum is present). The task of constructing the RHS generated by the Schr\"odinger equation was undertaken in Ref.~\cite{DIS}. The method proposed in \cite{DIS} has been applied to two simple potentials~\cite{JPA,FP02}. In this paper, we shall apply this method to the simplest example possible: the free Hamiltonian. We note that the results of this paper follow immediately from those in Refs.~\cite{JPA,FP02}, by making the value of the potential zero. Nevertheless, we think that the example of the free Hamiltonian provides a very transparent way to understand the essentials of the method of \cite{DIS}, because the calculations are reduced to the minimum. The time-independent Schr\"odinger equation for the free Hamiltonian $H_0$ reads, in the position representation, as \begin{equation} \label{delta} \langle \vec{x}|H_0|E\rangle = \frac{-\hbar^2}{2m} \nabla^2 \langle \vec{x}|E\rangle =E\langle \vec{x}|E\rangle \, , \end{equation} where $\nabla^2$ is the three-dimensional Laplacian. In spherical coordinates $\vec{x}\equiv (r,\theta, \phi)$, Eq.~(\ref{delta}) has the following form: \begin{equation} \langle r,\theta, \phi|H_0|E,l,m \rangle = \biggl( \frac{-\hbar^2}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r +\frac{\hbar^2l(l+1)}{2mr^2} \biggr) \langle r,\theta, \phi|E,l,m\rangle = E\langle r,\theta, \phi|E,l,m\rangle\,. \label{sphSe} \end{equation} By separating the radial and angular dependences, \begin{equation} \langle r,\theta, \phi|E,l,m\rangle \equiv \langle r|E\rangle_l \, \langle \theta, \phi |l,m\rangle \equiv \frac{1}{r}\chi_l(r;E)Y_{l,m}(\theta, \phi), \end{equation} where $Y_{l,m}(\theta, \phi)$ are the spherical harmonics, we obtain for the radial part \begin{equation} \biggl(\frac{-\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2l(l+1)}{2mr^2} \biggr) \chi_l(r;E)=E\chi_l(r;E)\,. \label{baba} \end{equation} In this paper, we shall restrict ourselves to the case of zero orbital angular momentum (the higher-order case can be treated analogously). We then write $\chi_{l=0}(r;E)\equiv \chi (r;E)$ and obtain \begin{equation} -\frac{\hbar^2}{2m} \frac{d^2}{dr^2}\chi(r;E)= E\chi (r;E)\,. \label{rSe0} \end{equation} We shall write this equation as \begin{equation} h_0\chi(r;E)= E\chi (r;E) \, , \label{rSe0s} \end{equation} where \begin{equation} h_0\equiv -\frac{\hbar ^2}{2m}\frac{d^2}{dr^2} \label{0doh} \end{equation} is the formal differential operator corresponding to the free Hamiltonian (for $l=0$). Our goal is to solve Eq.~(\ref{rSe0s}) and show that its solutions lie in a RHS rather than just in a Hilbert space. The basic tool necessary to solve Eq.~(\ref{rSe0s}) is the Sturm-Liouville theory~\cite{SL}. This theory provides the Hilbert space methods. As shown in many publications (cf.~\cite{DIS} and references therein), the Hilbert space methods do not provide us with all the tools needed in Quantum Mechanics when continuous spectrum is present. In particular, the Hilbert space cannot incorporate Dirac's bra-ket formalism. Therefore, an extension of the Hilbert space is needed. The extension that seems to be most suitable is the RHS (cf.~\cite{DIS} and references therein). In particular, the RHS incorporates Dirac's bra-ket formalism. The structure of the paper is as follows. In Section~\ref{sec:iseaexte}, we construct the domain and the self-adjoint extension of the differential operator (\ref{0doh}). In Section~\ref{sec:resoangf}, we obtain the free Green function, whose expression is used in Section~\ref{sec:sphspo} to calculate the spectrum of $H_0$. Section~\ref{sec:diaaeiexp} is devoted to the eigenfunction expansion and the direct integral decomposition of the Hilbert space. In Section~\ref{0sec:tneodRHS}, we construct the RHS of $H_0$. The Dirac basis vector expansion for $H_0$ is obtained in Section~\ref{sec:0DBVE}, and the energy representation of the RHS of $H_0$ is constructed in Section~\ref{sec:0ESrepr}. Finally, the results of the paper are summarized in the diagram of Eq.~(\ref{0diagramsavp}). \section{Self-Adjoint Extension} \setcounter{equation}{0} \label{sec:iseaexte} The first step is to define a linear operator on a Hilbert space corresponding to the formal differential operator (\ref{0doh}). In the radial position representation, the Hilbert space that belongs to the RHS of the free Hamiltonian is realized by the space $L^2([0,\infty ), dr)$ of square integrable functions $f(r)$ defined on the interval $[0,\infty )$ (see the diagram (\ref{0diagramsavp}) below). The domain ${\cal D}(H_0)$ of the free Hamiltonian must be a proper dense linear subspace of $L^2([0,\infty ), dr)$. The action of $h_0$ must be well defined on ${\cal D}(H_0)$, and this action must remain in $L^2([0,\infty ), dr)$. We need also a boundary condition that assures the self-adjointness of the Hamiltonian. The boundary conditions that select the possible self-adjoint extensions of $h_0$ are given by (see \cite{DS}, page 1306) \begin{equation} f(0)+\alpha \, f'(0)=0 \, , \quad -\infty < \alpha \leq \infty \, . \end{equation} Among all these boundary conditions, we choose $f(0)=0$. Therefore, the requirements that are to be fulfilled by the elements of the domain of $H_0$ are \begin{subequations} \label{bcthdpspr} \begin{eqnarray} &&f(r) \in L^2([0,\infty ), dr) \, , \label{HSc} \\ &&h_0f(r) \in L^2([0,\infty ), dr) \, , \label{reduce} \\ &&f(r) \in AC^2 [0,\infty ) \, , \label{ACc} \\ &&f(0)=0 \, , \label{sac} \end{eqnarray} \end{subequations} where $AC^2[0,\infty)$ denotes the space of functions whose derivative is absolutely continuous (for details on absolutely continuous functions, consult Refs.~\cite{DS,DIS}). The requirements in Eq.~(\ref{bcthdpspr}) yield the domain of $H_0$: \begin{equation} {\cal D}(H_0) =\{ f(r)\, | \ f(r), h_0f(r)\in L^2([0,\infty ), dr), \, f(r) \in AC^2[0,\infty ), \, f(0)=0 \} \, . \label{0domain} \end{equation} On ${\cal D}(H_0)$ the formal differential operator $h_0$ is self-adjoint. In choosing (\ref{0domain}) as the domain of our formal differential operator $h_0$, we define a linear operator $H_0$ by \begin{equation} H_0f(r) :=h_0f(r)=-\frac{\hbar ^2}{2m}\frac{d^2}{dr^2}f(r) \, , \quad f(r) \in {\cal D}(H_0) \, . \label{0operator} \end{equation} \section{Resolvent and Green Function} \setcounter{equation}{0} \label{sec:resoangf} The expression of the free Green function $G_0(r,s;E)$ is given in terms of eigenfunctions of the differential operator $h_0$ subject to certain boundary conditions (cf.~Theorem~1 in Appendix~\ref{sec:A1}). We shall divide the complex energy plane in three regions, and calculate $G_0(r,s;E)$ for each region separately. In all 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 $\mbox{Re}(E)<0$, $\mbox{Im}(E)\neq 0$} \label{sec:relolo} For $\mbox{Re}(E)<0$, $\mbox{Im}(E)\neq 0$, the free Green function (see Theorem~1 in Appendix~\ref{sec:A1}) is given by \begin{equation} G_0(r,s;E)=\left\{ \begin{array}{ll} -\frac{2m/\hbar ^2}{\sqrt{-2m/\hbar ^2 \, E}} \, \frac{\widetilde{\chi}(r;E) \, \widetilde{f}(s;E)}{2} \quad &rs \end{array} \right. \quad \mbox{Re}(E)<0 \, , \ \mbox{Im}(E)\neq 0 \, . \label{0-} \end{equation} The eigenfunction $\widetilde{\chi}(r;E)$ satisfies the Schr\"odinger equation, Eq.~(\ref{rSe0s}), and the boundary conditions \begin{subequations} \begin{eqnarray} && \widetilde{\chi} (0;E)=0 \, , \label{bca01} \\ && \widetilde{\chi}(r;E) {\rm \ is \ square \ integrable \ at \ } 0 \, , \label{sbca03} \end{eqnarray} \label{eigsbo0co} \end{subequations} which yield \begin{equation} \widetilde{\chi}(r;E) =e^{\sqrt{-\frac{2m}{\hbar ^2}E} \, r}- e^{-\sqrt{-\frac{2m}{\hbar ^2}E} \, r} \, , \quad 00$, $\mbox{Im}(E)>0$} \label{sec:regogo} When $\mbox{Re}(E)>0$, $\mbox{Im}(E)>0$, the expression of the free Green function is \begin{equation} G_0(r,s;E)=\left\{ \begin{array}{ll} -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \chi (r;E) \, f^+ (s;E) \quad &rs \end{array} \right. \quad \mbox{Re}(E)>0, \ \mbox{Im}(E)>0 \, . \label{0++} \end{equation} The eigenfunction $\chi (r;E)$ satisfies Eq.~(\ref{rSe0s}) and the boundary conditions (\ref{eigsbo0co}), which yield \begin{equation} \chi (r;E)=\sin ( \sqrt{\frac{2m}{\hbar ^2}E}\, r ) \, , \quad 00$, $\mbox{Im}(E)< 0$} \label{sec:regolo} In the region $\mbox{Re}(E)>0$, $\mbox{Im}(E)<0$, the free Green function reads \begin{equation} G_0(r,s;E)=\left\{ \begin{array}{ll} -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \chi (r;E) \, f^- (s;E) \quad &rs \end{array} \right. \quad \mbox{Re}(E)>0, \ \mbox{Im}(E)<0 \, . \label{0+-} \end{equation} The eigenfunction $\chi (r;E)$ is given by (\ref{0chi}), although now $E$ belongs to the fourth quadrant of the energy plane. The eigenfunction $f^-(r;E)$ satisfies Eq.~(\ref{rSe0s}) and the boundary condition (\ref{thetcoabejej}), which yield \begin{equation} f^-(r;E)=e^{-i\sqrt{\frac{2m}{\hbar ^2}E}\, r} \, , \quad 0s \, , \ \mbox{Re}(E)>0, \mbox{Im}(E)>0 \, . \label{0G++tofpuon} \end{equation} By substituting Eq.~(\ref{0Thet-inosigm}) into Eq.~(\ref{0+-}) we get to \begin{equation} G_0(r,s;E)= -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \sigma _1(s;E) \left[ -i\sigma _1(r;E)+\sigma _2(r;E) \right] , \ r>s \, , \ \mbox{Re}(E)>0, \mbox{Im}(E)<0\, . \label{0G+-tofpuon} \end{equation} Because \begin{equation} \overline{\sigma _1(s;\overline{E})}=\sigma _1(s;E) \, , \end{equation} Eq.~(\ref{0G++tofpuon}) leads to \begin{eqnarray} &&G_0(r,s;E)= -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \left[ i\sigma _1(r;E)\overline{\sigma _1(s;\overline{E})} +\sigma _2(r;E)\overline{\sigma _1(s;\overline{E})}\right] , \nonumber \\ &&\qquad \hskip6cm \mbox{Re}(E)>0, \mbox{Im}(E)>0 , \, r>s \, , \label{0redaot++} \end{eqnarray} whereas Eq.~(\ref{0G+-tofpuon}) leads to \begin{eqnarray} &&G_0(r,s;E)= -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \, \left[ -i\sigma _1(r;E)\overline{\sigma _1(s;\overline{E})} +\sigma _2(r;E)\overline{\sigma _1(s;\overline{E})}\right] , \nonumber \\ &&\qquad \hskip6cm \mbox{Re}(E)>0, \mbox{Im}(E)<0\, , \, r>s \, . \label{0redaot+-} \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:A1}) \begin{equation} G_0(r,s;E)=\sum_{i,j=1}^{2} \theta _{ij}^+ (E)\sigma _i(r;E)\overline{\sigma _j(s;\overline{E})}\, , \qquad r>s \, . \label{0GF++} \end{equation} By comparing (\ref{0GF++}) to (\ref{0redaot++}) we get to \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} i & -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \\ 0 & 0 \end{array} \right) , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)>0 \, . \label{0theta++} \end{equation} By comparing (\ref{0GF++}) to (\ref{0redaot+-}) we get to \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} i & -\frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}} \\ 0 & 0 \end{array} \right) , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)<0 \, . \label{0theta+-} \end{equation} From Eqs.~(\ref{0theta++}) and (\ref{0theta+-}) we can see that the measures $\varrho _{12}$, $\varrho _{21}$ and $\varrho _{22}$ in Theorem~4 of Appendix~\ref{sec:A1} are zero, and that the measure $\varrho _{11}$ is given by \begin{eqnarray} \varrho _{11}((E_1,E_2))&=&\lim _{\delta \to 0} \lim _{\varepsilon \to 0+} \frac{1}{2\pi i} \int_{E_1+\delta}^{E_2-\delta} \left[ \theta _{11}^+ (E-i\varepsilon ) -\theta _{11}^+ (E+i\varepsilon ) \right] dE \nonumber \\ &=&\int_{E_1}^{E_2} \frac{1}{\pi}\, \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}}\, dE \, , \end{eqnarray} which leads to \begin{equation} \varrho (E)\equiv \varrho _{11}(E)= \frac{1}{\pi}\, \frac{2m/\hbar ^2}{\sqrt{2m/\hbar ^2 \, E}}\, , \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 ${\rm Sp}(H_0)$. Because ${\rm Sp}(H_0)$ is a closed set, ${\rm Sp}(H_0)=[0,\infty )$. \section{Diagonalization and Eigenfunction Expansion} \setcounter{equation}{0} \label{sec:diaaeiexp} In the present section, we diagonalize $H_0$ and construct the expansion of the wave functions in terms of the eigenfunctions of the differential operator $h_0$. By Theorem~2 of Appendix~\ref{sec:A1}, there is a unitary map $\widetilde U_0$ defined by \begin{eqnarray} \widetilde{U}_0:L^2([0,\infty ),dr) &\longmapsto & L^2( (0,\infty ),\varrho (E)dE) \nonumber \\ f(r)& \longmapsto & \widetilde{f}(E)=\widetilde{U}_0f(E)=\int_0^{\infty}dr f(r) \overline{\chi (r;E)} \, , \label{0rhoU} \end{eqnarray} that brings ${\cal D}(H_0)$ onto the space \begin{equation} {\cal D}(\widetilde{H}_0)=\{ \widetilde{f}(E) \in L^2( (0,\infty ),\varrho (E)dE) \, | \ \int_0^{\infty}dE \, E^2|\widetilde{f}(E)|^2 \varrho (E) <\infty \} \, . \label{0rhospace} \end{equation} The unitary operator $\widetilde U_0$ provides a $\varrho $-normalization (cf.~Ref.~\cite{FP02}). In order to obtain a $\delta$-normalization, the measure $\varrho (E)$ must be absorbed in the definition of the eigenfunctions (cf.~Ref.~\cite{FP02}). This is why we define \begin{equation} \sigma (r;E):=\sqrt{\varrho (E)} \, \chi (r;E) \, , \label{0dnes} \end{equation} which is the $\delta$-normalized eigensolution of the differential operator $h_0$. If we define \begin{equation} \widehat{f}(E):=\sqrt{\varrho (E)}{\widetilde f}(E) \, , \quad \widetilde{f}(E) \in L^2( (0,\infty ),\varrho (E)dE) \, , \end{equation} and construct the unitary operator \begin{eqnarray} \widehat{U}_0:L^2((0,\infty),\varrho (E)dE) &\longmapsto & L^2((0,\infty),dE) \nonumber \\ {\widetilde f} &\longmapsto & \widehat{f}(E)= \widehat{U}_0{\widetilde f}(E):= \sqrt{\varrho (E)}{\widetilde f}(E) \, , \end{eqnarray} then the operator that $\delta$-diagonalizes our Hamiltonian is $U_0:={\widehat U}_0{\widetilde U}_0$, \begin{eqnarray} U_0:L^2([0,\infty),dr) &\longmapsto & L^2((0,\infty),dE) \nonumber \\ f &\longmapsto & U_0f:={\widehat f} \, . \end{eqnarray} The action of $U_0$ can be written as an integral operator: \begin{equation} \widehat{f}(E)=U_0f(E)= \int_0^{\infty}dr f(r) \overline{\sigma (r;E)} \, , \quad f(r)\in L^2([0,\infty ),dr) \, . \label{0inteexpre} \end{equation} The image of ${\cal D}(H_0)$ under the action of $U_0$ is \begin{equation} {\cal D}(\widehat{H}_0):=U{\cal D}(H_0)= \{ {\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_0:{\cal D}(H) \subset L^2([0,\infty ),dr) &\longmapsto & {\cal D}(\widehat{H}_0) \subset L^2((0,\infty ),dE) \nonumber \\ f &\longmapsto & \widehat{f}=U_0f \end{eqnarray} that transforms from the position representation into the energy representation (see diagram (\ref{0diagramsavp}) below). The operator $U_0$ diagonalizes the free Hamiltonian in the sense that $\widehat{H}_0\equiv U_0H_0U_0^{-1}$ is the multiplication operator. The inverse operator of $U_0$ is given by (see Theorem~3 of Appendix~\ref{sec:A1}) \begin{equation} f(r)=U_0^{-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{0invdiagonaliza} \end{equation} The operator $U_0^{-1}$ transforms from the energy representation into the position representation (see diagram (\ref{0diagramsavp}) below). The expressions (\ref{0inteexpre}) and (\ref{0invdiagonaliza}) provide the eigenfunction expansion of any square integrable function in terms of the $\delta$-normalized eigensolutions $\sigma (r;E)$ of $h_0$. \section{Construction of the RHS of the Free Hamiltonian} \setcounter{equation}{0} \label{0sec:tneodRHS} The Sturm-Liouville theory only provides a domain ${\cal D}(H_0)$ on which the Hamiltonian $H_0$ is self-adjoint and a unitary operator $U_0$ that diagonalizes $H_0$. This unitary operator induces a direct integral decomposition of the Hilbert space (cf.~Ref.~\cite{DIS} and references therein), \begin{eqnarray} {\cal H} &\longmapsto & U_0{\cal H} \equiv \widehat{\cal H}= \oplus \int_{{\rm Sp}(H_0)}{\cal H}(E)dE \nonumber \\ f &\longmapsto & U_0f\equiv \{ \widehat{f}(E) \}, \, \quad \widehat{f}(E) \in {\cal H}(E) \, . \label{0dirintdec} \end{eqnarray} As shown in Refs.~\cite{ANTOINE,DIS,JPA,FP02}, the direct integral decomposition does not provide us with all the tools needed in Quantum Mechanics. This is why we extend the Hilbert space to the RHS. We first need to construct a dense invariant domain ${\mathbf \Phi}_0$ on which all the powers and all the expectation values of $H_0$ are well defined, and on which the Dirac kets act as antilinear functionals. Before building ${\mathbf \Phi}_0$, we need to build the maximal invariant subspace ${\cal D}_0$ of $H_0$, \begin{equation} {\cal D}_0:= \bigcap _{n=0}^{\infty}{\cal D}(H_0^n) \, . \label{0misus} \end{equation} It is easy to check that \begin{eqnarray} {\cal D}_0=\{ \varphi \in L^2([0,\infty ),dr) \, | && \hskip-.5cm h_0^n\varphi (r)\in L^2([0,\infty ),dr),\ h_0^n\varphi (0)=0, n=0,1,2,\ldots , \nonumber \\ && \hskip-.5cm \varphi (r) \in C^{\infty}([0,\infty)) \} \, . \label{0mainisexi} \end{eqnarray} We can now construct the subspace ${\mathbf \Phi}_0$ on which the eigenkets $|E\rangle$ of $H_0$ are well defined as antilinear functionals. This subspace is given by \begin{equation} {\mathbf \Phi}_0=\{ \varphi \in {\cal D}_0 \, | \ \int_0^{\infty}dr \, \left| (r+1)^n(h_0+1)^m\varphi (r)\right| ^2<\infty, \quad n,m=0,1,2,\ldots \} \, . \end{equation} On ${\mathbf \Phi}_0$, we define the family of norms \begin{equation} \| \varphi \| _{n,m}:= \sqrt{\int_0^{\infty}dr \, \left| (r+1)^n(h_0+1)^m\varphi (r)\right| ^2} \ , \quad n,m=0,1,2,\ldots \label{0nmnorms} \end{equation} The quantities (\ref{0nmnorms}) fulfill the conditions to be a norm (see Proposition~1 of Appendix~\ref{sec:A2}), and can be used to define a countably normed topology $\tau _{{\mathbf \Phi}_0}$ on ${\mathbf \Phi}_0$ (for the definition of a countably normed topology, consult Ref.~\cite{DIS} and references therein), \begin{equation} \varphi _{\alpha}\, \mapupdown{\tau_{{\mathbf \Phi}_0}}{\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} The space ${\mathbf \Phi}_0$ is stable under the action of $H_0$, and $H_0$ is $\tau _{{\mathbf \Phi}_0}$-continuous (see Proposition~2 of Appendix~\ref{sec:A2}). Once we have constructed the space ${\mathbf \Phi}_0$, we can construct its topological dual ${\mathbf \Phi}^{\times}_0$ as the space of $\tau _{{\mathbf \Phi}_0}$-continuous antilinear functionals on ${\mathbf \Phi}_0$ and therewith the RHS of the free Hamiltonian (see diagram (\ref{0diagramsavp}) below): \begin{equation} {\mathbf \Phi}_0 \subset L^2([0,\infty ),dr) \subset {\mathbf \Phi}^{\times}_0 \, . \end{equation} For each $E\in {\rm Sp}(H_0)$, we can now associate a ket $|E\rangle$ to the generalized eigenfunction $\sigma (r;E)$ through \begin{eqnarray} |E\rangle :{\mathbf \Phi}_0 & \longmapsto & {\mathbb C} \nonumber \\ \varphi & \longmapsto & \langle \varphi |E\rangle := \int_0^{\infty}\overline{\varphi (r)}\sigma (r;E) \, dr =\overline{(U_0\varphi )(E)} \, . \label{0definitionket} \end{eqnarray} The ket $|E\rangle$ in Eq.~(\ref{0definitionket}) is a well-defined antilinear functional on ${\mathbf \Phi}_0$, i.e., $|E\rangle$ belongs to ${\mathbf \Phi}^{\times}_0$ (see Proposition~3 of Appendix~\ref{sec:A2}). The ket $|E\rangle$ is a generalized eigenvector of the free Hamiltonian $H_0$ (see Proposition~3 of Appendix~\ref{sec:A2}): \begin{equation} H_0^{\times}|E\rangle=E|E\rangle \, ; \end{equation} that is, \begin{equation} \langle \varphi |H_0^{\times}|E\rangle= \langle H_0^{\dagger}\varphi |E\rangle = E\langle \varphi|E\rangle \, , \quad \forall \varphi \in {\mathbf \Phi}_0 \, . \label{0afegenphis} \end{equation} \section{The Dirac Basis Vector Expansion for $H_0$} \setcounter{equation}{0} \label{sec:0DBVE} We are now in a position to derive the Dirac basis vector expansion for the free Hamiltonian. This derivation consists of the restriction of the Weyl-Kodaira expansions (\ref{0inteexpre}) and (\ref{0invdiagonaliza}) to the space ${\mathbf \Phi}_0$. 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 bra $\langle E|$ on $\varphi \in {\mathbf \Phi}_0$ as $\langle E| \varphi \rangle := \widehat{\varphi}(E)$, then Eq.~(\ref{0inteexpre}) becomes \begin{equation} \langle E|\varphi \rangle =\int_0^{\infty}dr \, \langle E |r \rangle \langle r|\varphi \rangle \, , \quad \varphi \in {\mathbf \Phi}_0 \, . \label{0DFeps} \end{equation} If we denote $\langle r|E \rangle \equiv \sigma (r;E)$, then Eq.~(\ref{0invdiagonaliza}) becomes \begin{equation} \langle r|\varphi \rangle =\int_0^{\infty}dE \, \langle r|E \rangle \langle E|\varphi \rangle \, , \quad \varphi \in {\mathbf \Phi}_0\, . \label{0inveqDva} \end{equation} This equation is the Dirac basis vector expansion of the wave function $\varphi$ in terms of the free eigenkets $|E\rangle$. We can also prove the Nuclear Spectral Theorem for the free Hamiltonian (see Proposition~4 of Appendix~\ref{sec:A2}), \begin{equation} (\varphi ,H_0^n \psi )=\int_0^{\infty}dE \, E^n \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in {\mathbf \Phi}_0 \, , n=1,2,\ldots \label{0GMT2a} \end{equation} \section{Energy Representation of the RHS of $H_0$} \setcounter{equation}{0} \label{sec:0ESrepr} We have already seen that in the energy representation, the Hamiltonian $H_0$ acts as the multiplication operator $\widehat{H}_0$. The energy representation of the space ${\mathbf \Phi}_0$ is defined as \begin{equation} \widehat{\mathbf \Phi}_0:= U_0{\mathbf \Phi}_0 \, . \end{equation} Obviously $\widehat{\mathbf \Phi}_0$ is a linear subspace of $L^2([0,\infty ),dE)$. In oder to endow $\widehat{\mathbf \Phi}_0$ with a topology $\tau _{\widehat{\mathbf \Phi }_0}$, we carry the topology on ${\mathbf \Phi}_0$ into $\widehat{\mathbf \Phi}_0$, \begin{equation} \tau _{\widehat{\mathbf \Phi }_0}:=U_0\tau _{{\mathbf \Phi}_0} \, . \end{equation} With this topology, the space $\widehat{\mathbf \Phi}_0$ is a linear topological space. If we denote the dual space of $\widehat{\mathbf \Phi}_0$ by $\widehat{\mathbf \Phi}_0^{\times}$, then we have \begin{equation} U_0^{\times}{\mathbf \Phi}_0^{\times}= (U_0 {\mathbf \Phi}_0)^{\times}= \widehat{\mathbf \Phi}_0^{\times} \, . \end{equation} If we denote $|\widehat{E}\rangle \equiv U_0^{\times}|E\rangle$, then we can prove that $|\widehat{E}\rangle$ is the antilinear Schwartz delta functional (see Proposition~5 of Appendix~\ref{sec:A2}), \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} It is very helpful to show the different realizations of the RHS through the following diagram: \begin{equation} \begin{array}{ccclccclccll} H_0; & \varphi (r) & \ & {\mathbf \Phi}_0 & \subset & L^2([0,\infty),dr) & \subset & {\mathbf \Phi}_0^{\times} & \ & |E\rangle & \ & {\rm position \ repr.} \nonumber \\ [2ex] & & \ & \downarrow U_0 & &\downarrow U_0 & & \downarrow U_0^{\times} & \ & & & \nonumber \\ [2ex] \widehat{H}_0; & \widehat{\varphi}(E) & \ & \widehat{\mathbf \Phi}_0 & \subset & L^2([0,\infty),dE) & \subset & \widehat{\mathbf \Phi}_0 ^{\times} & \ & |\widehat{E}\rangle &\ & {\rm energy \ repr.} \\ \end{array} \label{0diagramsavp} \end{equation} On the top line, we have the position representation of the Hamiltonian, the wave functions, the kets, and the RHS. On the bottom line, we have their energy representation counterparts. \section{Conclusions} \setcounter{equation}{0} \label{sec:concusios} We have constructed the RHS of $H_0$ (for the zero angular momentum case), and its energy representation. We have associated an eigenket $|E\rangle$ to each energy $E$ in the spectrum of $H_0$, and shown that $|E\rangle$ belongs to ${\mathbf \Phi}_0^{\times}$. We have seen that the energy representation of $|E\rangle$ is given by the antilinear Schwartz delta functional. We have also shown that the Dirac basis vector expansion holds within the RHS of $H_0$. Thus, we conclude that the natural setting for the solutions of the Schr\"odinger equation of $H_0$ is the Rigged Hilbert Space rather than just the Hilbert space. \section*{Acknowledgment} C.~Koeninger's advise on English style is gratefully acknowledged. This work was financially supported by the E.U.~TMR Contract No.~ERBFMRX-CT96-0087 ``The Physics of Quantum Information.'' \appendix \def\thesection{\Alph{section}} \section{Appendix~A: The Sturm-Liouville Theory} \setcounter{equation}{0} \label{sec:A1} The following theorem provides the procedure to compute the Green function of $H_0$ (cf.~Theorem~XIII.3.16 of Ref.~\cite{DS} and also Refs.~\cite{DIS,JPA,FP02} for some applications): \vskip0.5cm {\bf Theorem~1}\quad Let $H_0$ be the self-adjoint operator (\ref{0operator}) derived from the real formal differential operator (\ref{0doh}) by the imposition of the boundary condition (\ref{sac}). Let $\mbox{Im}(E) \neq 0$. Then there is exactly one solution $\chi (r;E)$ of $(h_0-E)\sigma =0$ square-integrable at $0$ and satisfying the boundary condition (\ref{sac}), and exactly one solution $f(r;E)$ of $(h_0-E)\sigma =0$ square-integrable at infinity. The resolvent $(E-H_0)^{-1}$ is an integral operator whose kernel $G_0(r,s;E)$ is given by \begin{equation} G_0(r,s;E)=\left\{ \begin{array}{ll} \frac{2m}{\hbar ^2} \, \frac{\chi (r;E) \, f(s;E)}{W(\chi ,f )} &rs \, , \end{array} \right. \label{exofGFA} \end{equation} where $W(\chi ,f )$ is the Wronskian of $\chi$ and $f$ \begin{equation} W(\chi ,f )=\chi f'-\chi ' f \, . \end{equation} \vskip0.5cm The theorem that provides the operator $U_0$ that diagonalizes $H_0$ is the following (cf.~Theorem XIII.5.13 of Ref.~\cite{DS} and also Refs.~\cite{DIS,JPA,FP02} for some applications): \vskip0.5cm {\bf Theorem~2} (Weyl-Kodaira)\quad Let $h_0$ be the formally self-adjoint differential operator (\ref{0doh}) defined on the interval $[0,\infty )$. Let $H_0$ be the self-adjoint operator (\ref{0operator}). 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 $h_0\sigma =E\sigma$. Then there exists a positive $2\times 2$ matrix measure $\{ \varrho _{ij} \}$ defined on $\Lambda$, such that \begin{enumerate} \item the limit \begin{equation} (U_0 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 $L^2(\Lambda ,\{ \varrho _{ij}\})$ for each $f$ in $L^2([0,\infty ),dr)$ and defines an isometric isomorphism $U_0$ of ${\sf E}(\Lambda )L^2([0,\infty ),dr)$ onto $L^2(\Lambda ,\{ \varrho _{ij}\})$, where ${\sf E}(\Lambda )$ is the spectral projection associated with $\Lambda$; \item for each Borel function $G$ defined on the real line and vanishing outside $\Lambda$, \begin{equation} U_0{\cal D}(G(H_0))=\{ [f_i]\in L^2(\Lambda ,\{ \varrho _{ij}\}) \, | \ [Gf_i]\in L^2(\Lambda ,\{ \varrho _{ij}\}) \} \end{equation} and \begin{equation} (U_0G(H_0)f)_i(E)=G(E)(U_0f)_i(E), \quad i=1,2, \, E\in \Lambda , \, f\in {\cal D}(G(H_0)) \, . \end{equation} \end{enumerate} \vskip0.5cm The theorem that provides the inverse of the operator $U_0$ is the following (cf.~Theorem XIII.5.14 of Ref.~\cite{DS} and also Refs.~\cite{DIS,JPA,FP02} for some applications): \vskip0.5cm {\bf Theorem~3} (Weyl-Kodaira)\quad Let $H_0$, $\Lambda$, $\{ \varrho _{ij} \}$, 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 $U_0$ of ${\sf E}(\Lambda )L^2([0,\infty ),dr)$ onto $L^2(\Lambda ,\{ \varrho _{ij}\})$ is given by the formula \begin{equation} (U_0^{-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)\varrho _{ij}(dE) \right) \end{equation} where $F=[F_1,F_2]\in L^2(\Lambda ,\{ \varrho _{ij}\})$, 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 $G(H_0)$ has the representation \begin{equation} G(H_0)f(r)=\int _0^{\infty}f(s)K(H_0,r,s)ds \, , \end{equation} where \begin{equation} K(H_0,r,s)=\sum_{i,j=1}^2 \int_e G(E)\overline{\sigma _i(s;E)}\sigma _j(r;E)\varrho _{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{DS} and also Refs.~\cite{DIS,JPA,FP02} for some applications): \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 ${\rm Re}(H_0)$ be the resolvent set of $H_0$. Let $\{ \sigma _1(r;E),\, \sigma _2(r;E)\}$ be a set of functions which form a basis for the solutions of the equation $h_0\sigma =E\sigma$, $E\in O$, and which are continuous on $(0,\infty )\times O$ and analytically dependent on $E$ for $E$ in $O$. Suppose that the kernel $G_0(r,s;E)$ for the resolvent $(E-H_0)^{-1}$ has a representation \begin{equation} G_0(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 ${\rm Re}(H_0)\cap O$, and that $\{ \varrho _{ij} \}$ is a positive matrix measure on $\Lambda$ associated with $H_0$ as in Theorem 2. Then the functions $\theta _{ij}^{\pm}$ are analytic in ${\rm Re}(H_0)\cap O$, 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} \varrho _{ij}((E_1,E_2))&=& \lim_{\delta \to 0}\lim_{\varepsilon \to 0+} \frac{1}{2\pi i}\int_{E_1+\delta}^{E_2-\delta} [ \theta _{ij}^-(E-i\varepsilon )-\theta _{ij}^-(E+i\varepsilon ) ]dE \\ \quad &=& \lim_{\delta \to 0}\lim_{\varepsilon \to 0+} \frac{1}{2\pi i}\int_{E_1+\delta}^{E_2-\delta} [ \theta _{ij}^+(E-i\varepsilon )-\theta _{ij}^+(E+i\varepsilon ) ]dE \, . \end{array} \end{equation} \vskip0.5cm \def\thesection{\Alph{section}} \section{Appendix~B: Auxiliary Propositions} \setcounter{equation}{0} \label{sec:A2} In this appendix, we list the propositions invoked throughout the paper. \vskip0.5cm {\bf Proposition~1} \quad The quantities \begin{equation} \| \varphi \| _{n,m} := \sqrt{\int_0^{\infty}dr \, \left| (r+1)^n(h_0+1)^m\varphi (r)\right| ^2} \ , \quad \varphi \in {\mathbf \Phi}_0 \, , \, 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{subequations} \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{subequations} The only condition that is somewhat difficult to prove is (\ref{homiensi}): if $\| \varphi \| _{n,m}=0$, then \begin{equation} (1+r)^n(h_0+1)^m\varphi (r)=0 \, , \end{equation} which yields \begin{equation} (h_0+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 $H_0$ whose corresponding eigenvector is $\varphi$. Since $-1$ is not an eigenvalue of $H_0$, $\varphi$ must be the zero vector. If $m>1$, the proof is similar. \vskip0.5cm {\bf Proposition~2}\quad The space ${\mathbf \Phi}_0$ is stable under the action of $H_0$, and $H_0$ is $\tau _{{\mathbf \Phi}_0}$-continuous. \vskip0.2cm {\it Proof}\quad In order to see that $H_0$ is $\tau _{{\mathbf \Phi}_0}$-continuous, we just have to realize that \begin{eqnarray} \| H_0\varphi \| _{n,m}&=&\| (H_0+I)\varphi -\varphi \| _{n,m} \nonumber \\ &\leq & \| (H_0+I)\varphi \| _{n,m}+ \| \varphi \| _{n,m} \nonumber \\ &=&\| \varphi \| _{n,m+1}+\| \varphi \| _{n,m} \, . \label{tauphiscont} \end{eqnarray} We now prove that ${\mathbf \Phi}_0$ is stable under the action of $H_0$. Let $\varphi \in {\mathbf \Phi}_0$. Saying that $\varphi \in {\mathbf \Phi}_0$ is equivalent to saying that $\varphi \in {\cal D}_0$ and that the norms $\| \varphi \| _{n,m}$ are finite for every $n,m=0,1,2,\ldots \ $ Since $H_0\varphi$ is also in ${\cal D}_0$, and since the norms $\| H_0\varphi \| _{n,m}$ are also finite (see Eq.~(\ref{tauphiscont})), the vector $H_0\varphi$ is also in ${\mathbf \Phi}_0$. \vskip0.5cm {\bf Proposition~3}\quad The function \begin{eqnarray} |E\rangle :{\mathbf \Phi}_0 & \longmapsto & {\mathbb C} \nonumber \\ \varphi & \longmapsto & \langle \varphi |E\rangle := \int_0^{\infty}\overline{\varphi (r)}\sigma (r;E)dr =\overline{(U_0\varphi )(E)} \, . \label{adefinitionket} \end{eqnarray} is an antilinear functional on ${\mathbf \Phi}_0$ and a generalized eigenvector of (the restriction to ${\mathbf \Phi}_0$ of) $H_0$. \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} Because \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 ${\mathbf \Phi}_0$ is endowed with the $\tau _{{\mathbf \Phi}_0}$ topology. In order to prove that $|E\rangle$ is a generalized eigenvector of $H_0$, we make use of the conditions (\ref{0mainisexi}) and (\ref{0nmnorms}) satisfied by the elements of ${\mathbf \Phi}_0$: \begin{eqnarray} \langle \varphi |H_0^{\times}|E\rangle &=& \langle H_0^{\dagger}\varphi |E\rangle \nonumber \\ &=& \int_0^{\infty}dr \, \left( -\frac{\hbar ^2}{2m}\frac{d^2}{dr^2} \overline{\varphi(r)} \right) \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} \sigma (r;E) \right) \nonumber \\ &=&E\langle \varphi |E\rangle \, . \end{eqnarray} Similarly, one can also prove that \begin{equation} \langle \varphi | (H_0^{\times})^n|E\rangle = E^n \langle \varphi |E\rangle \, . \end{equation} \vskip0.5cm {\bf Proposition~4} (Nuclear Spectral Theorem) \quad Let \begin{equation} {\mathbf \Phi}_0 \subset L^2([0,\infty ),dr)\subset {\mathbf \Phi}_0^{\times} \end{equation} be the RHS of $H_0$ such that ${\mathbf \Phi}_0$ remains invariant under $H_0$ and $H_0$ is a $\tau _{{\mathbf \Phi}_0}$-continuous operator on ${\mathbf \Phi}_0$. Then, for each $E$ in the spectrum of $H_0$ there is a generalized eigenvector $|E\rangle$ such that \begin{equation} H_0^{\times}|E\rangle =E|E\rangle \end{equation} and such that \begin{equation} (\varphi ,\psi )=\int_{ {\rm Sp}(H_0)}dE\, \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in {\mathbf \Phi}_0 \, , \label{GMT1} \end{equation} and \begin{equation} (\varphi ,H_0^n \psi )=\int_{ {\rm Sp}(H_0)}dE \, E^n \langle \varphi |E\rangle \langle E|\psi \rangle \, , \quad \forall \varphi ,\psi \in {\mathbf \Phi}_0 \, , n=1,2,\ldots \label{GMT2} \end{equation} \vskip0.2cm {\it Proof} \quad Let $\varphi$ and $\psi$ be in ${\mathbf \Phi}_0$. Since $U_0$ is unitary, \begin{equation} (\varphi ,\psi )=(U_0\varphi ,U_0\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_0)}dE \, \overline{ \widehat{\varphi}(E)} \widehat{\psi}(E) \, . \label{sphatvhaps} \end{equation} Because $\varphi$ and $\psi$ belong to ${\mathbf \Phi}_0$, the action of each eigenket $|E\rangle$ on them is well defined, \begin{subequations} \label{actionofEpsi} \begin{eqnarray} \langle \varphi |E\rangle =\overline{ \widehat{\varphi}(E)} \, ,\\ \langle E|\psi \rangle =\widehat{\psi}(E) \, . \end{eqnarray} \end{subequations} By plugging Eq.~(\ref{actionofEpsi}) into Eq.~(\ref{sphatvhaps}) and Eq.~(\ref{sphatvhaps}) into Eq.~(\ref{Usiuni}), we obtain Eq.~(\ref{GMT1}). The proof of (\ref{GMT2}) is similar: \begin{eqnarray} (\varphi ,H_0^n\psi )&=&(U_0\varphi , U_0H_0^nU_0^{-1}U_0\psi ) \nonumber \\ &=& (\widehat{\varphi} ,\widehat{H}_0^n\widehat{\psi} ) \nonumber \\ &=&\int_{{\rm Sp}(H_0)}dE \,\overline{ \widehat{\varphi}(E)} (\widehat{H}_0^n\widehat{\psi})(E) \nonumber \\ &=&\int_{{\rm Sp}(H_0)}dE\, E^n \overline{ \widehat{\varphi}(E)} \widehat{\psi}(E) \nonumber \\ &=& \int_{{\rm Sp}(H_0)}dE \, E^n \langle \varphi |E\rangle \langle E|\psi \rangle \, . \end{eqnarray} \vskip0.5cm {\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 Because \begin{eqnarray} \langle \widehat{\varphi }|U_0^{\times}|E\rangle &=& \langle U_0^{-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 $U_0^{\times}|E\rangle =|\widehat{E}\rangle$ is the antilinear Schwartz delta functional. \begin{thebibliography}{8.} \bibitem{ATKINSON} D.~Atkinson, P.~W.~Johnson, \emph{Quantum Field Theory -- a Self-Contained Introduction}, Rinton Press, Princeton (2002). \bibitem{BOGOLIOBOV} N.~N.~Bogolubov, A.~A.~Logunov, I.~T.~Todorov, \emph{Introduction to Axiomatic Quantum Field Theory}, Benjamin, Reading, Massachusetts (1975). \bibitem{BALLENTINE} L.~E.~Ballentine, \emph{Quantum Mechanics}, Prentice-Hall International, Inc., Englewood Cliffs, New Jersey (1990). \bibitem{BOHM} A.~Bohm, \emph{Quantum Mechanics: Foundations and Applications}, Springer-Verlag, New York (1994). \bibitem{CAPRI} A.~Z.~Capri, \emph{Nonrelativistic Quantum Mechanics}, Benjamin, Menlo Park, California (1985). \bibitem{DUBIN} D.~A.~Dubin, M.~A.~Hennings, \emph{Quantum Mechanics, Algebras and Distributions}, Longman, Harlow (1990). \bibitem{GALINDO} A.~Galindo, P.~Pascual, \emph{Mec\'anica Cu\'antica}, Universidad-Manuales, Eudema (1989); English translation by J.~D.~Garc\'\i a and L.~Alvarez-Gaum\'e, Springer-Verlag (1990). \bibitem{B70} A.~Bohm, \emph{The Rigged Hilbert Space and Quantum Mechanics}, Lecture Notes in Physics, vol.~{\bf 78}, Springer, New York (1978). \bibitem{DIS} R.~de la Madrid, {\it Quantum Mechanics in Rigged Hilbert Space Language}, PhD Thesis, Universidad de Valladolid (2001). Available at \texttt{http://www.isi.it/$\sim$rafa/}. \bibitem{JPA} R.~de la Madrid, J.~Phys.~A: Math.~Gen.~{\bf 35}, 319 (2002); {\sf quant-ph/0110165}. \bibitem{FP02} R.~de la Madrid, A.~Bohm, M.~Gadella, Fortschr.~Phys.~{\bf 50}, 185 (2002); {\sf quant-ph/0109154}. \bibitem{SL} We shall use the Sturm-Liouville theory in the form that appears in the treatise of Dunford and Schwartz~\cite{DS}. Applications of the Sturm-Liouville theory to simple potentials can be found in Refs.~\cite{JPA,FP02,BRANDAS1,BRANDAS,CSF}. \bibitem{DS} N.~Dunford, J.~Schwartz, {\it Linear Operators, Vol.~II}, Interscience Publishers, New York (1963). \bibitem{BRANDAS1} E.~Br\"andas, M.~Rittby, N.~Elander, J.~Math.~Phys.~{\bf 26}, 2648 (1985). \bibitem{BRANDAS} E.~Engdahl, E.~Br\"andas, M.~Rittby, N.~Elander, Phys.~Rev.~A~{\bf 37}, 3777 (1988) \bibitem{CSF} R.~de la Madrid, Chaos, Solitons \& Fractals~{\bf 12}, 2689 (2001); {\sf quant-ph/0107096}. \bibitem{ANTOINE} J.-P.~Antoine, J.~Math.~Phys.~{\bf 10}, 53 (1969); {\it ibid}.~{\bf 10}, 2276 (1969). \end{thebibliography} \end{document} ---------------0405310446565--