Content-Type: multipart/mixed; boundary="-------------0407231541763" This is a multi-part message in MIME format. ---------------0407231541763 Content-Type: text/plain; name="04-228.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-228.keywords" Rigged Hilbert spaces; Sturm-Liouville theory; Dirac's formalism; Fourier methods ---------------0407231541763 Content-Type: text/plain; name="iopams.sty" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="iopams.sty" %% %% This is file `iopams.sty' %% File to include AMS fonts and extra definitions for bold greek %% characters for use with iopart.cls %% \NeedsTeXFormat{LaTeX2e} \ProvidesPackage{iopams}[1997/02/13 v1.0] \RequirePackage{amsgen}[1995/01/01] \RequirePackage{amsfonts}[1995/01/01] \RequirePackage{amssymb}[1995/01/01] \RequirePackage{amsbsy}[1995/01/01] % \iopamstrue % \newif\ifiopams in iopart.cls & iopbk2e.cls % % allows optional text to be in author guidelines % % Bold lower case Greek letters % \newcommand{\balpha}{\boldsymbol{\alpha}} \newcommand{\bbeta}{\boldsymbol{\beta}} \newcommand{\bgamma}{\boldsymbol{\gamma}} \newcommand{\bdelta}{\boldsymbol{\delta}} \newcommand{\bepsilon}{\boldsymbol{\epsilon}} \newcommand{\bzeta}{\boldsymbol{\zeta}} \newcommand{\bfeta}{\boldsymbol{\eta}} \newcommand{\btheta}{\boldsymbol{\theta}} \newcommand{\biota}{\boldsymbol{\iota}} \newcommand{\bkappa}{\boldsymbol{\kappa}} \newcommand{\blambda}{\boldsymbol{\lambda}} \newcommand{\bmu}{\boldsymbol{\mu}} \newcommand{\bnu}{\boldsymbol{\nu}} \newcommand{\bxi}{\boldsymbol{\xi}} \newcommand{\bpi}{\boldsymbol{\pi}} \newcommand{\brho}{\boldsymbol{\rho}} \newcommand{\bsigma}{\boldsymbol{\sigma}} \newcommand{\btau}{\boldsymbol{\tau}} \newcommand{\bupsilon}{\boldsymbol{\upsilon}} \newcommand{\bphi}{\boldsymbol{\phi}} \newcommand{\bchi}{\boldsymbol{\chi}} \newcommand{\bpsi}{\boldsymbol{\psi}} \newcommand{\bomega}{\boldsymbol{\omega}} \newcommand{\bvarepsilon}{\boldsymbol{\varepsilon}} \newcommand{\bvartheta}{\boldsymbol{\vartheta}} \newcommand{\bvaromega}{\boldsymbol{\varomega}} \newcommand{\bvarrho}{\boldsymbol{\varrho}} \newcommand{\bvarzeta}{\boldsymbol{\varsigma}} %NB really sigma \newcommand{\bvarsigma}{\boldsymbol{\varsigma}} \newcommand{\bvarphi}{\boldsymbol{\varphi}} % % Bold upright capital Greek letters % \newcommand{\bGamma}{\boldsymbol{\Gamma}} \newcommand{\bDelta}{\boldsymbol{\Delta}} \newcommand{\bTheta}{\boldsymbol{\Theta}} \newcommand{\bLambda}{\boldsymbol{\Lambda}} \newcommand{\bXi}{\boldsymbol{\Xi}} \newcommand{\bPi}{\boldsymbol{\Pi}} \newcommand{\bSigma}{\boldsymbol{\Sigma}} \newcommand{\bUpsilon}{\boldsymbol{\Upsilon}} \newcommand{\bPhi}{\boldsymbol{\Phi}} \newcommand{\bPsi}{\boldsymbol{\Psi}} \newcommand{\bOmega}{\boldsymbol{\Omega}} % % Bold versions of miscellaneous symbols % \newcommand{\bpartial}{\boldsymbol{\partial}} \newcommand{\bell}{\boldsymbol{\ell}} \newcommand{\bimath}{\boldsymbol{\imath}} \newcommand{\bjmath}{\boldsymbol{\jmath}} \newcommand{\binfty}{\boldsymbol{\infty}} \newcommand{\bnabla}{\boldsymbol{\nabla}} \newcommand{\bdot}{\boldsymbol{\cdot}} % % Symbols for caption % \renewcommand{\opensquare}{\mbox{$\square$}} \renewcommand{\opentriangle}{\mbox{$\vartriangle$}} \renewcommand{\opentriangledown}{\mbox{$\triangledown$}} \renewcommand{\opendiamond}{\mbox{$\lozenge$}} \renewcommand{\fullsquare}{\mbox{$\blacksquare$}} \newcommand{\fulldiamond}{\mbox{$\blacklozenge$}} \newcommand{\fullstar}{\mbox{$\bigstar$}} \newcommand{\fulltriangle}{\mbox{$\blacktriangle$}} \newcommand{\fulltriangledown}{\mbox{$\blacktriangledown$}} \endinput %% %% End of file `iopams.sty'. ---------------0407231541763 Content-Type: application/x-tex; name="04jpa.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="04jpa.tex" \documentclass[12pt]{iopart} \eqnobysec % Uncomment next line if AMS fonts required \usepackage{iopams} \usepackage{amsthm} \newcommand{\Sw}{{\cal S}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swt}{{\cal S}^{\times}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swp}{{\cal S}^{\prime}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\rhsSwt}{\Sw \subset L^2(\mathbb R, \rmd x) \subset \Swt} \newcommand{\rhsSwp}{\Sw \subset L^2(\mathbb R, \rmd x) \subset \Swp} \newcommand{\Czi}{C_0^{\infty}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhpm}{\widehat{{\cal S}}_{\pm}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmt}{\widehat{{\cal S}}_{\pm}^{\times}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpml}{\widehat{{\cal S}}_{\pm;l}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmr}{\widehat{{\cal S}}_{\pm;r}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmlt}{\widehat{{\cal S}}_{\pm;l}^{\times}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmrt}{\widehat{{\cal S}}_{\pm;r}^{\times}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmp}{\widehat{{\cal S}}_{\pm}^{\prime}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmlp}{\widehat{{\cal S}}_{\pm;l}^{\prime}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmrp}{\widehat{{\cal S}}_{\pm;r}^{\prime}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swh}{ \widehat{{\cal S}}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swht}{ \widehat{{\cal S}}^{\times}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhp}{\widehat{{\cal S}}^{\prime}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhhpm}{\widehat{\widehat{{\cal S}}\,}\! _{\pm}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhhpmt}{\widehat{\widehat{{\cal S}}\,}\! _{\pm} \, \hskip-.32cm^{\times}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhhpmp}{\widehat{\widehat{{\cal S}}\,}\! _{\pm} \,\hskip-.22cm ^{\prime}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \begin{document} \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[The RHS of the 1D rectangular barrier potential]{The rigged Hilbert space of the algebra of the one-dimensional rectangular barrier potential} \author{Rafael de la Madrid} %\dag\ \footnote[3]{E-mail: %wtbdemor@lg.ehu.es}} \address{%\dag\ Departamento de F\'\i sica Te\'orica, Facultad de Ciencias, Universidad del Pa\'\i s Vasco \\ E-48080 Bilbao, Spain \\ E-mail: {\texttt{wtbdemor@lg.ehu.es}} \\ URL: {\texttt{http://www.ehu.es/$\sim$wtbdemor}}} \begin{abstract} The rigged Hilbert space of the algebra of the one-dimensional rectangular barrier potential is constructed. The one-dimensional rectangular potential provides another opportunity to show that the rigged Hilbert space fully accounts for Dirac's bra-ket formalism. The analogy between Dirac's formalism and Fourier methods is pointed out. \end{abstract} \pacs{03.65.-w, 02.30.Hq} %\submitto{\JPA} \maketitle \section{Introduction} One-dimensional (1D) models play a paramount role in Quantum Mechanics, because they enable us to understand a number of properties that also appear in more realistic situations. The simplicity of 1D models facilitates testing new hypothesis, approximation methods and theories without unnecessary and costly complications. In many cases, after proper calculations, it is possible to reduce an intricate problem to a Schr\"odinger equation in one dimension. For example: three-dimensional (3D) spherically symmetric Schr\"odinger equations can be reduced to 1D radial equations; time quantities such as tunneling or arrival times have in many cases been studied in 1D models~\cite{LEON,MUGA}; electrons in strong magnetic fields can be described by 1D potentials~\cite{BRUMMELHUIS}; some surface phenomena are described by 1D models~\cite{BY}; the application of the effective mass approximation to layered semiconductor structures leads to effective 1D systems~\cite{BASTARD}; the conductance of some semiconductor nanostructures can be obtained by solving 1D Schr\"odinger equations~\cite{WULF}. One-dimensional potentials have even practical interest, since advances in the microfabrication of semiconductors have allowed to design and control essentially 1D potentials~\cite{SAKAKI,KOLBAS}. One-dimensional models are also ideally suited to examine the mathematical foundations of Quantum Mechanics. In this paper, we shall take this foundational route. We shall construct the rigged Hilbert space (RHS) of the one-dimensional rectangular barrier potential, thereby showing that the mathematical setting of quantum mechanical systems with continuous spectrum is the RHS rather than just the Hilbert space. This paper follows up on Refs.~\cite{DIS,JPA,FP02}, where the RHSs of 3D spherical shell potentials were constructed~\cite{FNOTE1}, and on Ref.~\cite{IJTP}, where the RHS of the 3D free Hamiltonian was constructed. The present paper complements Refs.~\cite{DIS,JPA,FP02,IJTP} in the following ways: \begin{itemize} \item[$\bullet$] We treat a truly 1D model on the full real line, rather than the radial part of a 3D model. \item[$\bullet$] We construct the RHS of the algebra generated by the position, momentum and energy observables, rather than just the RHS of the Hamiltonian. \item[$\bullet$] We construct not only the Dirac kets but also the Dirac bras, thereby showing even more clearly that the RHS fully implements Dirac's bra-ket formalism. \end{itemize} The model we consider in this paper is supposed to represent a spinless particle moving in one dimension and impinging on a barrier. The relevant observables to this system are the position $Q$, the momentum $P$ and the Hamiltonian $H$. These observables are represented by the following differential operators: \begin{eqnarray} Qf(x)=xf(x) \, , \label{fdopp} \\ Pf(x)=-\rmi \hbar \frac{\rmd}{\rmd x}f(x) \, , \\ Hf(x)=-\frac{\hbar ^2}{2m}\frac{\rmd}{\rmd x^2}f(x)+V(x)f(x) \, , \label{fdoph} \end{eqnarray} where \begin{equation} V(x)=\left\{ \begin{array}{ll} 0 &-\infty 0$ has been introduced on dimensional grounds. It is straightforward to see that the only solution of Eq.~(\ref{defininequs}) that belongs to ${\cal D}(H^{\dagger}_{\rm min})$ is the zero solution, that is, $n_{\pm}(H)=0$. Thus, the only domain ${\cal D}(H)$ of $L^2 (\mathbb R,\rmd x )$ on which $h$ induces a self-adjoint operator coincides with the maximal domain: \begin{equation} {\cal D}(H)=\left\{ f\in L^2(\mathbb R,\rmd x) \, : \ f \in AC^2(\mathbb R,\rmd x), \ hf \in L^2(\mathbb R,\rmd x) \right\} \, . \label{domainH} \end{equation} By similar arguments, it can be shown that the only domain on which the multiplication operator induces a self-adjoint operator is given by \begin{equation} {\cal D}(Q)=\left\{ f\in L^2(\mathbb R,\rmd x) \, : xf \in L^2(\mathbb R,\rmd x) \right\} \, , \end{equation} and that the only domain on which the differential operator $-\rmi \hbar \rmd /\rmd x$ induces a self-adjoint operator is given by \begin{equation} {\cal D}(P)=\left\{ f\in L^2(\mathbb R, \rmd x) \, : \ f \in AC(\mathbb R, \rmd x), \ f' \in L^2(\mathbb R,\rmd x) \right\} \, . \end{equation} \section{The resolvent operator and the Green function} \label{sec:reopangreefu} In this section, we obtain the resolvent and the Green function of $H$, which can be easily calculated by way of the following theorem (cf.~Theorem~XIII.3.16 of Ref.~\cite{DUNFORDII}): \vskip0.5cm \theoremstyle{plain} \newtheorem*{Th1}{Theorem~1} \begin{Th1} Let $H$ be the self-adjoint Hamiltonian operator derived from the real formal differential operator (\ref{doh}) and the domain (\ref{domainH}). Let ${\rm Im}(E) \neq 0$. Then there is exactly one solution $\chi _{\rm r}(x;E)$ of $(h-E)\sigma =0$ square integrable at $-\infty$, and exactly one solution $\chi _{\rm l}(x;E)$ of $(h-E)\sigma =0$ square-integrable at $+\infty$. The resolvent $(E-H)^{-1}$ is an integral operator whose kernel $G(x,x';E)$ is given by \begin{equation} G(x,x';E)=\left\{ \begin{array}{ll} \frac{2m}{\hbar ^2} \, \frac{\chi _{\rm r}(x;E) \, \chi _{\rm l}(x';E)} {W(\chi _{\rm r},\chi _{\rm l})} &xx' \, , \end{array} \right. \label{exofGFA} \end{equation} where $W(\chi _{\rm r},\chi _{\rm l} )$ is the Wronskian of $\chi _{\rm r}$ and $\chi _{\rm l}$: \begin{equation} W(\chi _{\rm r},\chi _{\rm l} )= \chi _{\rm r}\chi _{\rm l}'-\chi _{\rm r}'\chi _{\rm l} \, . \end{equation} \end{Th1} \vskip0.5cm To obtain $G(x,x';E)$, we divide the complex $E$-plane in three regions (left half-plane, first quadrant, and fourth quadrant) and apply Theorem~1 to each of these regions separately. In our calculations, we shall use the following branch of the square root function: \begin{equation} \hskip-1.7cm \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} This branch is chosen because it grants the following relation: \begin{equation} \overline{\sqrt{\overline{z}\, }}=z \, , \quad z\in \mathbb C \, . \label{cczez} \end{equation} It is important to keep in mind that $\chi _{\rm r}(x;E)$ and $\chi _{\rm l}(x;E)$ of Theorem~1 are well defined for real as well as for complex energies. More precisely, the functions $\chi _{\rm r}(x;E)$ and $\chi _{\rm l}(x;E)$, which are derived for complex $E$ of nonzero imaginary part, have a well-defined limiting value when $E$ approaches the real line. The values of $\chi _{\rm r}(x;E)$ and $\chi _{\rm l}(x;E)$ for complex $E$ will be used in this section to calculate $G(x,x';E)$. The values of $\chi _{\rm r}(x;E)$ and $\chi _{\rm l}(x;E)$ for real $E$ will be used in Sec.~\ref{sec:consrhs} to construct the bras and kets associated to the energies in the spectrum of the Hamiltonian. \subsection{Left half-plane: ${\rm Re}(E)<0$, ${\rm Im}(E)\neq 0$} According Theorem~1, we need to obtain the eigensolutions $\widetilde{\chi}_{\rm r}$ and $\widetilde{\chi}_{\rm l}$ of the Schr\"odinger equation \begin{equation} \left( -\frac{\hbar ^2}{2m}\frac{\rmd ^2 \ }{\rmd x^2}+V(x)\right) \sigma (x;E)= E \sigma (x;E) \label{sde} \end{equation} that are square integrable at $-\infty$ and at $\infty$, respectively. Thus, the eigensolution $\widetilde{\chi}_{\rm r}(x;E)$ satisfies \numparts \begin{eqnarray} && \widetilde{\chi}_{\rm r} (x;E)\in AC^2(\mathbb R) \, , \label{bcchiac} \\ && \widetilde{\chi}_{\rm r}(x;E) {\rm \ is \ square \ integrable \ at \ } -\infty \, , \label{sbca03} \end{eqnarray} \label{eigsbo0co} \endnumparts whereas the eigensolution $\widetilde{\chi}_{\rm l}(x;E)$ satisfies \numparts \begin{eqnarray} &&\widetilde{\chi}_{\rm l}(x;E)\in AC^2(\mathbb R) \, , \label{bcainfty1} \\ &&\widetilde{\chi}_{\rm l}(x;E) \ {\rm is \ square \ integrable \ at \ } +\infty \, . \label{bcainfty2} \end{eqnarray} \label{thetcoabejej} \endnumparts Solving Eq.~(\ref{sde}) subjected to (\ref{bcchiac})-(\ref{sbca03}) yields \begin{equation} \widetilde{\chi}_{\rm r}(x;E)= \left( \frac{m}{2\pi \widetilde{k} \hbar ^2} \right)^{1/2} \times \left\{ \begin{array}{lc} \widetilde{T}(\widetilde{k})\rme ^{\widetilde{k}x} \quad &-\inftyx' \end{array} \right. \quad \mbox{Re}(E)<0 \, , \ \mbox{Im}(E)\neq 0 \, . \label{green-} \end{equation} \subsection{First quadrant: ${\rm Re}(E)>0$, ${\rm Im}(E)> 0$} \label{sec:region++} When $E$ belongs to the first quadrant, the eigensolution ${\chi}_{\rm r}^+$ that satisfies Eq.~(\ref{sde}) subjected to the boundary conditions (\ref{bcchiac})-(\ref{sbca03}) is given by \begin{equation} \chi _{\rm r}^+(x;E)= \left( \frac{m}{2\pi k \hbar ^2} \right)^{1/2} \times \left\{ \begin{array}{lc} T (k)\rme ^{-\rmi kx} &-\infty x' \end{array} \right. \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)>0 \, . \label{green++} \end{equation} \subsection{Fourth quadrant: ${\rm Re}(E)>0$, ${\rm Im}(E)< 0$} When $E$ belongs to the fourth quadrant, the eigensolution ${\chi}_{\rm r}^-$ that satisfies Eq.~(\ref{sde}) subjected to the boundary conditions (\ref{bcchiac})-(\ref{sbca03}) is given by \begin{equation} \chi _{\rm r}^-(x;E)= \left( \frac{m}{2\pi k \hbar ^2} \right)^{1/2} \times \left\{ \begin{array}{lc} T^*(k)\rme ^{\rmi kx} &-\infty x' \end{array} \right. \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)< 0 \, . \label{green+-} \end{equation} To finish this section, we recall the resolvents of $Q$ and $P$, which are well known and can be calculated by similar arguments. The resolvent of $Q$ is an integral operator whose kernel is \begin{equation} \langle x|\frac{1}{z-Q}|x'\rangle = \frac{1}{z-x} \delta (x-x') \, , \quad z \in \mathbb{C}/\mathbb{R} \, , \ x, x' \in \mathbb{R} \, . \end{equation} In the upper complex $p$-plane, the kernel of the resolvent of $P$ is given by~\cite{DUNFORDII} \begin{equation} \langle x|\frac{1}{p-P}|x'\rangle = \left\{ \begin{array}{cl} 0 & xx'\end{array} \right. \quad {\rm Im}(p)>0 \, , \end{equation} whereas in the lower complex $p$-plane it is given by~\cite{DUNFORDII} \begin{equation} \langle x|\frac{1}{p-P}|x'\rangle = \left\{ \begin{array}{cl} - \frac{1}{\rmi \hbar}\, \rme ^{\rmi p(x-x')/\hbar} & xx' \end{array} \right. \quad {\rm Im}(p)<0 \, . \end{equation} \section{Spectrum} \label{sec:spectrum} In this section, we obtain the spectrum of $H$, which we shall denote by $\mbox{Sp}(H)$. In order to obtain $\mbox{Sp}(H)$, we shall apply Theorem~3 below. Before stating Theorem~3, we need to state Theorem~2, which provides the unitary operators that diagonalize $H$ (cf.~Theorem XIII.5.13 of Ref.~\cite{DUNFORDII}): \vskip0.5cm \theoremstyle{plain} \newtheorem*{Th2}{Theorem~2} \begin{Th2} (Weyl-Kodaira) Let $h$ be the formally self-adjoint differential operator~(\ref{doh}). Let $H$ be the corresponding self-adjoint Hamiltonian. Let $\Lambda$ be an open interval of the real axis, and suppose that there is given a set $\{ \sigma _1(x;E),\, \sigma _2(x;E)\}$ of functions, defined and continuous on $\mathbb R \times \Lambda$, such that for each fixed $E$ in $\Lambda$, $\{ \sigma _1(x;E),\, \sigma _2(x;E)\}$ forms a basis for the space of solutions of $h\sigma =E\sigma$. Then there exists a positive $2\times 2$ matrix measure $\{ \varrho _{ij} \}$ defined on $\Lambda$, such that the limit \begin{equation} (Uf)_i(E)=\lim_{c\to 0}\lim_{d\to \infty} \left[ \int_c^d f(x) \overline{\sigma _i(x;E)}\rmd x \right] \end{equation} exists in the topology of $L^2(\Lambda ,\{ \varrho _{ij}\})$ for each $f$ in $L^2(\mathbb R,\rmd x)$ and defines an isometric isomorphism $U$ of ${\sf E}(\Lambda )L^2(\mathbb R,\rmd x)$ onto $L^2(\Lambda ,\{ \varrho _{ij}\})$, where ${\sf E}(\Lambda )$ is the spectral projection associated with $\Lambda$. \end{Th2} \vskip0.5cm The spectral measures $\{ \rho _{ij}\}$ are provided by the following theorem (cf.~Theorem XIII.5.18 of Ref.~\cite{DUNFORDII}): \vskip0.5cm \theoremstyle{plain} \newtheorem*{Th3}{Theorem~3} \begin{Th3} (Titchmarsh-Kodaira) Let $\Lambda$ be an open interval of the real axis and $O$ be an open set in the complex plane containing $\Lambda$. Let ${\rm re}(H)$ be the resolvent set of $H$. Let $\{ \sigma _1(x;E),\, \sigma _2(x;E)\}$ 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 $\mathbb R \times O$ and analytically dependent on $E$ for $E$ in $O$. Suppose that the kernel $G(x,x';E)$ for the resolvent $(E-H)^{-1}$ has a representation \begin{equation} G(x,x';E)=\left\{ \begin{array}{lll} \sum_{i,j=1}^2 \theta _{ij}^-(E)\sigma _i(x;E) \overline{\sigma _j(x';\overline{E})}\, , & \qquad & xx' \, , \end{array} \right. \label{greenfunitthes} \end{equation} for all $E$ in ${\rm re}(H)\cap O$, and that $\{ \varrho _{ij} \}$ 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} \hskip-1cm \varrho _{ij}((E_1,E_2))&=& \lim_{\delta \to 0}\lim_{\varepsilon \to 0+} \frac{1}{2\pi \rmi}\int_{E_1+\delta}^{E_2-\delta} [ \theta _{ij}^-(E-\rmi \varepsilon ) -\theta _{ij}^-(E+\rmi \varepsilon ) ]\rmd E \\ [1ex] \hskip-1cm \quad &=& \lim_{\delta \to 0}\lim_{\varepsilon \to 0+} \frac{1}{2\pi \rmi}\int_{E_1+\delta}^{E_2-\delta} [ \theta _{ij}^+(E-\rmi \varepsilon )- \theta _{ij}^+(E+\rmi \varepsilon ) ] \rmd E \, . \end{array} \label{specmesa} \end{equation} \end{Th3} \vskip0.5cm From Eq.~(\ref{specmesa}), it is clear that in order to obtain $\mbox{Sp}(H)$ and $\varrho _{ij}$, we need to see on what real $E$'s the functions $\theta _{ij}^{\pm}(E)$ fail to be analytic. We shall do so by taking $\Lambda$ in Theorem~3 to be $(-\infty ,0)$ and $(0,\infty )$. \subsection{Negative Energy Real Line: $\Lambda =(-\infty ,0)$} \label{sec:NeERlin} On the negative real line, we choose the basis $\{ \sigma _1, \sigma _2\}$ of Theorem~3 as \numparts \begin{eqnarray} &&\sigma _1(x;E)=\widetilde{\chi}_{\rm l}(x;E) \, , \label{tildsi1} \\ &&\sigma _2(x;E)=\widetilde{\chi}_{\rm r}(x;E) \, . \label{tildsi2} \end{eqnarray} \endnumparts From Eqs.~(\ref{cczez}) and (\ref{tildethetfunc}) it follows that \begin{equation} \overline{\widetilde{\chi}_l(x';\overline{E})}= \widetilde{\chi}_l(x';E) \, . \label{croscctil} \end{equation} Now, by taking advantage of Eqs.~(\ref{tildsi1}), (\ref{tildsi2}) and (\ref{croscctil}), we write Eq.~(\ref{green-}) as \begin{equation} \hskip-1.7cm G(x,x';E)= -2\pi \, \frac{\sigma _2(x;E) \overline{\sigma _1(x';\overline{E})}} {\widetilde{T}(E)} \, , \quad x0, \mbox{Im}(E)>0\, , \, x>x' \, . \label{redaot++} \end{eqnarray} After substituting Eq.~(\ref{chil-s1s2}) into Eq.~(\ref{green+-}) and after some calculations, we get to \begin{eqnarray} && \hskip-0.4cm G(x,x';E)= \frac{2\pi}{\rmi} \, \left[ \frac{R_{\rm r}(E)}{T(E)}\sigma _1(x;E) \overline{\sigma _2 (x';\overline{E})} -\sigma _2(x;E) \overline{\sigma _2 (x';\overline{E})} \right] \, , \nonumber \\ &&\qquad \hskip5.3cm \mbox{Re}(E)>0, \mbox{Im}(E)<0\, , \, x>x' \, . \label{redaot+-} \end{eqnarray} By comparing (\ref{greenfunitthes}) to (\ref{redaot++}) we obtain \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} \frac{2\pi}{\rmi} & -\frac{2\pi}{\rmi} \frac{R_{\rm l}^*(E)}{T^*(E)}\\ 0 & 0 \end{array} \right) , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)>0 \, . \label{theta++} \end{equation} By comparing (\ref{greenfunitthes}) to (\ref{redaot+-}) we obtain \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} 0 & \frac{2\pi}{\rmi} \frac{R_{\rm r}(E)}{T(E)} \\ 0 & -\frac{2\pi}{\rmi} \end{array} \right) , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)<0 \, . \label{theta+-} \end{equation} Substitution of Eqs.~(\ref{theta++}) and (\ref{theta+-}) into Eq.~(\ref{specmesa}) yield the spectral measures $\varrho _{ij}$ of Theorem~3. The measure $\varrho _{21}$ is clearly zero. So is the measure $\varrho _{12}$, since \begin{eqnarray} \varrho _{12}((E_1,E_2))&=& \lim _{\delta \to 0} \lim _{\varepsilon \to 0+} \frac{1}{2\pi \rmi} \int_{E_1+\delta}^{E_2-\delta} \left[ \theta _{12}^+ (E-\rmi \varepsilon ) -\theta _{12}^+ (E+\rmi \varepsilon ) \right] \rmd E \nonumber \\ &=&\int_{E_1}^{E_2} - \left( \frac{R_{\rm r}(E)}{T(E)}+\frac{R_{\rm l}^*(E)}{T^*(E)}\right) \rmd E \nonumber \\ &=&0 \, , \end{eqnarray} where in the last step we have used the relation \begin{equation} R_{\rm r}(E)T^*(E)+T(E)R_{\rm l}^*(E)=0 \, . \end{equation} The measures $\varrho _{11}$ and $\varrho _{22}$ are just the Lebesgue measure, since \begin{eqnarray} \varrho _{11}((E_1,E_2))&=& \lim _{\delta \to 0} \lim _{\varepsilon \to 0+} \frac{1}{2\pi \rmi} \int_{E_1+\delta}^{E_2-\delta} \left[ \theta _{11}^+ (E-\rmi \varepsilon ) -\theta _{11}^+ (E+\rmi \varepsilon ) \right] \rmd E \nonumber \\ &=&\int_{E_1}^{E_2} \rmd E =E_2-E_1 \, , \end{eqnarray} and \begin{eqnarray} \varrho _{22}((E_1,E_2))&=& \lim _{\delta \to 0} \lim _{\varepsilon \to 0+} \frac{1}{2\pi \rmi} \int_{E_1+\delta}^{E_2-\delta} \left[ \theta _{22}^+ (E-\rmi \varepsilon ) -\theta _{22}^+ (E+\rmi \varepsilon ) \right] \rmd E \nonumber \\ &=&\int_{E_1}^{E_2} \rmd E =E_2-E_1 \, . \end{eqnarray} Clearly, the functions $\theta _{11}^+(E)$ and $\theta _{22}^+(E)$ both have a branch cut along $(0,\infty)$, and therefore $(0,\infty )$ is included in ${\rm Sp}(H)$. Since ${\rm Sp}(H)$ is a closed set, it must hold that \begin{equation} \mbox{Sp}(H)= [0,\infty ) \, . \end{equation} We note that, instead of the ``initial'' basis (\ref{sigma1=sci})-(\ref{sigam2cos}), we could use the ``final'' basis: \numparts \begin{eqnarray} &&\sigma _1(x;E)=\chi _{\rm l}^-(x;E)\, , \\ &&\sigma _2(x;E)= \chi _{\rm r}^-(x;E)\, . \end{eqnarray} \endnumparts This basis produces the same spectrum (as it should be) and the same spectral measures (cf.~\ref{sec:compfor-}). To finish this section, we recall that the spectra of the position and momentum observables coincide with the full real line: \begin{equation} {\rm Sp}(Q)= {\rm Sp}(P) = (-\infty , \infty ) \, . \end{equation} The spectra of $Q$ and $P$ are simple, whereas the spectrum of $H$ is doubly degenerate. Indeed, to each energy $E\in [0,\infty )$ there correspond two linearly independent eigenfunctions, $\chi _{\rm l}^+$ and $\chi _{\rm r}^+$ (or, equivalently, $\chi _{\rm l}^-$ and $\chi _{\rm r}^-$). \section{Diagonalization and eigenfunction expansion} \label{sec:diagonalization} Theorem~2 of Sec.~\ref{sec:spectrum} provides the means to construct two unitary operators $U_{\pm}$ that diagonalize $H$. The operator $U_+$ is associated with the basis $\{ \chi _{\rm l}^+, \chi _{\rm r}^+ \}$, whereas $U_-$ is associated with $\{ \chi _{\rm l}^-, \chi _{\rm r}^- \}$. These unitary operators transform from the position into the energy representation, and they induce two eigenfunction expansions and two direct integral decompositions of the Hilbert space. For the sake of brevity, we shall present each calculation associated with $U_+$ together with the corresponding calculation associated with $U_-$. By Theorem~2, the mappings $U_{\pm}$ are given by \begin{equation} \begin{array}{rcl} \hskip-0.5cm U_{\pm}: L^2(\mathbb R, \rmd x) &\longmapsto & L^2([0,\infty),\rmd E)\oplus L^2( [0,\infty ),\rmd E) \nonumber \\ \hskip-2cm f(x) &\longmapsto & \widehat{f}^{\pm}(E)\equiv U_{\pm}f(E) \equiv \left[ (U_{\pm}f)_{\rm l}(E),(U_{\pm}f)_{\rm r}(E) \right] \, , \end{array} \label{operatoruplu} \end{equation} where \begin{eqnarray} \widehat{f}_{\rm l}^{\pm}(E)\equiv (U_{\pm}f)_{\rm l}(E)= \int_{-\infty}^{\infty}\rmd x \, f(x) \overline{\chi _{\rm l}^{\pm}(x;E)} \, , \label{hatEl+} \\ \widehat{f}_{\rm r}^{\pm}(E)\equiv (U_{\pm}f)_{\rm r}(E)= \int_{-\infty}^{\infty}\rmd x \, f(x) \overline{\chi _{\rm r}^{\pm}(x;E)} \, . \label{hatEr+} \end{eqnarray} Note that $\widehat{f}^{\pm}(E)\equiv U_{\pm}f(E)$ are two-component vectors, because the spectrum of $H$ is doubly degenerate. The inverses of $U_{\pm}$ can be obtained from the following theorem (cf.~Theorem XIII.5.14 of Ref.~\cite{DUNFORDII}): \vskip0.5cm \theoremstyle{plain} \newtheorem*{Th4}{Theorem~4} \begin{Th4} (Weyl-Kodaira) Let $H$, $\Lambda$, $\{ \varrho _{ij} \}$, etc., be as in Theorem~2. Let $E_0$ and $E_1$ be the end points of $\Lambda$. Then the inverse of the isometric isomorphism $U$ of ${\sf E}(\Lambda )L^2(\mathbb R,\rmd x)$ onto $L^2(\Lambda ,\{ \varrho _{ij}\})$ is given by the formula \begin{equation} (U^{-1}F)(x)=\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(x;E)\varrho _{ij}(\rmd E) \right) \end{equation} where $F=[F_1,F_2]\in L^2(\Lambda ,\{ \varrho _{ij}\})$, the limit existing in the topology of $L^2(\mathbb R,\rmd x)$. \end{Th4} \vskip0.5cm By Theorem~4, the inverses of $U_{\pm}$ are given by \begin{equation} \hskip-1cm f(x)=(U_{\pm}^{-1}\widehat{f})(x)= \int_{0}^{\infty}\rmd E\, \widehat{f}_{\rm l}^{\pm}(E) \chi _{\rm l}^{\pm}(x;E) +\int_{0}^{\infty}\rmd E\, \widehat{f}_{\rm r}^{\pm}(E) \chi _{\rm r}^{\pm}(x;E) \, , \label{invdiagonaliza} \end{equation} where \begin{equation} \widehat{f}_{\rm l}^{\pm}(E), \ \widehat{f}_{\rm r}^{\pm}(E) \in L^2([0,\infty ),\rmd E) \, . \end{equation} The operators $U_{\pm}^{-1}$ transform from $L^2([0,\infty ),\rmd E)\oplus L^2([0,\infty ),\rmd E)$ onto $L^2(\mathbb R,\rmd x)$. Note that Eq.~(\ref{invdiagonaliza}) can also be seen as the eigenfunction expansions of any element $f(x)$ of $L^2(\mathbb R, \rmd x )$ in terms of the basis $\{ \chi _{\rm l}^{\pm}, \chi _{\rm r}^{\pm} \}$. Similarly, we can write the two-component vectors $U_{\pm}f$, Eq.~(\ref{operatoruplu}), as \begin{equation} \widehat{f}^{\pm}(E)\equiv \int_{-\infty}^{\infty}\rmd x \, f(x) \overline{\chi _{\rm l}^{\pm}(x;E)} \dotplus \int_{-\infty}^{\infty}\rmd x \, f(x) \overline{\chi _{\rm r}^{\pm}(x;E)} \, , \label{eigenfucfinphi} \end{equation} which provide the eigenfunction expansions of any element $\widehat{f}^{\pm}(E)$ of $L^2([0,\infty ),\rmd E )\oplus L^2([0,\infty ),\rmd E)$ in terms of $\{ \chi _{\rm l}^{\pm}, \chi _{\rm r}^{\pm} \}$. (The symbol $\dotplus$ in Eq.~(\ref{eigenfucfinphi}) intends to mean that, from a mathematical point of view, this equation should be seen as a two-component vector equality rather than as an actual sum.) A straightforward calculation shows that \begin{equation} \hskip-1.5cm \widehat{H}\widehat{f}^{\pm}(E)= U_{\pm}HU_{\pm}^{-1}\widehat{f}^{\pm}(E)=E\widehat{f}^{\pm}(E) \equiv [ E\widehat{f}_{\rm l}^{\pm}(E), E\widehat{f}_{\rm r}^{\pm}(E) ] \, , \quad f\in {\cal D}(H) \, . \label{idagoplofh} \end{equation} The direct integral decompositions of the Hilbert space induced by $U_{\pm}$ read as \begin{equation} \begin{array}{rcl} U_{\pm}:\mathcal{H} &\longmapsto & \widehat{\cal H}= \int_{0}^{\infty} \widehat{\cal H}_{\rm l}(E)\rmd E \oplus \int_{0}^{\infty} \widehat{\cal H}_{\rm r}(E)\rmd E \\ f &\longmapsto & U_{\pm}f:= [(U_{\pm}f)_{\rm l},(U_{\pm}f)_{\rm r}] \, , \end{array} \label{dirintdec} \end{equation} where ${\cal H}$ is realized by $L^2(\mathbb R,\rmd x)$, and $\widehat{\cal H}$ is realized by $L^2([0,\infty ),\rmd E)\oplus L^2([0,\infty ),\rmd E)$. The Hilbert spaces $\widehat{\mathcal H}_{\rm l}(E)$ and $\widehat{\mathcal H}_{\rm r}(E)$, which are associated to each energy $E$ in the spectrum of $H$, are realized by $\mathbb C$. The scalar product on $\widehat{\cal H}$ can be written as \begin{equation} \left( \widehat{f},\widehat{g} \right) _{\widehat{\cal H}}= \int_0^{\infty}\rmd E\, \overline{\widehat{f}_{\rm l}^{\pm}(E)} \, \widehat{g}_{\rm l}^{\pm}(E) +\int_0^{\infty}\rmd E\, \overline{\widehat{f}_{\rm r}^{\pm}(E)} \, \widehat{g}_{\rm r}^{\pm}(E) \, . \end{equation} It is worthwhile noticing the similarities between $U_{\pm}$ and the Fourier transform $\cal F$, which is given by \begin{equation} \begin{array}{rcl} {\cal F}: L^2({\mathbb R},\rmd x) &\longmapsto & L^2({\mathbb R},\rmd p) \\ f(x) &\longmapsto & {\cal F}f(p)=\frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty}\rmd x \, f(x) \rme ^{-\rmi px/\hbar} \, . \end{array} \label{fourt} \end{equation} The operators $U_{\pm}$ transform between the position and the energy representations, and $\cal F$ transforms between the position and the momentum representations. The kernels of $U_{\pm}$, $\chi ^{\pm}_{\rm l,r}$, are eigenfunctions of the energy operator, and the kernel of $\cal F$, $\frac{1}{\sqrt{2\pi \hbar}} \rme ^{-\rmi px/\hbar}$, is an eigenfunction of the momentum operator. Like $\cal F$, $U_{\pm}$ are unitary operators. Thus, $U_{\pm}$ are Fourier-like transforms. %The calculations associated to the basis %$\left\{ \chi _{\rm l}^-, \chi _{\rm r}^-\right\}$ and $U_-$ are the same as %those for $\left\{ \chi _{\rm l}^+, \chi _{\rm r}^+\right\}$ and $U_+$. The %unitary operator $U_-$ from $L^2(\mathbb R,\rmd x)$ %onto $L^2([0,\infty ),\rmd E )\oplus L^2([0,\infty ),\rmd E)$ that acts as %\begin{equation} %\begin{array}{rcl} % \hskip-0.5cm % U_-: L^2(\mathbb R, \rmd x) % &\longmapsto & L^2([0,\infty),\rmd E)\oplus L^2( [0,\infty ),\rmd E) % \nonumber \\ % \hskip-2cm f(x) &\longmapsto & \widehat{f}^-(E)\equiv U_-f(E)\equiv % \left[ (U_-f)_{\rm l}(E),(U_-f)_{\rm r}(E) \right] \, , %\end{array} % \label{operatoruplu-} %\end{equation} %where %\begin{equation} % U_-f(E)=\widehat{f}^-(E)\equiv \int_{-\infty}^{\infty}\rmd x \, % f(x) \overline{\chi _{\rm l}^-(x;E)} \dotplus % \int_{-\infty}^{\infty}\rmd x \, % f(x) \overline{\chi _{\rm r}^-(x;E)} \, . % \label{eigenfucfinphi-t} %\end{equation} %This equation provides the eigenfunction expansion of any element %$\widehat{f}^-(E)$ of $L^2([0,\infty ),\rmd E )\oplus L^2([0,\infty ),\rmd E)$ %in terms of $\chi _{\rm l}^-(x;E)$ and $\chi _{\rm r}^-(x;E)$. (As in %Eq.~(\ref{eigenfucfinphi}), the $\dotplus$ in Eq.~(\ref{eigenfucfinphi-t}) %indicates a two-component vector equality rather than an actual sum.) The %inverse of $U_-$ is given by %\begin{equation} % \hskip-1cm f(x)=(U_-^{-1}\widehat{f})(x)= % \int_{0}^{\infty}\rmd E\, \widehat{f}_{\rm l}^-(E) % \chi _{\rm l}^-(x;E) % +\int_{0}^{\infty}\rmd E\, \widehat{f}_{\rm r}^-(E) % \chi _{\rm r}^-(x;E) \, , % \label{invdiagonaliza-t} %\end{equation} %where %\begin{equation} % \widehat{f}_{\rm l}^-(E), \ \widehat{f}_{\rm r}^-(E) % \in L^2([0,\infty ),\rmd E) \, . %\end{equation} %The operator $U_-^{-1}$ transforms %from $L^2([0,\infty ),\rmd E)\oplus L^2([0,\infty ),\rmd E)$ onto %$L^2(\mathbb R,\rmd x)$. Equation~(\ref{invdiagonaliza-t}) can be %seen as the eigenfunction expansion of any element $f(x)$ of %$L^2(\mathbb R, \rmd x )$ in terms of %$\chi _{\rm l}^-(x;E)$ and $\chi _{\rm r}^-(x;E)$. %Then, $U_-$ is defined as %\begin{equation} %\begin{array}{rcl} % \hskip-0.5cm % U_-: L^2(\mathbb R, \rmd x) % &\longmapsto & L^2([0,\infty),\rmd E)\oplus L^2( [0,\infty ),\rmd E) % \nonumber \\ % \hskip-2cm f(x) &\longmapsto & \widehat{f}^-(E)\equiv U_-f(E)\equiv % \left[ (U_-f)_{\rm l}(E),(U_-f)_{\rm r}(E) \right] \, , %\end{array} % \label{operatoruplu-} %\end{equation} %where %\begin{eqnarray} % \widehat{f}_{\rm l}^-(E)\equiv (U_-f)_{\rm l}(E)= % \int_{-\infty}^{\infty}\rmd x \, % f(x) \overline{\chi _{\rm l}^-(x;E)} \, , \label{hatEl-} \\ % \widehat{f}_{\rm r}^-(E)\equiv (U_-f)_{\rm r}(E)= % \int_{-\infty}^{\infty}\rmd x \, % f(x) \overline{\chi _{\rm r}^-(x;E)} \, . \label{hatEr-} %\end{eqnarray} % %The inverse of $U_-$ can be obtained from Theorem~4: %\begin{equation} % \hskip-1cm f(x)=(U_-^{-1}\widehat{f})(x)= % \int_{0}^{\infty}\rmd E\, \widehat{f}_{\rm l}^-(E) % \chi _{\rm l}^-(x;E) % +\int_{0}^{\infty}\rmd E\, \widehat{f}_{\rm r}^-(E) % \chi _{\rm r}^-(x;E) \, , % \label{invdiagonaliza-} %\end{equation} %where %\begin{equation} %% \widehat{f}_{\rm l}^-(E), \ \widehat{f}_{\rm r}^-(E) % \in L^2([0,\infty ),\rmd E) \, . %\end{equation} %The two-component vector $U_-f$, Eq.~(\ref{operatoruplu-}), can be written as %\begin{equation} % \widehat{f}^-(E)\equiv \int_{-\infty}^{\infty}\rmd x \, % f(x) \overline{\chi _{\rm l}^-(x;E)} \dotplus % \int_{-\infty}^{\infty}\rmd x \, % f(x) \overline{\chi _{\rm r}^-(x;E)} \, , % \label{eigenfucfinphi-} %\end{equation} % %Once we have constructed $U_-^{-1}$, we can see that %\begin{equation} % \hskip-1.5cm \widehat{H}\widehat{f}^-(E)= % U_-HU_-^{-1}\widehat{f}^-(E)=E\widehat{f}^-(E) \equiv % [ E\widehat{f}_{\rm l}^-(E), E\widehat{f}_{\rm r}^-(E) ] \, , \quad % f\in {\cal D}(H) \, ; % \label{diagonahmi} %\end{equation} %that is, the operator $U_-$ diagonalizes the Hamiltonian.% % %The construction of $U_-$ is tantamount to the construction of a direct %integral decomposition of the Hilbert space associated to the Hamiltonian $H$: %\begin{equation} %\begin{array}{rcl} % U_-:\mathcal{H} &\longmapsto & % \widehat{\cal H}= % \int_{0}^{\infty} \widehat{\cal H}_{\rm l}(E)\rmd E % \oplus \int_{0}^{\infty} \widehat{\cal H}_{\rm r}(E)\rmd E \\ % f &\longmapsto & U_-f:= [(U_-f)_{\rm l},(U_-f)_{\rm r}] \, , %\end{array} % \label{dirintdec-} %\end{equation} %where ${\cal H}$ is realized by $L^2(\mathbb R,\rmd x)$, and %$\widehat{\cal H}$ is realized by %$L^2([0,\infty ),\rmd E)\oplus L^2([0,\infty ),\rmd E)$. The Hilbert spaces %$\widehat{\mathcal H}_{\rm l}(E)$ and $\widehat{\mathcal H}_{\rm r}(E)$, which %are associated to each energy $E$ in the spectrum of $H$, are realized by %$\mathbb C$. The scalar product on $\widehat{\cal H}$ can be written as %\begin{equation} % \left( \widehat{f},\widehat{g} \right) _{\widehat{\cal H}}= % \int_0^{\infty}\rmd E\, % \overline{\widehat{f}_{\rm l}^-(E)} \, \widehat{g}_{\rm l}^-(E) % +\int_0^{\infty}\rmd E\, % \overline{\widehat{f}_{\rm r}^-(E)} \, \widehat{g}_{\rm r}^-(E) % \, . %\end{equation} \section{The rigged Hilbert space and Dirac's formalism} \label{sec:consrhs} In the previous sections, we have exhausted the Sturm-Liouville theory (i.e., the Hilbert space mathematics) when applied to the rectangular barrier. In this section, we equip the Sturm-Liouville theory with distribution theory, thereby constructing the equipped (i.e., rigged) Hilbert space of the rectangular barrier. \subsection{Construction of the rigged Hilbert space} \label{sec:constr} As explained in the Introduction, we need a subdomain of the Hilbert space on which algebraic operations (sums, multiplications and commutation relations) involving $P$, $Q$ and $H$ are well defined and on which expectation values are finite. Essentially, that subdomain should remain stable under the action of any algebraic operation involving $P$, $Q$ and $H$. The largest of such subdomains is the maximal invariant subspace of the algebra $\cal A$~\cite{ROBERTSJMP,ROBERTSCMP}, which we shall denote by $\cal D$. Clearly, the elements of ${\cal D}$ must fulfill the following conditions: \begin{itemize} \item[$\bullet$] they are infinitely differentiable, so the differentiation operation can be applied as many times as wished, \item[$\bullet$] they vanish at $x=a$ and $x=b$, so differentiation is meaningful at the discontinuities of the potential~\cite{ROBERTSCMP}, \item[$\bullet$] the action of any power of the multiplication operator, of the differentiation operator and of $h$ is square integrable. \end{itemize} Hence, \begin{eqnarray} \hskip-1cm {\cal D} =\{ \varphi \in L^2 (\mathbb R, \rmd x) \, : \ \varphi \in C^{\infty}(\mathbb R), \ \varphi ^{(n)}(a)=\varphi ^{(n)}(b)=0 \, , \ n=0,1,\ldots \, , \nonumber \\ \hskip3.6cm \frac{\rmd ^n}{\rmd x^n}x^mh^l\varphi (x) \in L^2 (\mathbb R, \rmd x) \, , \ n,m,l=0,1, \ldots \} \, . \label{ddomain} \end{eqnarray} The algebra of observables induces a natural topology on $\cal D$, whose definition of convergence is as follows: \begin{equation} \varphi _{\alpha}\, \mapupdown{\tau_{\mathbf \Phi}}{\alpha \to \infty} \, \varphi \quad {\rm iff} \quad \| \varphi _{\alpha }-\varphi \| _{n,m,l} \, \mapupdown{}{\alpha \to \infty}\, 0 \, , \quad n,m,l=0,1, \ldots \, , \end{equation} where the norms $\| \, \cdot \, \|_{n,m,l}$ are defined as \begin{equation} \| \varphi \| _{n,m,l} := \sqrt{\int_{-\infty}^{\infty}\rmd x \, \left| P^nQ^mH^l\varphi (x)\right| ^2} \, , \quad n,m,l=0,1,\ldots \, . \label{nmnorms} \end{equation} When the space $\cal D$ is topologized by these norms, we obtain the locally convex space of test functions $\mathbf \Phi$. On $\mathbf \Phi$, the expectation values \begin{equation} (\varphi , A^n\varphi ) \, , \quad \varphi \in \mathbf \Phi \, , \ A=P, Q, H \end{equation} are finite, and the commutation relations (\ref{cr1})-(\ref{cr3}) are well defined. (Note that, when acting on $\varphi \in {\mathbf \Phi}$, the commutation relation~(\ref{cr3}) becomes $[H,P]=0$, due to the vanishing of the derivatives of $\varphi$ at the discontinuities of the potential.) Moreover, the restrictions of $P$, $Q$ and $H$ to $\mathbf \Phi$ are essentially self-adjoint, $\tau _{\mathbf \Phi}$-continuous operators. Equations~(\ref{ddomain}) and (\ref{nmnorms}) show that $\mathbf \Phi$ is very similar to the Schwartz space, the major differences being that the derivatives of the elements of $\mathbf \Phi$ vanish at $x=a,b$ and that $\mathbf \Phi$ is invariant not only under $P$ and $Q$ but also under $H$. This is why we shall write \begin{equation} \mathbf \Phi \equiv \Sw \, . \end{equation} Once we have constructed the space $\mathbf \Phi$, we can construct its topological antidual $\mathbf \Phi ^{\times}$ as the space of $\tau _{\mathbf \Phi}$-continuous {\it antilinear} functionals on $\mathbf \Phi$, and therewith the RHS corresponding to the algebra of the 1D rectangular barrier potential, \begin{equation} \mathbf \Phi \subset {\cal H}\subset \mathbf \Phi ^{\times} \, , \label{RHSCONT} \end{equation} which in the position representation is realized by \begin{equation} \rhsSwt \, . \label{RHSCONTpr} \end{equation} The space $\mathbf \Phi ^{\times}$ is meant to contain the eigenkets $|p \rangle$, $|x\rangle$ and $|E^{\pm}\rangle _{\rm l,r}$ of $P$, $Q$ and $H$. The definition of these eigenkets is borrowed from the theory of distributions. The eigenket $|p\rangle$ is defined as an integral operator whose kernel is the eigenfunction of the differential operator $-\rmi \hbar \rmd /\rmd x$ with eigenvalue $p$: \begin{equation} \begin{array}{rcl} |p\rangle :\mathbf \Phi & \longmapsto & {\mathbb C} \\ \varphi & \longmapsto & \langle \varphi |p\rangle := \int_{-\infty}^{\infty}\rmd x \, \overline{\varphi (x)} \frac{1}{\sqrt{2\pi \hbar}} \rme ^{\rmi px/\hbar} =\overline{({\cal F}\varphi) (p)} \, . \end{array} \label{definitionketp} \end{equation} Note that, although the eigenfunctions $\frac{1}{\sqrt{2\pi \hbar}} \rme ^{\rmi px/\hbar}$ are in principle well defined for any complex $p$, the momentum in Eq.~(\ref{definitionketp}) runs only over ${\rm Sp}(P)=(-\infty ,\infty )$, because we are interested in assigning kets $|p\rangle$ only to the momenta in the spectrum of $P$, which are the only momenta that participate in the Dirac basis expansion associated to $P$. The eigenfunctions corresponding to the multiplication operator are just the delta function $\delta (x-x')$, and therefore the ket corresponding to each $x \in {\rm Sp}(Q)$ is defined as \begin{equation} \begin{array}{rcl} |x\rangle :\mathbf \Phi & \longmapsto & {\mathbb C} \\ \varphi & \longmapsto & \langle \varphi |x\rangle := \int_{-\infty}^{\infty}\rmd x' \, \overline{\varphi (x')} \delta (x-x') =\overline{\varphi (x)} \, . \end{array} \label{definitionketx} \end{equation} Similarly, we define the eigenkets corresponding to the Hamiltonian: \begin{equation} \hskip-0.7cm \begin{array}{rcl} |E^{\pm}\rangle _{\rm l,r} :\mathbf \Phi & \longmapsto & {\mathbb C} \\ \varphi & \longmapsto & \langle \varphi |E^{\pm}\rangle _{\rm l,r} := \int_{-\infty}^{\infty}\rmd x \, \overline{\varphi (x)} \chi _{\rm l,r}^{\pm}(x;E) =\overline{(U_{\pm}\varphi) _{\rm l,r}(E)} \, . \end{array} \label{definitionketE} \end{equation} Note that in Eq.~(\ref{definitionketE}) we have defined four different kets. Note also that, although the eigenfunctions $\chi _{\rm l,r}^{\pm}(x;E)$ are in principle well defined for any complex $E$, the energy in Eq.~(\ref{definitionketE}) runs only over ${\rm Sp}(H)=[0,\infty )$, because we are interested in assigning kets $|E^{\pm}\rangle _{\rm l,r}$ only to the energies in the spectrum of $H$, which are the only energies that participate in the Dirac basis expansion associated to $H$. The following proposition, whose proof can be found in \ref{sec:proofprop}, summarizes the results of this subsection: \vskip0.5cm \theoremstyle{plain} \newtheorem*{Prop1}{Proposition~1} \begin{Prop1} The triplet of spaces (\ref{RHSCONTpr}) is a rigged Hilbert space, and it satisfies all the requirements demanded in the Introduction. More specifically, \begin{itemize} \item[(i)] The quantities (\ref{nmnorms}) fulfill the conditions to be a norm. \item[(ii)] The space $\Sw$ is stable under the action of $P$, $Q$ and $H$. The restrictions of $P$, $Q$ and $H$ to $\Sw$ are essentially self-adjoint, $\tau _{\mathbf \Phi}$-continuous operators. The space $\Sw$ is dense in $L^2(\mathbb R,\rmd x)$. \item[(iii)] The kets $|p\rangle$, $|x\rangle$ and $|E^{\pm}\rangle _{\rm l,r}$ are well-defined antilinear functionals on $\Sw$, i.e., they belong to $\Swt$. \item[(iv)] The kets $|p\rangle$ are generalized eigenvectors of $P$, \begin{equation} P|p\rangle=p|p\rangle \, , \quad p\in \mathbb R \, ; \end{equation} the kets $|x\rangle$ are generalized eigenvectors of $Q$, \begin{equation} Q|x\rangle=x|x\rangle \, , \quad x \in \mathbb R \, ; \end{equation} the kets $|E^{\pm}\rangle _{\rm l,r}$ are generalized eigenvectors of $H$, \begin{equation} H|E^{\pm}\rangle_{\rm l,r} =E|E^{\pm}\rangle _{\rm l,r} \, , \quad E\in [0,\infty ) \, . \end{equation} \end{itemize} \end{Prop1} \vskip0.5cm (Note that $|p\rangle$ and $|x\rangle$ are in particular tempered distributions, whereas $|E^{\pm}\rangle_{\rm l,r}$ are not.) \subsection{The Dirac bras} We have constructed the kets $|p\rangle$, $|x\rangle$ and $|E^{\pm}\rangle _{\rm l,r}$, and we have shown that they belong to the space of {\it antilinear} functionals over $\Sw$, which we denoted by $\Swt$. In this subsection, we construct the corresponding bras $\langle x|$, $\langle p|$ and $_{\rm l,r}\langle ^{\pm}E|$, and we show that they belong to the space of {\it linear} functionals over $\Sw$, which we shall denote by $\Swp$. The triplet of spaces \begin{equation} \rhsSwp \label{rhsforbras} \end{equation} or, equivalently, \begin{equation} \mathbf \Phi \subset {\cal H} \subset \mathbf \Phi ^{\prime} \end{equation} is also a rigged Hilbert space, although now suitable to contain the eigenbras of the observables. The definition of the bra $\langle p|$ is as follows: \begin{equation} \begin{array}{rcl} \langle p| :\mathbf \Phi & \longmapsto & {\mathbb C} \\ \varphi & \longmapsto & \langle p| \varphi \rangle := \int_{-\infty}^{\infty}\rmd x \, \varphi (x) \frac{1}{\sqrt{2\pi \hbar}} \rme ^{-\rmi px/\hbar} =({\cal F}\varphi) (p) \, . \end{array} \label{definitionbrap} \end{equation} Comparison with Eq.~(\ref{definitionketp}) shows that the action of $\langle p|$ is the complex conjugate of the action of $|p \rangle$: \begin{equation} \langle p| \varphi \rangle = \overline{\langle \varphi |p \rangle} \, . \end{equation} The bra $\langle x|$ is defined as \begin{equation} \begin{array}{rcl} \langle x| :\mathbf \Phi & \longmapsto & {\mathbb C} \\ \varphi & \longmapsto & \langle x| \varphi \rangle := \int_{-\infty}^{\infty}\rmd x' \, \varphi (x') \delta (x-x') =\varphi (x) \, . \end{array} \label{definitionbrax} \end{equation} Comparison with Eq.~(\ref{definitionketx}) shows that the action of $\langle x|$ is complex conjugated to the action of $|x \rangle$: \begin{equation} \langle x| \varphi \rangle = \overline{\langle \varphi |x \rangle} \, . \end{equation} Analogously, the eigenbras of the Hamiltonian are defined as \begin{equation} \hskip-0.7cm \begin{array}{rcl} _{\rm l,r}\langle ^{\pm}E| :\mathbf \Phi & \longmapsto & {\mathbb C} \\ \varphi & \longmapsto & _{\rm l,r}\langle ^{\pm}E|\varphi\rangle := \int_{-\infty}^{\infty}\rmd x \, \varphi (x) \overline{\chi _{\rm l,r}^{\pm}(x;E)} =(U_{\pm}\varphi) _{\rm l,r}(E) \, . \end{array} \label{definitionbraE} \end{equation} (Note that in Eq.~(\ref{definitionbraE}) we have defined four different bras.) Comparison with Eq.~(\ref{definitionketE}) shows that the actions of the bras $_{\rm l,r}\langle ^{\pm}E|$ are the complex conjugates of the actions of the kets $|E ^{\pm} \rangle _{\rm l,r}$: \begin{equation} _{\rm l,r}\langle ^{\pm}E|\varphi\rangle = \overline{\langle \varphi |E ^{\pm} \rangle}_{\rm l,r} \, . \label{braketccE} \end{equation} The bras $\langle p|$, $\langle x|$ and $_{\rm l,r}\langle ^{\pm}E|$ are eigenvectors of $P$, $Q$ and $H$, respectively, as the following proposition shows: \vskip0.5cm \theoremstyle{plain} \newtheorem*{Prop2}{Proposition~2} \begin{Prop2} Within the rigged Hilbert space (\ref{rhsforbras}), it holds that \begin{itemize} \item[(i)] The bras $\langle p|$, $\langle x|$ and $_{\rm l,r} \langle ^{\pm}E|$ are well-defined linear functionals over $\Sw$, i.e., they belong to $\Swp$. \item[(ii)] The bras $\langle p|$ are generalized left-eigenvectors of $P$, \begin{equation} \langle p|P=p\langle p| \, , \quad p\in \mathbb R \, ; \end{equation} the bras $\langle x|$ are generalized left-eigenvectors of $Q$, \begin{equation} \langle x|Q=x\langle x| \, , \quad x \in \mathbb R \, ; \end{equation} the bras $_{\rm l,r} \langle ^{\pm}E|$ are generalized left-eigenvectors of $H$, \begin{equation} _{\rm l,r} \langle ^{\pm}E|H= E \hskip0.12cm _{\rm l,r} \langle ^{\pm}E| \, , \quad E\in [0,\infty ) \, . \end{equation} \end{itemize} \end{Prop2} \vskip0.5cm The proof of Proposition~2 is included in \ref{sec:proofprop}. Note that, in particular, and in accordance with Dirac's prescription, there is a one-to-one correspondence between bras and kets: Given an observable $A$, to each element $a$ in the spectrum of $A$, there corresponds a bra $\langle a|$ that is a left-eigenvector of $A$ and also a ket $|a\rangle$ that is a right-eigenvector of $A$. The bra $\langle a|$ belongs to $\mathbf \Phi ^{\prime}$, whereas the ket $|a\rangle$ belongs to $\mathbf \Phi ^{\times}$. \subsection{The Dirac basis vector expansions} Another important aspect of Dirac's formalism is that the bras and kets form a complete basis system such that [see also Eq.~(\ref{diracbve})] \begin{equation} \sum_{\alpha}\int_{{\rm Sp}(A)} \rmd a \, |a\rangle _{\alpha} \, _{\alpha}\langle a| = I \, . \label{resonident} \end{equation} In the present subsection, we derive various Dirac basis vector expansions for the algebra of the 1D rectangular barrier potential. We start by writing \begin{equation} \langle x|E^{\pm}\rangle _{\rm l,r} := \chi ^{\pm}_{\rm l,r}(x;E) \, , \end{equation} and \begin{equation} _{\rm l,r} \langle ^{\pm}E|x \rangle := \overline{\chi ^{\pm}_{\rm l,r}(x;E)} \, . \end{equation} Then, the restriction of (\ref{invdiagonaliza}) to $\Sw$ yields the following basis expansions: \begin{equation} \langle x|\varphi \rangle = \int_0^{\infty}\rmd E \, \langle x|E^{\pm}\rangle _{\rm l}\, _{\rm l}\langle ^{\pm}E|\varphi \rangle + \int_0^{\infty}\rmd E \, \langle x|E^{\pm}\rangle _{\rm r}\, _{\rm r}\langle ^{\pm}E|\varphi \rangle \, . \label{inveqDva+} \end{equation} %Although the eigenfunction expansions (\ref{invdiagonaliza}) and %(\ref{invdiagonaliza}) are valid for every element of the Hilbert space, the %Dirac basis vector expansions (\ref{inveqDva+}) and (\ref{inveqDva-}) are %valid only for functions $\varphi \in \Sw$, %because only on the elements of $\Sw$ the actions %of the bras $_{\rm l,r}\langle ^{\pm} E|$ and the kets %$|E^{\pm}\rangle _{\rm l,r}$ are well defined. Thus, the Dirac basis vector %expansions acquire meaning within the rigged Hilbert space. The restriction of Eqs.~(\ref{hatEl+}) and (\ref{hatEr+}) to $\Sw$ yields four other basis expansions: \begin{equation} _{\rm l,r}\langle ^{\pm}E|\varphi \rangle = \int_{-\infty}^{\infty}\rmd x \ _{\rm l,r}\langle ^{\pm}E|x\rangle \langle x|\varphi \rangle \, . \label{inveqDvaeix} \end{equation} The basis vector expansions (\ref{inveqDva+})-(\ref{inveqDvaeix}) are very similar to those given by the restriction of the Fourier transform to $\Sw$. If we define \begin{equation} \langle x|p\rangle := \frac{1}{\sqrt{2\pi \hbar }}\rme ^{\rmi px /\hbar} \, , \end{equation} \begin{equation} \langle p|x\rangle := \frac{1}{\sqrt{2\pi \hbar }}\rme ^{-\rmi px /\hbar} \, , \end{equation} then the restriction of (\ref{fourt}) to $\Sw$ yields \begin{equation} \langle p|\varphi \rangle = \int_{-\infty}^{\infty}\rmd x \ \langle p|x\rangle \langle x|\varphi \rangle \, , \label{inveqDvaeipx} \end{equation} whereas the restriction of the inverse of (\ref{fourt}) to $\Sw$ yields \begin{equation} \langle x|\varphi \rangle = \int_{-\infty}^{\infty}\rmd p \ \langle x|p\rangle \langle p|\varphi \rangle \, . \label{inveqDvaeixp} \end{equation} The similarity between the Dirac basis vector expansions (\ref{inveqDva+})-(\ref{inveqDvaeix}) and the Fourier expansions (\ref{inveqDvaeipx})-(\ref{inveqDvaeixp}) is another facet of the parallel between Dirac's formalism and Fourier methods. For the sake of completeness, we include the 1D rectangular potential version of the Nuclear Spectral Theorem~\cite{GELFAND} (see \ref{sec:proofprop} for its proof): \vskip0.5cm \theoremstyle{plain} \newtheorem*{Prop3}{Proposition~3} \begin{Prop3} (Nuclear Spectral Theorem) Let \begin{equation} \rhsSwt \end{equation} be the RHS of the 1D rectangular barrier algebra such that $\Sw$ remains invariant under the action of the algebra $\cal A$, and such that the operators of $\cal A$ are $\tau _{\mathbf \Phi}$-continuous, essentially self-adjoint operators over $\Sw$. Then, for each element in the spectrum of $P$, $Q$ or $H$, there is a generalized eigenvector such that \begin{eqnarray} &P|p\rangle =p|p\rangle \, , \quad &p \in \mathbb R \, , \label{eigePket} \\ &Q|x\rangle =x|x\rangle \, , \quad &x \in \mathbb R \, , \label{eigeQket} \\ &H|E^{\pm}\rangle _{\rm l,r}=E|E^{\pm} \rangle _{\rm l,r} \, , \quad &E\in [0,\infty ) \, , \label{eigeHket} \end{eqnarray} and such that for all $\varphi ,\psi \in \Sw$ \begin{eqnarray} (\varphi ,\psi ) &=& \int_0^{\infty}\rmd E\, \langle \varphi |E^{\pm}\rangle_{\rm l}\, _{\rm l}\langle ^{\pm}E|\psi \rangle + \int_0^{\infty}\rmd E\, \langle \varphi |E^{\pm}\rangle_{\rm r}\, _{\rm r}\langle ^{\pm}E|\psi \rangle \label{spinofEbk} \\ &=&\int_{-\infty}^{\infty}\rmd p \, \langle \varphi |p\rangle \langle p|\psi \rangle \label{spinofpbk} \\ &=&\int_{-\infty}^{\infty}\rmd x \, \langle \varphi |x\rangle \langle x|\psi \rangle \, , \label{spinofxbk} \end{eqnarray} and for all $\varphi ,\psi \in \Sw$, $n=1,2, \ldots$ \begin{equation} \hskip-1cm (\varphi ,H^n \psi )= \int_0^{\infty} \rmd E \, E^n \langle \varphi |E^{\pm}\rangle_{\rm l}\, _{\rm l}\langle ^{\pm}E|\psi \rangle + \int_0^{\infty}\rmd E\, E^n \langle \varphi |E^{\pm}\rangle_{\rm r}\, _{\rm r}\langle ^{\pm}E|\psi \rangle \, , \label{GMT2H} \end{equation} \begin{equation} (\varphi ,P^n \psi )=\int_{-\infty}^{\infty}\rmd p \, p^n \langle \varphi |p\rangle \langle p|\psi \rangle \, , \label{GMT2P} \end{equation} \begin{equation} (\varphi ,Q^n \psi )=\int_{-\infty}^{\infty}\rmd x \, x^n \langle \varphi |x\rangle \langle x|\psi \rangle \, . \label{GMT2Q} \end{equation} \end{Prop3} \vskip0.5cm If we ``sandwich'' Eq.~(\ref{resonident}) in between two elements $\varphi$ and $\psi$ of $\Sw$, then we obtain the expansions (\ref{spinofEbk})-(\ref{GMT2Q}), when $A=H^n, P^n, Q^n$, $n=0,1,2,\ldots$. If we ``sandwich'' Eq.~(\ref{resonident}) in between an element $\varphi$ of $\Sw$ and a bra $\langle x|$, $_{\rm l,r}\langle ^{\pm}E|$ or $\langle p|$, then we obtain the expansions (\ref{inveqDva+})-(\ref{inveqDvaeix}) and (\ref{inveqDvaeipx})-(\ref{inveqDvaeixp}). This ``sandwiching,'' however, is not valid when $\varphi$ or $\psi$ lies outside $\Sw$, because then the action of the bras and kets is not well defined. Thus, the RHS, rather than just the Hilbert space, fully justifies Dirac's formalism. \subsection{Energy, momentum and wave-number representations of the rigged Hilbert space} In subsection~\ref{sec:constr}, we constructed the position representation of the rigged Hilbert space of the 1D rectangular barrier algebra [see Eq.~(\ref{RHSCONTpr})]. In this subsection, we construct three spectral representations of (\ref{RHSCONTpr}): the energy, the momentum and the wave-number representations. We start with the energy representation. The unitary operators $U_{\pm}$ of Eq.~(\ref{dirintdec}) afford two energy representations of the RHS (\ref{RHSCONTpr}). The energy representations of $\Sw$ will be denoted as \begin{equation} \Swhpm \equiv U_{\pm} \Sw \, . \label{spaenrep} \end{equation} On $\Swhpm$, the Hamiltonian acts as the multiplication operator, as Eq.~(\ref{idagoplofh}) shows. The spaces $\Swhpm$ are linear subspaces of $L^2([0,\infty ),\rmd E)\oplus L^2([0,\infty ),\rmd E)$. In order to endow $\Swhpm$ with a topology, we carry the topology on $\Sw$ onto $\Swhpm$, \begin{equation} \tau _{\widehat{\mathbf \Phi}_{\pm}}:= U_{\pm} \tau _{\mathbf \Phi} \, . \end{equation} Endowed with these topologies, $\Swhpm$ are linear topological spaces. If we denote the antidual spaces of $\Swhpm$ by $\Swhpmt$, then we have \begin{equation} U_{\pm}^{\times} \Swt = \left[ U_{\pm} \Sw \right] ^{\times}= \Swhpmt \, . \end{equation} We can further split the energy representations of the RHS into left and right components, which are associated to left and right incidences. In order to do so, we need to recall the definition of the left and right components of the wave functions [see Eqs.~(\ref{hatEl+}) and (\ref{hatEr+})]: \begin{equation} \widehat{\varphi}_{\rm l,r}^{\pm}(E)=(U_{\pm}\varphi )_{\rm l,r}(E) \, . \end{equation} Any element $\widehat{\varphi}^{\pm}$ of $\Swhpm$ can therefore be written as a two-component vector, \begin{equation} \widehat{\varphi}^{\pm} \equiv \left[ \widehat{\varphi}_{\rm l}^{\pm}, \widehat{\varphi}_{\rm r}^{\pm} \right] \, , \end{equation} which is equivalent to write the spaces (\ref{spaenrep}) as sums of left and right components: \begin{equation} \Swhpm \equiv \Swhpml \oplus \Swhpmr \, . \label{splishat} \end{equation} Their antiduals can be split in a similar way, \begin{equation} \Swhpmt \equiv \Swhpmlt \oplus \Swhpmrt \, . \label{splishatandual} \end{equation} The energy representation of the kets $|E^{\pm}\rangle _{\rm l,r}$ is given by a familiar distribution. If we denote the energy representation of these kets by $|\widehat{E}^{\pm}\rangle _{\rm l,r}$, then the following equalities \begin{eqnarray} \langle \widehat{\varphi}_{\rm l,r}^{\pm}| \widehat{E}^{\pm}\rangle _{\rm l,r} &=& \langle \widehat{\varphi}_{\rm l,r}^{\pm}| U_{\pm}^{\times}|E^{\pm}\rangle _{\rm l,r} \\ &=& \langle U_{\pm}^{-1}\widehat{\varphi}_{\rm l,r}^{\pm}| E^{\pm}\rangle_{\rm l,r} \nonumber \\ &=& \int_{-\infty}^{\infty}\rmd x \, \overline{\varphi (x)} \chi_{\rm l,r}^{\pm} (x;E) \nonumber \\ &=& \overline{\widehat{\varphi}_{\rm l,r}^{\pm}(E)} \, \end{eqnarray} show that $|\widehat{E}^{\pm}\rangle _{\rm l}$ and $|\widehat{E}^{\pm}\rangle _{\rm r}$ act as the {\it antilinear} Schwartz delta functional over the spaces $\Swhpml$ and $\Swhpmr$, respectively. The different realizations of the RHS are easily visualized through the following diagram: \begin{equation} \hskip-2.5cm \begin{array}{cccccccccc} H; \ \varphi & & \Sw & \subset & L^2(\mathbb R, \rmd x) & \subset & \Swt & & |E^{\pm}\rangle _{\rm l,r} \nonumber \\ & & \downarrow U_{\pm} & & \downarrow U_{\pm} & & \downarrow U_{\pm}^{\times} & & \nonumber \\ \widehat{H}; \ \widehat{\varphi}_{\pm} & & \Swhpm & \subset & \oplus L^2([0,\infty ), \rmd E) & \subset & \Swhpmt & & |\widehat{E}^{\pm}\rangle _{\rm l,r} \\ [1ex] \end{array} \label{diagramsavp} \end{equation} where $\oplus L^2([0,\infty ), \rmd E)$ denotes $L^2([0,\infty ), \rmd E)\oplus L^2([0,\infty ), \rmd E)$. The top line of diagram~(\ref{diagramsavp}) displays the Hamiltonian, the wave functions, the RHS and the Dirac kets in the position representation. The bottom line displays their energy representation counterparts. We can also construct the energy representation of the eigenbras $_{\rm l,r} \langle ^{\pm}E|$. To this end, we first construct the energy representation of $\Swp$, which we shall denote by $\Swhpmp$. These two spaces are related as follows: \begin{equation} U_{\pm}^{\prime} \Swp = \left[ U_{\pm} \Sw \right] ^{\prime}= \Swhpmp \, . \end{equation} Similarly to Eqs.~(\ref{splishat}) and (\ref{splishatandual}), the dual space can be split into left and right components, \begin{equation} \Swhpmp =\Swhpmlp \oplus \Swhpmrp \, . \end{equation} Now, if we denote the energy representation of the energy eigenbras by $_{\rm l,r} \langle ^{\pm}\widehat{E}|$, then the following equalities \begin{eqnarray} _{\rm l,r} \langle ^{\pm}\widehat{E}| \widehat{\varphi}_{\rm l,r}^{\pm} \rangle &=& _{\rm l,r} \langle ^{\pm}E|U_{\pm}^{\prime}| \widehat{\varphi}_{\rm l,r}^{\pm} \rangle \\ &=& _{\rm l,r} \langle ^{\pm}E|U_{\pm}^{-1} \widehat{\varphi}_{\rm l,r}^{\pm} \rangle \nonumber \\ &=& \int_{-\infty}^{\infty}\rmd x \, \varphi (x) \overline{\chi_{\rm l,r}^{\pm} (x;E)} \nonumber \\ &=& \widehat{\varphi}_{\rm l,r}^{\pm}(E) \end{eqnarray} show that $_{\rm l} \langle ^{\pm}\widehat{E}|$ and $_{\rm r} \langle ^{\pm}\widehat{E}|$ are the {\it linear} Schwartz delta functional over the spaces $\Swhpml$ and $\Swhpmr$, respectively. The diagram corresponding to the bras is as follows: \begin{equation} \hskip-2cm \begin{array}{cccccccccc} H; \ \varphi & & \Sw & \subset & L^2(\mathbb R, \rmd x) & \subset & \Swp & & _{\rm l,r} \langle ^{\pm}E| \nonumber \\ & & \downarrow U_{\pm} & & \downarrow U_{\pm} & & \downarrow U_{\pm}^{\prime} & & \nonumber \\ \widehat{H}; \ \widehat{\varphi}_{\pm} & & \Swhpm & \subset & \oplus L^2([0,\infty ), \rmd E) & \subset & \Swhpmp & & _{\rm l,r} \langle ^{\pm} \widehat{E}| \\ [1ex] \end{array} \label{diagramsavpbra} \end{equation} The energy representations of $P$ and $Q$ have not been included, since they are fairly complicated. The momentum representation of the RHS (\ref{RHSCONTpr}) can be constructed in a similar fashion, by way of the Fourier transform ${\cal F}$. We shall not reproduce the calculations here but only provide the resulting diagrams. The diagram corresponding to the position and momentum kets reads as \begin{equation} \hskip-1cm \begin{array}{cccccccccc} P,Q; \ \varphi & & \Sw & \subset & L^2(\mathbb R, \rmd x) & \subset & \Swt & & |p\rangle , \, |x\rangle \nonumber \\ & & \downarrow {\cal F} & & \downarrow {\cal F} & & \downarrow {\cal F}^{\times} & & \nonumber \\ \widehat{P},\widehat{Q}; \ \widehat{\varphi} & & \Swh & \subset & L^2(\mathbb R , \rmd p) & \subset & \Swht & & |\widehat{p}\rangle , \, |\widehat{x}\rangle \\ [1ex] \end{array} \label{diagramsavpFk} \end{equation} where $\widehat{P}$ acts as the multiplication operator by $p$, $\widehat{Q}$ acts as the differential operator $\rmi \hbar \rmd / \rmd p$, $|\widehat{p}\rangle$ is the {\it antilinear} Schwartz delta functional, and $|\widehat{x}\rangle$ is the {\it antilinear} functional whose kernel is $(2\pi \hbar)^{-1/2} \exp \left(-\rmi px/\hbar\right)$. The momentum diagram for the position and momentum bras is \begin{equation} \hskip-1cm \begin{array}{cccccccccc} P,Q; \ \varphi & & \Sw & \subset & L^2(\mathbb R, \rmd x) & \subset & \Swp & & \langle p| , \, \langle x| \nonumber \\ & & \downarrow {\cal F} & & \downarrow {\cal F} & & \downarrow {\cal F}^{\prime} & & \nonumber \\ \widehat{P},\widehat{Q}; \ \widehat{\varphi} & & \Swh& \subset & L^2(\mathbb R , \rmd p) & \subset & \Swhp & & \langle\widehat{p}| , \, \langle \widehat{x}| \\ [1ex] \end{array} \label{diagramsavpFb} \end{equation} where $\langle\widehat{p}|$ is the {\it linear} Schwartz delta functional, and $\langle \widehat{x}|$ is the {\it linear} functional with kernel $(2\pi \hbar) ^{-1/2} \exp \left(\rmi px/\hbar\right)$. The momentum representation of $H$ has not been included, since in the momentum representation $H$ has a complicated expression. The momentum representation should not be confused with the wave number representation, which we construct in the remainder of this subsection. The eigenfunctions of the Schr\"odinger differential operator, the Green function and the transmission and reflection coefficients depend on the square root of the energy rather than on the energy itself. Thus, the wave number, which is defined as \begin{equation} k:=\sqrt{\frac{2m}{\hbar ^2} \, E} \, , \label{momenuks} \end{equation} is a more convenient variable. In terms of $k$, the $\delta$-normalized eigensolutions of the differential operator (\ref{doh}) read as \begin{equation} \langle x|k^{\pm}\rangle _{\rm l,r} \equiv \sqrt{\frac{\hbar ^2k}{m}\ } \, \chi _{\rm l,r}^{\pm}(x;E) \, . \label{continuoseign} \end{equation} These eigensolutions can be used to obtain the unitary operators $V_{\pm}$ that transform between the position and the wave-number representations, \begin{equation} \hskip-1.2cm \widehat{\widehat{f}\,}_{\pm}(k)=(V_{\pm}f)(k)= \int_{-\infty}^{\infty}\rmd x \, f(x) \overline{\langle x|k^{\pm}\rangle} _{\rm l} + \int_{-\infty}^{\infty}\rmd x \, f(x) \overline{\langle x|k^{\pm}\rangle} _{\rm r} \, , \ f\in {\cal H} \, , \label{Vcontinuoseign} \end{equation} where ``$\widehat{\widehat{\quad}}$'' denotes the $k$-representation. On this representation, the Hamiltonian acts as multiplication by $\frac{\hbar ^2}{2m}k^2$. To each $k\in [0,\infty )$, there correspond four eigenkets $|k^{\pm}\rangle_{\rm l,r}$ that act on $\Sw$ as the following integral operators: \begin{equation} \hskip-0.5cm \langle \varphi |k^{\pm}\rangle_{\rm l,r} := \int_{-\infty}^{\infty}\rmd x \, \langle \varphi |x\rangle \langle x|k^{\pm}\rangle _{\rm l,r} = \overline{(V_{\pm}\varphi )_{\rm l,r} (k)}\, , \quad \varphi \in \Sw \, . \end{equation} These eigenkets are generalized eigenvectors of the Hamiltonian with eigenvalue $\frac{\hbar ^2}{2m}k^2$. The following diagram provides the $k$-representation counterpart of~(\ref{diagramsavp}): \begin{equation} \hskip-2.5cm \begin{array}{cccccccccc} H; \ \varphi & & \Sw & \subset & L^2(\mathbb R, \rmd x) & \subset & \Swt & & |k^{\pm}\rangle _{\rm l,r} \nonumber \\ & & \downarrow V_{\pm} & & \downarrow V_{\pm} & & \downarrow V_{\pm}^{\times} & & \nonumber \\ \widehat{\widehat{H}}; \ \widehat{\widehat{\varphi}\,}_{\pm} & & \Swhhpm& \subset & \oplus L^2([0,\infty ), \rmd k) & \subset & \Swhhpmt & & |\widehat{\widehat{k}}\,^{\pm}\rangle _{\rm l,r} \\ [1ex] \end{array} \label{kdiagramsavp} \end{equation} where $\oplus L^2([0,\infty ), \rmd k)$ denotes $L^2([0,\infty ), \rmd k) \oplus L^2([0,\infty ), \rmd k)$, and $|\widehat{\widehat{k}} \, ^{\pm}\rangle _{\rm l,r}$ act as the {\it antilinear} Schwartz delta functional. The $k$-representation counterpart of~(\ref{diagramsavpbra}) is given by the following diagram: \begin{equation} \hskip-2.5cm \begin{array}{cccccccccc} H; \ \varphi & & \Sw & \subset & L^2(\mathbb R, \rmd x) & \subset & \Swp & & _{\rm l,r}\langle ^{\pm} k| \nonumber \\ & & \downarrow V_{\pm} & & \downarrow V_{\pm} & & \downarrow V_{\pm}^{\prime} & & \nonumber \\ \widehat{\widehat{H}}; \ \widehat{\widehat{\varphi}\,}\!_{\pm} & & \Swhhpm & \subset & \oplus L^2([0,\infty ), \rmd k) & \subset & \Swhhpmp & & _{\rm l,r}\langle ^{\pm}\widehat{\widehat{k}}| \\ [1ex] \end{array} \label{kdiagramsavpbra} \end{equation} where $_{\rm l,r}\langle ^{\pm}\widehat{\widehat{k}}|$ act as the {\it linear} Schwartz delta functional. The wave number is particularly useful in writing the Green function in a simple, compact form, as we are going to see now. Expressions (\ref{tk}) for $T(k)$ and (\ref{tstark}) for $T^*(k)$ yield \begin{equation} T(-k)=T^*(k) \, , \quad k>0 \, . \label{relttstar} \end{equation} From Eqs.~(\ref{chir+}) and (\ref{chir-}) it results that \begin{equation} \chi _{\rm r}^+(x;-k) = \rmi \chi _{\rm r}^-(x;k) \, , \quad k>0 \, , \end{equation} and from Eqs.~(\ref{chil+}) and (\ref{chil-}) it results that \begin{equation} \chi _{\rm l}^+(x;-k) = \rmi \chi _{\rm l}^-(x;k) \, , \quad k>0 \, . \end{equation} We can use the last three equations to write the Green function for all values of $k$ (and therefore for all values of $E$): \begin{equation} G(x,x';k)=\frac{2\pi}{\rmi} \, \frac{\chi _{\rm r}^+(x_<;k)\chi _{\rm l}^+(x_>;k)}{T(k)} \, , \quad k\in {\mathbb C} \, , \label{grenkfunc} \end{equation} where $x_< , x_>$ refer to the smaller and to the bigger of $x$ and $x'$, respectively. \section{Conclusions} \label{sec:conclusions} We have explicitly constructed the RHSs of the algebra of the 1D rectangular potential. In the position representation, these RHSs are given by \begin{equation} \Sw \subset L^2(\mathbb{R},\rmd x) \subset \Swt \, , \end{equation} \begin{equation} \Sw \subset L^2(\mathbb{R},\rmd x) \subset \Swp \, . \end{equation} On $\Sw$, the observables are essentially self-adjoint, continuous operators. Algebraic operations such as commutation relations are well defined on $\Sw$. We have also constructed the Dirac bras and kets of each observable of the algebra, as well as the basis expansions generated by the bras and kets. By doing so, we have shown (once again) that the RHS fully accounts for Dirac's formalism. By comparing the results for the Fourier transform $\cal F$ with those for the unitary operators $U_{\pm}$, we have seen that Dirac's formalism can be viewed as an extension of Fourier methods: Monoenergetic eigenfunctions extend the notion of monochromatic plane waves, $U_{\pm}$ extend the notion of Fourier transform, and Dirac's basis expansions extend the notion of Fourier decomposition. The results of this paper can be applied to many other algebras, at least when resonance eigenvalues are not considered. In general, the space of test functions $\mathbf \Phi$ is given by the maximal invariant subspace of the algebra, and the spaces of distributions $\mathbf \Phi ^{\times}$ and $\mathbf \Phi ^{\prime}$ are given by the antidual and dual spaces of $\mathbf \Phi$. As a corollary to the results of this paper, we can derive the RHSs of the algebra of the 1D free Hamiltonian. By making $V_0$ tend to zero, we can see that these RHSs are given by \begin{equation} \mathcal{S}(\mathbb{R}) \subset L^2(\mathbb{R},\rmd x) \subset \mathcal{S}^{\times}(\mathbb{R}) \, , \end{equation} \begin{equation} \mathcal{S}(\mathbb{R}) \subset L^2(\mathbb{R},\rmd x) \subset \mathcal{S}^{\prime}(\mathbb{R}) \, , \end{equation} where $\mathcal{S}(\mathbb{R})$ is the Schwartz space. Finally, of mathematical interest is the introduction of a new space of test functions, the Schwartz-like space $\Sw$, and new spaces of distributions, the spaces of tempered-like distributions $\Swt$ and $\Swp$. \ack Research supported by the Basque Government through reintegration fellowship No.~BCI03.96. \appendix \setcounter{section}{0} \section{Auxiliary functions} \label{sec:appauxfunc} For the sake of completeness, we provide the explicit expressions of the coefficients of the eigenfunctions: %\numparts \begin{eqnarray} &\widetilde{T}(\widetilde{k})=\rme ^{\widetilde{k}(b-a)} \frac{ 4\widetilde{Q}/\widetilde{k}} {(1+\widetilde{Q}/\widetilde{k})^2 \rme ^{\widetilde{Q}(b-a)}- (1-\widetilde{Q}/\widetilde{k})^2 \rme ^{-\widetilde{Q}(b-a)}} \\ [1ex] &\widetilde{A}_{\rm r}(\widetilde{k})=\frac{-2\rme ^{\widetilde{k}b} \rme ^{\widetilde{Q}a} (1-\widetilde{Q}/\widetilde{k})} {(1+\widetilde{Q}/\widetilde{k})^2 \rme ^{\widetilde{Q}(b-a)}- (1-\widetilde{Q}/\widetilde{k})^2 \rme ^{-\widetilde{Q}(b-a)}} \\ [1ex] &\widetilde{B}_{\rm r}(\widetilde{k})=\frac{2\rme ^{\widetilde{k}b} \rme ^{-\widetilde{Q}a} (1+\widetilde{Q}/\widetilde{k})} {(1+\widetilde{Q}/\widetilde{k})^2 \rme ^{\widetilde{Q}(b-a)}- (1-\widetilde{Q}/\widetilde{k})^2 \rme ^{-\widetilde{Q}(b-a)}} \\ [1ex] &\widetilde{R}_{\rm r}(\widetilde{k})=\rme ^{2\widetilde{k}b} \frac{(1-(\widetilde{Q}/\widetilde{k})^2 ) \rme ^{\widetilde{Q}(b-a)}-( 1-(\widetilde{Q}/\widetilde{k})^2 ) \rme ^{-\widetilde{Q}(b-a)}} {(1+\widetilde{Q}/\widetilde{k})^2 \rme ^{\widetilde{Q}(b-a)}- (1-\widetilde{Q}/\widetilde{k})^2 \rme ^{-\widetilde{Q}(b-a)}} \\ [1ex] &\widetilde{R}_{\rm l}(\widetilde{k})=\rme ^{-2\widetilde{k}a} \frac{( 1-(\widetilde{Q}/\widetilde{k})^2 ) \rme ^{\widetilde{Q}(b-a)} - ( 1-(\widetilde{Q}/\widetilde{k})^2 ) \rme ^{-\widetilde{Q}(b-a)}} {(1+\widetilde{Q}/\widetilde{k})^2 \rme ^{\widetilde{Q}(b-a)}- (1-\widetilde{Q}/\widetilde{k})^2 \rme ^{-\widetilde{Q}(b-a)}} \\ [1ex] &\widetilde{A}_{\rm l}(\widetilde{k})=\frac{2\rme ^{-\widetilde{k}a} \rme ^{\widetilde{Q}b}(1+\widetilde{Q}/\widetilde{k})} {(1+\widetilde{Q}/\widetilde{k})^2 \rme ^{\widetilde{Q}(b-a)}- (1-\widetilde{Q}/\widetilde{k})^2 \rme ^{-\widetilde{Q}(b-a)}} \\ [1ex] &\widetilde{B}_{\rm l}(\widetilde{k})=\frac{-2\rme ^{-\widetilde{k}a} \rme ^{-\widetilde{Q}b}(1-\widetilde{Q}/\widetilde{k})} {(1+\widetilde{Q}/\widetilde{k})^2 \rme ^{\widetilde{Q}(b-a)}- (1-\widetilde{Q}/\widetilde{k})^2 \rme ^{-\widetilde{Q}(b-a)} } \end{eqnarray} %\endnumparts %\numparts \begin{eqnarray} &T(k)=\rme ^{-\rmi k(b-a)} \frac{-4Q/k} {(1-Q/k)^2 \rme ^{\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{-\rmi Q(b-a)}} \label{tk} \\ [1ex] &A_{\rm r}(k)=\frac{2\rme ^{-\rmi kb} \rme ^{-\rmi Qa}(1-Q/k)} {(1-Q/k)^2 \rme ^{\rmi Q(b-a)}-(1+Q/k)^2 \rme ^{-\rmi Q(b-a)} } \\ [1ex] &B_{\rm r}(k)=\frac{-2\rme ^{-\rmi kb} \rme ^{\rmi Qa}(1+Q/k)} {(1-Q/k)^2 \rme ^{\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{-\rmi Q(b-a)}} \\ [1ex] &R_{\rm r}(k)=\rme ^{-2\rmi kb} \frac{ \left( 1-(Q/k)^2 \right) \rme ^{\rmi Q(b-a)} -\left( 1-(Q/k)^2 \right) \rme ^{-\rmi Q(b-a)}} {(1-Q/k)^2 \rme ^{\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{-\rmi Q(b-a)}} \\ [1ex] &R_{\rm l}(k)=\rme ^{2\rmi ka} \frac{ \left( 1-(Q/k)^2 \right) \rme ^{\rmi Q(b-a)} -\left( 1-(Q/k)^2 \right) \rme ^{-\rmi Q(b-a)}} {(1-Q/k)^2 \rme ^{\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{-\rmi Q(b-a)}} \\ [1ex] &A_{\rm l}(k)=\frac{-2\rme ^{\rmi ka} \rme ^{-\rmi Qb}(1+Q/k)} {(1-Q/k)^2 \rme ^{\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{-\rmi Q(b-a)}} \\ [1ex] &B_{\rm l}(k)=\frac{2\rme ^{\rmi ka} \rme ^{\rmi Qb}(1-Q/k)} {(1-Q/k)^2 \rme ^{\rmi Q(b-a)}-(1+Q/k)^2 \rme ^{-\rmi Q(b-a)} } \end{eqnarray} %\endnumparts \begin{eqnarray} &T^*(k)=\rme ^{\rmi k(b-a)} \frac{-4Q/k} {(1-Q/k)^2 \rme ^{-\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{\rmi Q(b-a)}} \label{tstark} \\ [1ex] &A_{\rm r}^*(k)=\frac{2\rme ^{\rmi kb} \rme ^{\rmi Qa}(1-Q/k)} {(1-Q/k)^2 \rme ^{-\rmi Q(b-a)}-(1+Q/k)^2 \rme ^{\rmi Q(b-a)} } \\ [1ex] &B_{\rm r}^*(k)=\frac{-2\rme ^{\rmi kb} \rme ^{-\rmi Qa}(1+Q/k)} {(1-Q/k)^2 \rme ^{-\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{\rmi Q(b-a)}} \\ [1ex] &R_{\rm r}^*(k)=\rme ^{2\rmi kb} \frac{ \left( 1-(Q/k)^2 \right) \rme ^{-\rmi Q(b-a)} -\left( 1-(Q/k)^2 \right) \rme ^{\rmi Q(b-a)}} {(1-Q/k)^2 \rme ^{-\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{\rmi Q(b-a)}} \\ [1ex] &R_{\rm l}^*(k)=\rme ^{-2\rmi ka} \frac{ \left( 1-(Q/k)^2 \right) \rme ^{-\rmi Q(b-a)} -\left( 1-(Q/k)^2 \right) \rme ^{\rmi Q(b-a)}} {(1-Q/k)^2 \rme ^{-\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{\rmi Q(b-a)}} \\ [1ex] &A_{\rm l}^*(k)=\frac{-2\rme ^{-\rmi ka} \rme ^{\rmi Qb}(1+Q/k)} {(1-Q/k)^2 \rme ^{-\rmi Q(b-a)}- (1+Q/k)^2 \rme ^{\rmi Q(b-a)}} \\ [1ex] &B_{\rm l}^*(k)=\frac{2\rme ^{-\rmi ka} \rme ^{-\rmi Qb}(1-Q/k)} {(1-Q/k)^2 \rme ^{-\rmi Q(b-a)}-(1+Q/k)^2 \rme ^{\rmi Q(b-a)} } \end{eqnarray} \section{Spectral measures associated to $\{ \chi _{\rm l}^-, \chi _{\rm r}^-\}$} \label{sec:compfor-} The ``final'' basis $\{ \chi _{\rm l}^-,\chi _{\rm r}^-\}$ can be used as well as the ``initial'' basis $\{ \chi _{\rm l}^+,\chi _{\rm r}^+\}$ to calculate ${\rm Sp}(H)$. This calculation, which follows the procedure of Sec.~\ref{sec:spectrum}, is provided in this appendix. If we choose \begin{eqnarray} &&\sigma _1(x;E)=\chi _{\rm l}^-(x;E) \label{sigma1=sci-} \\ &&\sigma _2(x;E)= \chi _{\rm r}^-(x;E) \label{sigam2cos-} \end{eqnarray} as the basis of Theorem~3, then Eqs.~(\ref{chil+}), (\ref{chir-}), (\ref{chil-}), (\ref{comcomr+r-}), (\ref{sigma1=sci-}) and (\ref{sigam2cos-}) lead to %\numparts \begin{eqnarray} &&\chi _{\rm l}^+(x;E)=-\frac{T(E)R_{\rm r}^*(E)}{T^*(E)}\sigma _1(x;E)+ T(E)\sigma _2(x;E)\, , \label{chir+s1s2-} \\ &&\chi _{\rm r}^-(x';E)=T^*(E)\overline{\sigma _1(x';\overline{E})} -\frac{R_{\rm l}(E)T^*(E)}{T(E)}\overline{\sigma _2(x';\overline{E})} \, . \label{chil-s1s2-} \end{eqnarray} %\endnumparts By substituting Eq.~(\ref{chir+s1s2-}) into Eq.~(\ref{green++}) and after some calculations, we get to \begin{eqnarray} \hskip-0.6cm G(x,x';E)= \frac{2\pi}{\rmi} \, \left[-\frac{R_{\rm r}^*(E)}{T^*(E)} \sigma _1 (x;E) \overline{\sigma _2(x';\overline{E})}+ \sigma _2(x;E) \overline{\sigma _2(x';\overline{E})} \right] \, , \nonumber \\ \qquad \hskip5.2cm \mbox{Re}(E)>0, \mbox{Im}(E)>0\, , \, x>x' \, . \label{redaot++-} \end{eqnarray} By substituting Eq.~(\ref{chil-s1s2-}) into Eq.~(\ref{green+-}) and after some calculations, we get to \begin{eqnarray} && \hskip-0.4cm G(x,x';E)= \frac{2\pi}{\rmi} \, \left[ -\sigma _1(x;E) \overline{\sigma _1 (x';\overline{E})} +\frac{R_{\rm l}(E)}{T(E)} \sigma _1(x;E) \overline{\sigma _2 (x';\overline{E})} \right] \, , \nonumber \\ &&\qquad \hskip5.3cm \mbox{Re}(E)>0, \mbox{Im}(E)<0\, , \, x>x' \, . \label{redaot+--} \end{eqnarray} By comparing (\ref{greenfunitthes}) to (\ref{redaot++-}) we obtain \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} 0 & -\frac{2\pi}{\rmi} \frac{R_{\rm r}^*(E)}{T^*(E)}\\ 0 & \frac{2\pi}{\rmi} \end{array} \right) \, , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)>0 \, . \label{theta++-} \end{equation} By comparing (\ref{greenfunitthes}) to (\ref{redaot+--}) we obtain \begin{equation} \theta _{ij}^+(E)= \left( \begin{array}{cc} -\frac{2\pi}{\rmi} & \frac{2\pi}{\rmi} \frac{R_{\rm l}(E)}{T(E)} \\ 0 & 0 \end{array} \right) \, , \quad \mbox{Re}(E)>0 \, , \ \mbox{Im}(E)<0 \, . \label{theta+--} \end{equation} As expected, the functions $\theta _{11}^+(E)$ and $\theta _{22}^+(E)$ both have a branch cut along the spectrum of $H$. The measures $\varrho _{ij}$ of Theorem~3 can be readily obtained from Eqs.~(\ref{theta++-}) and (\ref{theta+--}). The measure $\varrho _{21}$ is clearly zero. So is the measure $\varrho _{12}$, since \begin{equation} \varrho _{12}((E_1,E_2)) =\int_{E_1}^{E_2} - \left( \frac{R_{\rm l}(E)}{T(E)}-\frac{R_{\rm r}^*(E)}{T^*(E)}\right) \rmd E = 0 \, . \end{equation} The measures $\varrho _{11}$ and $\varrho _{22}$ are simply the Lebesgue measure: \begin{equation} \varrho _{11}((E_1,E_2))=\varrho _{22}((E_1,E_2)) =\int_{E_1}^{E_2} \rmd E = E_2-E_1 \, . \end{equation} \section{Proofs of propositions} \label{sec:proofprop} In this appendix, we provide the proofs of some propositions we stated in the main body of the paper. For the sake of clarity, in the proofs we shall denote the antidual and dual extensions of $H$ by respectively $H^{\times}$ and $H^{\prime}$. \begin{proof}[Proof of Proposition~1] ({\it i}) This is immediate. \vskip0.3cm ({\it ii}) From the definition of $\cal D$, Eq.~(\ref{ddomain}), and from the expressions of the differential operators associated to $P$, $Q$ and $H$, it can be seen after straightforward (though tedious) calculations that $\cal D$ is stable under the algebra of observables. It is also easy to see that $\cal D$ is indeed the largest subdomain of $L^2(\mathbb R,\rmd x)$ that remains stable under the action of the algebra of observables, i.e., $\cal D$ is the maximal invariant subspace of $\cal A$. That $P$, $Q$ and $H$ are essentially self-adjoint over $\Sw$ is obvious, since their only possible self-adjoint extensions are those with domains ${\cal D}(P)$, ${\cal D}(Q)$ and ${\cal D}(H)$. In order to prove that $H$ is $\tau _{\mathbf \Phi}$-continuous, we just have to realize that \begin{eqnarray} \| H \varphi \| _{n,m,l}&=& \| P^nQ^mH^lH\varphi \| \nonumber \\ &= & \| \varphi \| _{n,m,l+1} \, . \label{tauphiscont} \end{eqnarray} In order to prove that $P$ and $Q$ are $\tau _{\mathbf \Phi}$-continuous, we need the following commutation relations: \begin{equation} [H^n,Q]= -\frac{1}{\rm m}n\rmi \hbar PH^{n-1} \, , \ n=1,2, \ldots \label{crHnQ} \end{equation} where $\rm m$ refers to the mass, \begin{equation} [Q^n,P]=n\rmi \hbar Q^{n-1} \, , \ n=1,2, \ldots \label{crQnP} \end{equation} and \begin{equation} [H^n,P]=0 \, , \ n=1,2, \ldots . \label{crHnP} \end{equation} (Note that all these commutation relations are well defined on $\mathbf \Phi$.) Then, the $\tau _{\mathbf \Phi}$-continuity of $P$ follows from \begin{eqnarray} \| P \varphi \| _{n,m,l}&=& \| P^nQ^mH^lP\varphi \| \nonumber \\ &=& \| P^nQ^mPH^l\varphi \| \hskip1cm \mbox{from (\ref{crHnP})} \nonumber \\ &=& \| P^n(PQ^m+m\rmi \hbar Q^{m-1})H^l\varphi \| \hskip1cm \mbox{from (\ref{crQnP})} \nonumber \\ &\leq & \| P^{n+1}Q^mH^l\varphi \| + m\hbar \| P^{n}Q^{m-1}H^l\varphi \| \nonumber \\ &=& \| \varphi \|_{n+1,m,l} +m \hbar \| \varphi \|_{n,m-1,l} \, , \end{eqnarray} and the $\tau _{\mathbf \Phi}$-continuity of $Q$ follows from \begin{eqnarray} \hskip-1cm \| Q \varphi \| _{n,m,l}&=& \| P^nQ^mH^lQ\varphi \| \nonumber \\ &=& \| P^nQ^m (QH^l-\frac{1}{\rm m} l\rmi \hbar PH^{l-1})\varphi \| \hskip1cm \mbox{from (\ref{crHnQ})} \nonumber \\ &\leq & \| P^nQ^{m+1}H^l\varphi \| + \frac{1}{\rm m}l\hbar \| P^nQ^mPH^{l-1}\varphi \| \nonumber \\ &=& \| \varphi \| _{n,m+1,l} + \frac{1}{\rm m}l\hbar \| P^n(PQ^m+m\rmi \hbar Q^{m-1})H^{l-1}\varphi \| \hskip1cm \mbox{from (\ref{crQnP})} \nonumber \\ &\leq & \| \varphi \| _{n,m+1,l} + \frac{1}{\rm m}l\hbar \left( \| P^{n+1}Q^mH^{l-1}\varphi \| + m\hbar \| P^nQ^{m-1}H^{l-1}\varphi \| \right) \nonumber \\ &=& \| \varphi \| _{n,m+1,l} + \frac{1}{\rm m}l\hbar \| \varphi \|_{n+1,m,l-1} + \frac{1}{\rm m}l\hbar ^2 m \| \varphi \|_{n,m-1,l-1} \, . \end{eqnarray} In order to show that $\Sw$ is dense in $L^2(\mathbb R, \rmd x)$, we need to define the space of infinitely differentiable functions with compact support that vanish at $x=a,b$ along with all their derivatives~\cite{ROBERTSCMP}: \begin{eqnarray} \Czi := \{ f \in L^2(\mathbb R, \rmd x) \, : \ f\in C^{\infty}(\mathbb R ) \, , f^{(n)}(a)=f^{(n)}(b)=0 \, , \nonumber \\ \hskip3.3cm f \ \mbox{has compact support} \} \, . \end{eqnarray} Because \begin{equation} \Czi \subset \Sw \, , \end{equation} and because $\Czi$ is dense in $L^2(\mathbb R, \rmd x)$~\cite{ROBERTSCMP}, the space $\Sw$ is dense in $L^2(\mathbb R, \rmd x)$. \vskip0.3cm ({\it iii}) From definition (\ref{definitionketE}), it is pretty easy to see that $|E ^{\pm}\rangle _{\rm l,r}$ are antilinear functionals. In order to show that $|E ^{\pm}\rangle _{\rm l,r}$ are continuous, we define \begin{equation} {\cal M}^{\pm}_{\rm l,r}(E):= \sup _{x\in \mathbb R} \left| \chi ^{\pm}_{\rm l,r}(x;E) \right| \end{equation} and \begin{equation} C^{\pm}_{\rm l,r}(E):= {\cal M}^{\pm}_{\rm l,r}(E) \left( \int_{-\infty}^{\infty}\rmd x \, \frac{1}{(1+x^2)^2} \right) ^{1/2} \, . \end{equation} Since \begin{eqnarray} |\langle \varphi |E^{\pm}\rangle _{\rm l,r}| &=& \left| \int_{-\infty}^{\infty}\rmd x \, \overline{\varphi (x)}\chi^{\pm}_{\rm l,r}(x;E)\right| \nonumber \\ &\leq & {\cal M}^{\pm}_{\rm l,r}(E) \int _{-\infty}^{\infty} \rmd x \, |\varphi (x)| \nonumber \\ &=& {\cal M}^{\pm}_{\rm l,r}(E) \int_{-\infty}^{\infty}\rmd x \, \frac{1}{1+x^2} (1+x^2) |\varphi (x)| \nonumber \\ &\leq & {\cal M}^{\pm}_{\rm l,r}(E) \left( \int_{-\infty}^{\infty}\rmd x \, \frac{1}{(1+x^2)^2} \right) ^{1/2} \left( \int_{-\infty}^{\infty}\rmd x \, \left| (1+x^2) \varphi (x) \right| ^2 \right) ^{1/2} \nonumber \\ &=&C^{\pm}_{\rm l,r}(E) \, \| (1+Q^2)\varphi \| \nonumber \\ &\leq &C^{\pm}_{\rm l,r}(E) \, (\| \varphi \| + \|Q^2\varphi \|) \nonumber \\ &=&C^{\pm}_{\rm l,r}(E) \, ( \| \varphi \|_{0,0,0} + \|\varphi \| _{0,2,0}) \, , \label{contekets} \end{eqnarray} the functionals $|E^{\pm}\rangle _{\rm l,r}$ are continuous when $\mathbf \Phi$ is endowed with the topology $\tau _{\mathbf \Phi}$. The proof that $|p\rangle$ and $|x\rangle$ are also continuous antilinear functionals over $\mathbf \Phi$ is similar. \vskip0.3cm ({\it iv}) In order to prove that $|E^{\pm}\rangle _{\rm l,r}$ are generalized eigenvectors of $H$, we make use of the conditions (\ref{ddomain}) and (\ref{nmnorms}) satisfied by the elements of $\mathbf \Phi$, \begin{eqnarray} \langle \varphi |H^{\times}|E^{\pm}\rangle _{\rm l,r} &=& \langle H^{\dagger}\varphi |E^{\pm}\rangle _{\rm l,r} \nonumber \\ &=& \int_{-\infty}^{\infty}\rmd x \, \left( -\frac{\hbar ^2}{2m}\frac{\rmd ^2}{\rmd x^2}+V(x) \right) \overline{\varphi (x)} \chi ^{\pm}_{\rm l,r}(x;E) \nonumber \\ &=&-\frac{\hbar ^2}{2m} \left[ \frac{\rmd \overline{\varphi (x)}}{\rmd x} \chi ^{\pm}_{\rm l,r}(x;E) \right] _{-\infty}^{\infty} +\frac{\hbar ^2}{2m} \left[ \overline{\varphi (x)} \frac{\rmd \chi^{\pm}_{\rm l,r}(x;E)}{\rmd x} \right] _{-\infty}^{\infty} \nonumber \\ &&+ \int_{-\infty}^{\infty}\rmd x \, \overline{\varphi (x)} \left( -\frac{\hbar ^2}{2m}\frac{\rmd ^2}{\rmd x^2}+V(x) \right) \chi ^{\pm}_{\rm l,r}(x;E) \nonumber \\ &=&E \int_{-\infty}^{\infty}\rmd x \, \overline{\varphi (x)} \chi ^{\pm}_{\rm l,r}(x;E) \nonumber \\ &=&E\langle \varphi |E^{\pm}\rangle _{\rm l,r} \, . \label{ketsareegofH} \end{eqnarray} The proof that $|p\rangle$ and $|x\rangle$ are generalized eigenvectors of $P$ and $Q$, respectively, is similar. \end{proof} \vskip0.5cm \begin{proof}[Proof of Proposition~2] ({\it i}) It is clear from definitions (\ref{definitionbrap}), (\ref{definitionbrax}) and (\ref{definitionbraE}) that $\langle p|$, $\langle x|$ and $_{\rm l,r} \langle ^{\pm}E|$ are {\it linear} functionals over $\mathbf \Phi$. Because \begin{eqnarray} \left| _{\rm l,r} \langle ^{\pm}E|\varphi \rangle \right| &=& \left| \langle \varphi |E^{\pm}\rangle _{\rm l,r} \right| \hskip1cm \mbox{from (\ref{braketccE})} \nonumber \\ &\leq & C^{\pm}_{\rm l,r}(E) \, \left( \| \varphi \|_{0,0,0} + \|\varphi \| _{0,2,0} \right) \, , \hskip1cm \mbox{from (\ref{contekets})} \end{eqnarray} the bras $_{\rm l,r} \langle ^{\pm}E|$ are continuous. That $|p\rangle$ and $|x\rangle$ are also continuous over $\mathbf \Phi$ can be proved in a similar way. \vskip0.3cm ({\it ii}) Because \begin{eqnarray} _{\rm l,r} \langle ^{\pm}E|H^{\prime}|\varphi \rangle &=& _{\rm l,r} \langle ^{\pm}E|H^{\dagger}\varphi \rangle \nonumber \\ &=& \overline{\langle H^{\dagger}\varphi| E^{\pm}\rangle}_{\rm l,r} \hskip1cm \mbox{from (\ref{braketccE})} \nonumber \\ &=& E \, \overline{\langle \varphi| E^{\pm}\rangle}_{\rm l,r} \hskip1cm \mbox{from (\ref{ketsareegofH})} \nonumber \\ &=& E \hskip0.12cm _{\rm l,r} \langle ^{\pm}E|\varphi \rangle \, , \hskip1cm \mbox{from (\ref{braketccE})} \end{eqnarray} the bras $_{\rm l,r} \langle ^{\pm}E|$ are generalized left-eigenvectors of $H$. Similarly, it can be proved that $\langle p|$ and $\langle x|$ are generalized left-eigenvectors of respectively $P$ and $Q$. \end{proof} \vskip0.5cm \begin{proof}[Proof of Proposition~3] We only need to prove Eqs.~(\ref{spinofEbk})-(\ref{GMT2Q}), since Eqs.~(\ref{eigePket})-(\ref{eigeHket}) were proved in Proposition~1. Let us start with Eq.~(\ref{spinofEbk}). Take $\varphi$ and $\psi$ in $\Sw$. Because $U_{\pm}$ of Eq.~(\ref{dirintdec}) are unitary, we have that \begin{equation} (\varphi ,\psi )=(U_{\pm}\varphi ,U_{\pm}\psi )= (\widehat{\varphi}^{\pm} ,\widehat{\psi}^{\pm} ) \, . \label{Usiuni} \end{equation} Since $\widehat{\varphi}^{\pm}$ and $\widehat{\psi}^{\pm}$ are in particular elements of $L^2([0,\infty ),\rmd E)\oplus L^2([0,\infty ),\rmd E)$, their scalar product is given by \begin{equation} (\widehat{\varphi}^{\pm} ,\widehat{\psi}^{\pm})= \int_0^{\infty}\rmd E \, \overline{\widehat{\varphi}_{\rm l}^{\pm}(E)}\, \widehat{\psi}_{\rm l}^{\pm}(E) + \int_0^{\infty}\rmd E \, \overline{\widehat{\varphi}_{\rm r}^{\pm}(E)}\, \widehat{\psi}_{\rm r}^{\pm}(E) \, . \label{sphatvhaps} \end{equation} Since $\varphi$ and $\psi$ belong to $\Sw$, the actions of the eigenkets and eigenbras of $H$ are well defined on them: \begin{eqnarray} \langle \varphi |E^{\pm} \rangle _{\rm l,r} = \overline{ \widehat{\varphi} _{\rm l,r}^{\pm}(E)} \, , \label{actionofEphi}\\ _{\rm l,r}\langle ^{\pm}E|\psi \rangle = \widehat{\psi} _{\rm l,r}^{\pm}(E) \, . \label{actionofEpsi} \end{eqnarray} By plugging Eqs.~(\ref{actionofEphi}) and (\ref{actionofEpsi}) into Eq.~(\ref{sphatvhaps}), and Eq.~(\ref{sphatvhaps}) into Eq.~(\ref{Usiuni}), we get to Eq.~(\ref{spinofEbk}). It is clear that the trick to prove (\ref{spinofEbk}) was to go to the energy representation by way of $U_{\pm}$, in which representation $H$ acts as the multiplication operator. The same trick can be used to prove Eq.~(\ref{GMT2H}). A similar trick applies to the proof of equations~(\ref{spinofpbk}) and (\ref{GMT2P}), although instead of $U_{\pm}$ we must use the Fourier transform to go to the momentum representation, where $P$ acts as the multiplication operator. The calculations are straightforward and will not be reproduced here. Finally, Eqs.~(\ref{spinofxbk}) and (\ref{GMT2Q}) are immediate. \end{proof} \section*{References} \begin{thebibliography}{99} \bibitem{LEON} J.~Le\'on, J.~Julve, P.~Pitanga, F.~J.~de Urries, Phys.~Rev.~A~{\bf 61}, 062101 (2000); {\sf quant-ph/0002011}. \bibitem{MUGA} J.~G.~Muga, ``Characteristic times in one dimensional scattering,'' in ``Time in Quantum Mechanics,'' edited by J.~G.~Muga, R.~Sala Mayato, and I.~L.~Egusquiza, Springer, Berlin (2002), p.~29; {\sf quant-ph/0105081}. \bibitem{BRUMMELHUIS} R.~Brummelhuis, M.~B.~Ruskai, ``One-dimensional models for atoms in strong magnetic fields, II: Antisymmetry in the Landau levels,'' {\sf quant-ph/0308040}. \bibitem{BY} Y.~B.~By, S.~Efrima, Phys.~Rev.~B~{\bf 28}, 4126 (1983). \bibitem{BASTARD} G.~Bastard, ``Wave mechanics applied to semiconductors heterostructures,'' Les Editions de Physique, Paris (1998). \bibitem{WULF} E.~R.~Racec, U.~Wulf, %``Resonant quantum transport in %semiconductor nanostructures,'' Phys.~Rev.~B~{\bf 64}, 115318 (2001). \bibitem{SAKAKI} S.~Sakaki, ``Advances in Microfabrication and Microstructure Physics,'' in {\it Proc.~Int.~Symp.~on Foundations of Quantum Mechanics in the Light of New Technology}, edited by S.~Kamefuchi {\it et al.}, Phys.~Soc.~Japan (1984), pp.~94-110. \bibitem{KOLBAS} R.~M.~Kolbas, N.~Holonyak, Jr., %``Man-made quantum %wells: A new perspective on the finite square-well problem,'' Am.~J.~Phys.~{\bf 52}, 431 (1984). \bibitem{DIS} R.~de la Madrid, {\it Quantum Mechanics in Rigged Hilbert Space Language,} Ph.D.~Thesis, Universidad de Valladolid, Valladolid (2001). Available at \texttt{http://www.ehu.es/$\sim$wtbdemor/}. \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{FNOTE1} Note that, because we dealt with the radial part of a spherical shell potential for zero angular momentum, in Refs.~\cite{DIS,FP02} we used the unfortunate term {\it square barrier potential}, although we should have used {\it spherical shell potential}. Analogously, in Ref.~\cite{JPA}, we used the unfortunate term {\it square well-barrier potential}. \bibitem{IJTP} R.~de la Madrid, Int.~J.~Theo.~Phys.~{\bf 42}, 2441 (2003); {\sf quant-ph/0210167}. \bibitem{FORBES} G.~W.~Forbes, M.~A.~Alonso, Am.~J.~Phys.~{\bf 69}, 340 (2001). \bibitem{ROBERTSJMP} J.~E.~Roberts, J.~Math.~Phys.~{\bf 7}, 1097 (1966). \bibitem{ROBERTSCMP} J.~E.~Roberts, Commun.~Math.~Phys.~{\bf 3}, 98 (1966). \bibitem{BG} A.~Bohm and M.~Gadella, {\it Dirac kets, Gamow Vectors, and Gelfand Triplets}, Springer Lecture Notes in Physics Vol.~348 (Springer, Berlin, 1989). \bibitem{BOLLINI} C.~G.~Bollini, O.~Civitarese, A.~L.~DePaoli, M.~C.~Rocca, J.~Math.~Phys.~{\bf 37}, 4235 (1996). \bibitem{GALAPON} E.~A.~Galapon, J.~Math.~Phys.~{\bf 45}, 3180 (2004); {\sf quant-ph/0207044}. \bibitem{AT93} I.~Antoniou, S.~Tasaki, Int.~J.~Quant.~Chem.~{\bf 44}, 425 (1993). \bibitem{SUCHANECKI} Z.~Suchanecki, I.~Antoniou, S.~Tasaki, O.~F.~Brandtlow, J.~Math.~Phys.~{\bf 37}, 5837 (1996). \bibitem{BOHM} A.~Bohm, I.~Antoniou, P.~Kielanowski, %``The %preparation-registration arrow of time in quantum mechanics,'' Phys.~Lett.~A~{\bf 189}, 442-448 (1994). \bibitem{DUNFORDII} N.~Dunford, J.~Schwartz, \emph{Linear operators}, vol.~II, Interscience Publishers, New York (1963). %\bibitem{LEVINE} R.~D.~Levine, ``Quantum Mechanics of Molecular Rate %Processes,'' Oxford University Press, Oxford (1969). \bibitem{GELFAND} I.~M.~Gelfand, N.~Y.~ Vilenkin, \emph{Generalized Functions}~Vol.~IV, New York, Academic Press (1964); K.~Maurin, \emph{Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups,}Warsaw, Polish Scientific Publishers, (1968). %\bibitem{RESS} Note that this result holds automatically anyway, since it %also holds on the Schwartz space ${\cal S}(\mathbb R)$ and since %$\Sw \subset {\cal S}(\mathbb R)$. %\bibitem{NEWTON} Newton~R~G~1966 %{\it Scattering Theory of Waves and Particles} (New York: McGraw-Hill); %2nd edition 1982 (New York: Springer-Verlag). %\bibitem{TAYLOR} Taylor~J~R~1972 {\it Scattering theory} %(New York: Jhon Wiley \& Sons, Inc.). \end{thebibliography} \end{document} ---------------0407231541763 Content-Type: application/x-tex; name="iopart.cls" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="iopart.cls" %% %% This is file `iopart.cls' %% %% This file is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! 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\def\ftype@table{2} \def\ext@table{lot} \def\fnum@table{\tablename~\thetable} \newenvironment{table}{\footnotesize\rm\@float{table}}% {\end@float\normalsize\rm} \newenvironment{table*}{\footnotesize\rm\@dblfloat{table}}% {\end@dblfloat\normalsize\rm} \newlength\abovecaptionskip \newlength\belowcaptionskip \setlength\abovecaptionskip{10\p@} \setlength\belowcaptionskip{0\p@} % % Added redefinition of \@caption so captions are not written to % aux file therefore less need to \protect fragile commands % \long\def\@caption#1[#2]#3{\par\begingroup \@parboxrestore \normalsize \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par \endgroup} % \long\def\@makecaption#1#2{\vskip \abovecaptionskip \begin{indented} \item[]{\bf #1.} #2 \end{indented}\vskip\belowcaptionskip} \let\@portraitcaption=\@makecaption \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} 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\large \bfseries #1\hfil \hbox to\@pnumwidth{\hss #2}}\par \nobreak \if@compatibility \global\@nobreaktrue \everypar{\global\@nobreakfalse\everypar{}} \fi \endgroup \fi} \newcommand\l@section[2]{% \ifnum \c@tocdepth >\z@ \addpenalty{\@secpenalty}% \addvspace{1.0em \@plus\p@}% \setlength\@tempdima{1.5em}% \begingroup \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth \leavevmode \bfseries \advance\leftskip\@tempdima \hskip -\leftskip #1\nobreak\hfil \nobreak\hbox to\@pnumwidth{\hss #2}\par \endgroup \fi} \newcommand\l@subsection {\@dottedtocline{2}{1.5em}{2.3em}} \newcommand\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}} \newcommand\l@paragraph {\@dottedtocline{4}{7.0em}{4.1em}} \newcommand\l@subparagraph {\@dottedtocline{5}{10em}{5em}} \newcommand\listoffigures{% \section*{\listfigurename \@mkboth{\uppercase{\listfigurename}}% {\uppercase{\listfigurename}}}% \@starttoc{lof}% } \newcommand\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} \newcommand\listoftables{% \section*{\listtablename \@mkboth{\uppercase{\listtablename}}{\uppercase{\listtablename}}}% \@starttoc{lot}% } \let\l@table\l@figure \newenvironment{theindex} {\if@twocolumn \@restonecolfalse \else \@restonecoltrue \fi \columnseprule \z@ \columnsep 35\p@ \twocolumn[\section*{\indexname}]% \@mkboth{\uppercase{\indexname}}% {\uppercase{\indexname}}% \thispagestyle{plain}\parindent\z@ \parskip\z@ \@plus .3\p@\relax \let\item\@idxitem} {\if@restonecol\onecolumn\else\clearpage\fi} \newcommand\@idxitem {\par\hangindent 40\p@} \newcommand\subitem {\par\hangindent 40\p@ \hspace*{20\p@}} \newcommand\subsubitem{\par\hangindent 40\p@ \hspace*{30\p@}} \newcommand\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} \newcommand\contentsname{Contents} \newcommand\listfigurename{List of Figures} \newcommand\listtablename{List of Tables} \newcommand\refname{References} \newcommand\indexname{Index} \newcommand\figurename{Figure} \newcommand\tablename{Table} \newcommand\partname{Part} \newcommand\appendixname{Appendix} \newcommand\abstractname{Abstract} \newcommand\today{\number\day\space\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\year} \setlength\columnsep{10\p@} \setlength\columnseprule{0\p@} \newcommand{\Tables}{\clearpage\section*{Tables and table captions} \def\fps@table{hp}\noappendix} \newcommand{\Figures}{\clearpage\section*{Figure captions} \def\fps@figure{hp}\noappendix} % \newcommand{\Figure}[1]{\begin{figure} \caption{#1} \end{figure}} % \newcommand{\Table}[1]{\begin{table} \caption{#1} \begin{indented} \lineup \item[]\begin{tabular}{@{}l*{15}{l}}} \def\endTable{\end{tabular}\end{indented}\end{table}} \let\endtab=\endTable % \newcommand{\fulltable}[1]{\begin{table} \caption{#1} \lineup \begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus 12pt}}l}}} \def\endfulltable{\end{tabular*}\end{table}} % % \newcommand{\Bibliography}[1]{\section*{References}\par\numrefs{#1}} \newcommand{\References}{\section*{References}\par\refs} \def\thebibliography#1{\list {\hfil[\arabic{enumi}]}{\topsep=0\p@\parsep=0\p@ \partopsep=0\p@\itemsep=0\p@ \labelsep=5\p@\itemindent=-10\p@ \settowidth\labelwidth{\footnotesize[#1]}% \leftmargin\labelwidth \advance\leftmargin\labelsep \advance\leftmargin -\itemindent \usecounter{enumi}}\footnotesize \def\newblock{\ } \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \let\endthebibliography=\endlist \def\numrefs#1{\begin{thebibliography}{#1}} \def\endnumrefs{\end{thebibliography}} \let\endbib=\endnumrefs % \def\thereferences{\list{}{\topsep=0\p@\parsep=0\p@ \partopsep=0\p@\itemsep=0\p@\labelsep=0\p@\itemindent=-18\p@ \labelwidth=0\p@\leftmargin=18\p@ }\footnotesize\rm \def\newblock{\ } \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax } % \let\endthereferences=\endlist \newlength{\indentedwidth} \newdimen\mathindent \indentedwidth=\mathindent % % Macro to used for references in the Harvard system % \newenvironment{harvard}{\list{}{\topsep=0\p@\parsep=0\p@ \partopsep=0\p@\itemsep=0\p@\labelsep=0\p@\itemindent=-18\p@ \labelwidth=0\p@\leftmargin=18\p@ }\footnotesize\rm \def\newblock{\ } \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax}{\endlist} % \def\refs{\begin{harvard}} \def\endrefs{\end{harvard}} % \newenvironment{indented}{\begin{indented}}{\end{indented}} \newenvironment{varindent}[1]{\begin{varindent}{#1}}{\end{varindent}} % \def\indented{\list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@ \labelwidth=0\p@\leftmargin=\mathindent\topsep=0\p@\partopsep=0\p@ \parsep=0\p@\listparindent=15\p@}\footnotesize\rm} \let\endindented=\endlist \def\varindent#1{\setlength{\varind}{#1}% \list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@ \labelwidth=0\p@\leftmargin=\varind\topsep=0\p@\partopsep=0\p@ \parsep=0\p@\listparindent=15\p@}\footnotesize\rm} \let\endvarindent=\endlist \def\[{\relax\ifmmode\@badmath\else \begin{trivlist} \@beginparpenalty\predisplaypenalty \@endparpenalty\postdisplaypenalty \item[]\leavevmode \hbox to\linewidth\bgroup$ \displaystyle \hskip\mathindent\bgroup\fi} \def\]{\relax\ifmmode \egroup $\hfil \egroup \end{trivlist}\else \@badmath \fi} \def\equation{\@beginparpenalty\predisplaypenalty \@endparpenalty\postdisplaypenalty \refstepcounter{equation}\trivlist \item[]\leavevmode \hbox to\linewidth\bgroup $ \displaystyle \hskip\mathindent} \def\endequation{$\hfil \displaywidth\linewidth\@eqnnum\egroup \endtrivlist} % \@namedef{equation*}{\[} \@namedef{endequation*}{\]} % \def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation \global\@eqnswtrue \global\@eqcnt\z@\tabskip\mathindent\let\\=\@eqncr \abovedisplayskip\topsep\ifvmode\advance\abovedisplayskip\partopsep\fi \belowdisplayskip\abovedisplayskip \belowdisplayshortskip\abovedisplayskip \abovedisplayshortskip\abovedisplayskip $$\halign to \linewidth\bgroup\@eqnsel$\displaystyle\tabskip\z@ {##{}}$&\global\@eqcnt\@ne $\displaystyle{{}##{}}$\hfil &\global\@eqcnt\tw@ $\displaystyle{{}##}$\hfil \tabskip\@centering&\llap{##}\tabskip\z@\cr} \def\endeqnarray{\@@eqncr\egroup \global\advance\c@equation\m@ne$$\global\@ignoretrue } \mathindent = 6pc % \def\eqalign#1{\null\vcenter{\def\\{\cr}\openup\jot\m@th \ialign{\strut$\displaystyle{##}$\hfil&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} % \def\eqalignno#1{\displ@y \tabskip\z@skip \halign to\displaywidth{\hspace{5pc}$\@lign\displaystyle{##}$% \tabskip\z@skip &$\@lign\displaystyle{{}##}$\hfill\tabskip\@centering &\llap{$\@lign\hbox{\rm##}$}\tabskip\z@skip\crcr #1\crcr}} % \newif\ifnumbysec \def\theequation{\ifnumbysec \arabic{section}.\arabic{equation}\else \arabic{equation}\fi} \def\eqnobysec{\numbysectrue\@addtoreset{equation}{section}} \newcounter{eqnval} \def\numparts{\addtocounter{equation}{1}% \setcounter{eqnval}{\value{equation}}% \setcounter{equation}{0}% \def\theequation{\ifnumbysec \arabic{section}.\arabic{eqnval}{\it\alph{equation}}% \else\arabic{eqnval}{\it\alph{equation}}\fi}} \def\endnumparts{\def\theequation{\ifnumbysec \arabic{section}.\arabic{equation}\else \arabic{equation}\fi}% \setcounter{equation}{\value{eqnval}}} % \def\cases#1{% \left\{\,\vcenter{\def\\{\cr}\normalbaselines\openup1\jot\m@th% \ialign{\strut$\displaystyle{##}\hfil$&\tqs \rm##\hfil\crcr#1\crcr}}\right.}% % \newcommand{\e}{\mathrm{e}} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\rmd}{\mathrm{d}} \renewcommand{\qquad}{\hspace*{25pt}} \newcommand{\tdot}[1]{\stackrel{\dots}{#1}} % Added 1/9/94 \newcommand{\tqs}{\hspace*{25pt}} \newcommand{\fl}{\hspace*{-\mathindent}} \newcommand{\Tr}{\mathop{\mathrm{Tr}}\nolimits} \newcommand{\tr}{\mathop{\mathrm{tr}}\nolimits} \newcommand{\Or}{\mathord{\mathrm{O}}} %changed from \mathop 20/1/95 \newcommand{\lshad}{[\![} \newcommand{\rshad}{]\!]} \newcommand{\case}[2]{{\textstyle\frac{#1}{#2}}} \def\pt(#1){({\it #1\/})} \newcommand{\dsty}{\displaystyle} \newcommand{\tsty}{\textstyle} \newcommand{\ssty}{\scriptstyle} \newcommand{\sssty}{\scriptscriptstyle} \def\lo#1{\llap{${}#1{}$}} \def\eql{\llap{${}={}$}} \def\lsim{\llap{${}\sim{}$}} \def\lsimeq{\llap{${}\simeq{}$}} \def\lequiv{\llap{${}\equiv{}$}} % \newcommand{\eref}[1]{(\ref{#1})} \newcommand{\sref}[1]{section~\ref{#1}} \newcommand{\fref}[1]{figure~\ref{#1}} \newcommand{\tref}[1]{table~\ref{#1}} \newcommand{\Eref}[1]{Equation (\ref{#1})} \newcommand{\Sref}[1]{Section~\ref{#1}} \newcommand{\Fref}[1]{Figure~\ref{#1}} \newcommand{\Tref}[1]{Table~\ref{#1}} \newcommand{\opencircle}{\mbox{\Large$\circ\,$}} % moved Large outside maths \newcommand{\opensquare}{\mbox{$\rlap{$\sqcap$}\sqcup$}} \newcommand{\opentriangle}{\mbox{$\triangle$}} \newcommand{\opentriangledown}{\mbox{$\bigtriangledown$}} \newcommand{\opendiamond}{\mbox{$\diamondsuit$}} \newcommand{\fullcircle}{\mbox{{\Large$\bullet\,$}}} % moved Large outside maths \newcommand{\fullsquare}{\,\vrule height5pt depth0pt width5pt} \newcommand{\dotted}{\protect\mbox{${\mathinner{\cdotp\cdotp\cdotp\cdotp\cdotp\cdotp}}$}} \newcommand{\dashed}{\protect\mbox{-\; -\; -\; -}} \newcommand{\broken}{\protect\mbox{-- -- --}} \newcommand{\longbroken}{\protect\mbox{--- --- ---}} \newcommand{\chain}{\protect\mbox{--- $\cdot$ ---}} \newcommand{\dashddot}{\protect\mbox{--- $\cdot$ $\cdot$ ---}} \newcommand{\full}{\protect\mbox{------}} \def\;{\protect\psemicolon} \def\psemicolon{\relax\ifmmode\mskip\thickmuskip\else\kern .3333em\fi} \def\lineup{\def\0{\hbox{\phantom{\footnotesize\rm 0}}}% \def\m{\hbox{$\phantom{-}$}}% \def\-{\llap{$-$}}} % %%%%%%%%%%%%%%%%%%%%% % Tables rules % %%%%%%%%%%%%%%%%%%%%% \newcommand{\boldarrayrulewidth}{1\p@} % Width of bold rule in tabular environment. \def\bhline{\noalign{\ifnum0=`}\fi\hrule \@height \boldarrayrulewidth \futurelet \@tempa\@xhline} \def\@xhline{\ifx\@tempa\hline\vskip \doublerulesep\fi \ifnum0=`{\fi}} % % Rules for tables with extra space around % \newcommand{\br}{\ms\bhline\ms} \newcommand{\mr}{\ms\hline\ms} % \newcommand{\centre}[2]{\multispan{#1}{\hfill #2\hfill}} \newcommand{\crule}[1]{\multispan{#1}{\hspace*{\tabcolsep}\hrulefill \hspace*{\tabcolsep}}} \newcommand{\fcrule}[1]{\ifnum\thetabtype=1\multispan{#1}{\hrulefill \hspace*{\tabcolsep}}\else\multispan{#1}{\hrulefill}\fi} % % Extra spaces for tables and displayed equations % \newcommand{\ms}{\noalign{\vspace{3\p@ plus2\p@ minus1\p@}}} \newcommand{\bs}{\noalign{\vspace{6\p@ plus2\p@ minus2\p@}}} \newcommand{\ns}{\noalign{\vspace{-3\p@ plus-1\p@ minus-1\p@}}} \newcommand{\es}{\noalign{\vspace{6\p@ plus2\p@ minus2\p@}}\displaystyle}% % \newcommand{\etal}{{\it et al\/}\ } \newcommand{\dash}{------} \newcommand{\nonum}{\par\item[]} %\par added 1/9/93 \newcommand{\mat}[1]{\underline{\underline{#1}}} % % abbreviations for IOPP journals % \newcommand{\CQG}{{\it Class. Quantum Grav.} } \newcommand{\CTM}{{\it Combust. Theory Modelling\/} } \newcommand{\DSE}{{\it Distrib. Syst. Engng\/} } \newcommand{\EJP}{{\it Eur. J. Phys.} } \newcommand{\HPP}{{\it High Perform. Polym.} } % added 4/5/93 \newcommand{\IP}{{\it Inverse Problems\/} } \newcommand{\JHM}{{\it J. Hard Mater.} } % added 4/5/93 \newcommand{\JO}{{\it J. Opt.} } \newcommand{\JOA}{{\it J. Opt. A: Pure Appl. Opt.} } \newcommand{\JOB}{{\it J. Opt. B: Quantum Semiclass. Opt.} } \newcommand{\JPA}{{\it J. Phys. A: Math. Gen.} } \newcommand{\JPB}{{\it J. Phys. B: At. Mol. Phys.} } %1968-87 \newcommand{\jpb}{{\it J. Phys. B: At. Mol. Opt. Phys.} } %1988 and onwards \newcommand{\JPC}{{\it J. Phys. C: Solid State Phys.} } %1968--1988 \newcommand{\JPCM}{{\it J. Phys.: Condens. Matter\/} } %1989 and onwards \newcommand{\JPD}{{\it J. Phys. D: Appl. Phys.} } \newcommand{\JPE}{{\it J. Phys. E: Sci. Instrum.} } \newcommand{\JPF}{{\it J. Phys. F: Met. Phys.} } \newcommand{\JPG}{{\it J. Phys. G: Nucl. Phys.} } %1975--1988 \newcommand{\jpg}{{\it J. Phys. G: Nucl. Part. Phys.} } %1989 and onwards \newcommand{\MSMSE}{{\it Modelling Simulation Mater. Sci. Eng.} } \newcommand{\MST}{{\it Meas. Sci. Technol.} } %1990 and onwards \newcommand{\NET}{{\it Network: Comput. Neural Syst.} } \newcommand{\NJP}{{\it New J. Phys.} } \newcommand{\NL}{{\it Nonlinearity\/} } \newcommand{\NT}{{\it Nanotechnology} } \newcommand{\PAO}{{\it Pure Appl. Optics\/} } \newcommand{\PM}{{\it Physiol. Meas.} } % added 4/5/93 \newcommand{\PMB}{{\it Phys. Med. Biol.} } \newcommand{\PPCF}{{\it Plasma Phys. Control. Fusion\/} } % added 4/5/93 \newcommand{\PSST}{{\it Plasma Sources Sci. Technol.} } \newcommand{\PUS}{{\it Public Understand. Sci.} } \newcommand{\QO}{{\it Quantum Opt.} } \newcommand{\QSO}{{\em Quantum Semiclass. Opt.} } \newcommand{\RPP}{{\it Rep. Prog. Phys.} } \newcommand{\SLC}{{\it Sov. Lightwave Commun.} } % added 4/5/93 \newcommand{\SST}{{\it Semicond. Sci. Technol.} } \newcommand{\SUST}{{\it Supercond. Sci. Technol.} } \newcommand{\WRM}{{\it Waves Random Media\/} } \newcommand{\JMM}{{\it J. Micromech. Microeng.\/} } % % Other commonly quoted journals % \newcommand{\AC}{{\it Acta Crystallogr.} } \newcommand{\AM}{{\it Acta Metall.} } \newcommand{\AP}{{\it Ann. Phys., Lpz.} } \newcommand{\APNY}{{\it Ann. Phys., NY\/} } \newcommand{\APP}{{\it Ann. Phys., Paris\/} } \newcommand{\CJP}{{\it Can. J. Phys.} } \newcommand{\JAP}{{\it J. Appl. Phys.} } \newcommand{\JCP}{{\it J. Chem. Phys.} } \newcommand{\JJAP}{{\it Japan. J. Appl. Phys.} } \newcommand{\JP}{{\it J. Physique\/} } \newcommand{\JPhCh}{{\it J. Phys. Chem.} } \newcommand{\JMMM}{{\it J. Magn. Magn. Mater.} } \newcommand{\JMP}{{\it J. Math. Phys.} } \newcommand{\JOSA}{{\it J. Opt. Soc. Am.} } \newcommand{\JPSJ}{{\it J. Phys. Soc. Japan\/} } \newcommand{\JQSRT}{{\it J. Quant. Spectrosc. Radiat. Transfer\/} } \newcommand{\NC}{{\it Nuovo Cimento\/} } \newcommand{\NIM}{{\it Nucl. Instrum. Methods\/} } \newcommand{\NP}{{\it Nucl. Phys.} } \newcommand{\PL}{{\it Phys. Lett.} } \newcommand{\PR}{{\it Phys. Rev.} } \newcommand{\PRL}{{\it Phys. Rev. Lett.} } \newcommand{\PRS}{{\it Proc. R. Soc.} } \newcommand{\PS}{{\it Phys. Scr.} } \newcommand{\PSS}{{\it Phys. Status Solidi\/} } \newcommand{\PTRS}{{\it Phil. Trans. R. Soc.} } \newcommand{\RMP}{{\it Rev. Mod. Phys.} } \newcommand{\RSI}{{\it Rev. Sci. Instrum.} } \newcommand{\SSC}{{\it Solid State Commun.} } \newcommand{\ZP}{{\it Z. Phys.} } % \pagestyle{headings} \pagenumbering{arabic} % Arabic page numbers \raggedbottom \onecolumn \endinput %% %% End of file `iopart.cls'. ---------------0407231541763 Content-Type: application/x-tex; name="iopart10.clo" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="iopart10.clo" %% %% This is file `iopart10.clo' %% %% This file is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \ProvidesFile{iopart10.clo}[1997/01/13 v1.0 IOP Book file (size option)] \renewcommand\normalsize{% \@setfontsize\normalsize\@xpt\@xiipt \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ \belowdisplayskip \abovedisplayskip \let\@listi\@listI} \normalsize \newcommand\small{% \@setfontsize\small\@ixpt{11}% \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ \abovedisplayshortskip \z@ \@plus2\p@ \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 4\p@ \@plus2\p@ \@minus2\p@ \parsep 2\p@ \@plus\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\footnotesize{% \@setfontsize\footnotesize\@viiipt{9.5}% \abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@ \abovedisplayshortskip \z@ \@plus\p@ \belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 3\p@ \@plus\p@ \@minus\p@ \parsep 2\p@ \@plus\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt} \newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt} \newcommand\large{\@setfontsize\large\@xiipt{14}} \newcommand\Large{\@setfontsize\Large\@xivpt{18}} \newcommand\LARGE{\@setfontsize\LARGE\@xviipt{22}} \newcommand\huge{\@setfontsize\huge\@xxpt{25}} \newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}} \if@twocolumn \setlength\parindent{12\p@} \else \setlength\parindent{15\p@} \fi \setlength\headheight{12\p@} \setlength\headsep {12\p@} \setlength\topskip {10\p@} \setlength\footskip{20\p@} \setlength\maxdepth{.5\topskip} \setlength\@maxdepth\maxdepth \setlength\textwidth{31pc} \setlength\textheight{49pc} \setlength\oddsidemargin {24\p@} \setlength\evensidemargin {24\p@} \setlength\marginparwidth {72\p@} \setlength\marginparsep {10\p@} \setlength\marginparpush{5\p@} \setlength\topmargin{\z@} \setlength\footnotesep{6.65\p@} \setlength{\skip\footins} {9\p@ \@plus 4\p@ \@minus 2\p@} \setlength\floatsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\textfloatsep {20\p@ \@plus 2\p@ \@minus 4\p@} \setlength\intextsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\dblfloatsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\dbltextfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@} \setlength\@fptop{0\p@} \setlength\@fpsep{8\p@ \@plus 2fil} \setlength\@fpbot{0\p@} \setlength\@dblfptop{0\p@} \setlength\@dblfpsep{8\p@ \@plus 2fil} \setlength\@dblfpbot{0\p@} \setlength\partopsep{2\p@ \@plus 1\p@ \@minus 1\p@} \def\@listI{\leftmargin\leftmargini \parsep=\z@ \topsep=5\p@ \@plus3\p@ \@minus3\p@ \itemsep=3\p@ \@plus2\p@ \@minus\p@} \let\@listi\@listI \@listi \def\@listii {\leftmargin\leftmarginii \labelwidth\leftmarginii \advance\labelwidth-\labelsep \topsep=2\p@ \@plus2\p@ \@minus\p@ \parsep=\z@ \itemsep=\parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii \advance\labelwidth-\labelsep \topsep=\z@ \parsep=\z@ \partopsep=\z@ \itemsep=\z@} \def\@listiv {\leftmargin\leftmarginiv \labelwidth\leftmarginiv \advance\labelwidth-\labelsep} \def\@listv {\leftmargin\leftmarginv \labelwidth\leftmarginv \advance\labelwidth-\labelsep} \def\@listvi {\leftmargin\leftmarginvi \labelwidth\leftmarginvi \advance\labelwidth-\labelsep} \endinput %% %% End of file `iopart.clo'. ---------------0407231541763 Content-Type: application/x-tex; name="iopart12.clo" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="iopart12.clo" %% %% This is file `iopart12.clo' %% %% This file is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! 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Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \ProvidesFile{iopart12.clo}[1997/01/15 v1.0 LaTeX2e file (size option)] \renewcommand\normalsize{% \@setfontsize\normalsize\@xiipt{16}% \abovedisplayskip 12\p@ \@plus3\p@ \@minus7\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ \belowdisplayskip \abovedisplayskip \let\@listi\@listI} \normalsize \newcommand\small{% \@setfontsize\small\@xipt{14}% \abovedisplayskip 11\p@ \@plus3\p@ \@minus6\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ \def\@listi{\leftmargin\leftmargini \topsep 9\p@ \@plus3\p@ \@minus5\p@ \parsep 4.5\p@ \@plus2\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\footnotesize{% % \@setfontsize\footnotesize\@xpt\@xiipt \@setfontsize\footnotesize\@xpt{13}% \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ \def\@listi{\leftmargin\leftmargini \topsep 6\p@ \@plus2\p@ \@minus2\p@ \parsep 3\p@ \@plus2\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9.5}} \newcommand\tiny{\@setfontsize\tiny\@vipt\@viipt} \newcommand\large{\@setfontsize\large\@xivpt{18}} \newcommand\Large{\@setfontsize\Large\@xviipt{22}} \newcommand\LARGE{\@setfontsize\LARGE\@xxpt{25}} \newcommand\huge{\@setfontsize\huge\@xxvpt{30}} \let\Huge=\huge \if@twocolumn \setlength\parindent{14\p@} \else \setlength\parindent{18\p@} \fi \setlength\headheight{14\p@} \setlength\headsep{14\p@} \setlength\topskip{12\p@} \setlength\footskip{24\p@} \setlength\maxdepth{.5\topskip} \setlength\@maxdepth\maxdepth \setlength\textwidth{37.2pc} \setlength\textheight{56pc} \setlength\oddsidemargin {\z@} \setlength\evensidemargin {\z@} \setlength\marginparwidth {72\p@} \setlength\marginparsep{10\p@} \setlength\marginparpush{5\p@} \setlength\topmargin{-12pt} \setlength\footnotesep{8.4\p@} \setlength{\skip\footins} {10.8\p@ \@plus 4\p@ \@minus 2\p@} \setlength\floatsep {14\p@ \@plus 2\p@ \@minus 4\p@} \setlength\textfloatsep {24\p@ \@plus 2\p@ \@minus 4\p@} \setlength\intextsep {16\p@ \@plus 4\p@ \@minus 4\p@} \setlength\dblfloatsep {16\p@ \@plus 2\p@ \@minus 4\p@} \setlength\dbltextfloatsep{24\p@ \@plus 2\p@ \@minus 4\p@} \setlength\@fptop{0\p@} \setlength\@fpsep{10\p@ \@plus 1fil} \setlength\@fpbot{0\p@} \setlength\@dblfptop{0\p@} \setlength\@dblfpsep{10\p@ \@plus 1fil} \setlength\@dblfpbot{0\p@} \setlength\partopsep{3\p@ \@plus 2\p@ \@minus 2\p@} \def\@listI{\leftmargin\leftmargini \parsep=\z@ \topsep=6\p@ \@plus3\p@ \@minus3\p@ \itemsep=3\p@ \@plus2\p@ \@minus1\p@} \let\@listi\@listI \@listi \def\@listii {\leftmargin\leftmarginii \labelwidth\leftmarginii \advance\labelwidth-\labelsep \topsep=3\p@ \@plus2\p@ \@minus\p@ \parsep=\z@ \itemsep=\parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii \advance\labelwidth-\labelsep \topsep=\z@ \parsep=\z@ \partopsep=\z@ \itemsep=\z@} \def\@listiv {\leftmargin\leftmarginiv \labelwidth\leftmarginiv \advance\labelwidth-\labelsep} \def\@listv{\leftmargin\leftmarginv \labelwidth\leftmarginv \advance\labelwidth-\labelsep} \def\@listvi {\leftmargin\leftmarginvi \labelwidth\leftmarginvi \advance\labelwidth-\labelsep} \endinput %% %% End of file `iopart12.clo'. ---------------0407231541763 Content-Type: application/x-tex; name="setstack.sty" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="setstack.sty" %% %% This is file `setstack.sty', created by VIK 15 Dec 1998 %% Reproduces useful macros from amsmath.sty, thus avoiding the need to load the entire %% amsmath.sty and run into conflicts %% Adds definitions for \overset, \underset, \sideset, \substack, \boxed, \leftroot, %% \uproot, \dddot, \ddddot, \varrow, \harrow (see The LateX Companion, pp 225-227) \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1998/12/15]% LaTeX date must December 1998 or later \ProvidesPackage{setstack} \DeclareRobustCommand{\text}{% \ifmmode\expandafter\text@\else\expandafter\mbox\fi} \let\nfss@text\text \def\text@#1{\mathchoice {\textdef@\displaystyle\f@size{#1}}% {\textdef@\textstyle\tf@size{\firstchoice@false #1}}% {\textdef@\textstyle\sf@size{\firstchoice@false #1}}% {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}% \check@mathfonts } \def\textdef@#1#2#3{\hbox{{% \everymath{#1}% \let\f@size#2\selectfont #3}}} % adds underset, overset, sideset and % substack features from amsmath.sty (Companion p. 226) \def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag} \def\dft@tag{\invalid@tag{\string\tag\space not allowed here}} \def\default@tag{\let\tag\dft@tag} \def\Let@{\let\\\math@cr} \def\restore@math@cr{\def\math@cr@@@{\cr}} \def\overset#1#2{\binrel@{#2}% \binrel@@{\mathop{\kern\z@#2}\limits^{#1}}} \def\underset#1#2{\binrel@{#2}% \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}} \def\sideset#1#2#3{% \@mathmeasure\z@\displaystyle{#3}% \global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@ \setbox\tw@\box\@ne \@mathmeasure4\displaystyle{\copy\tw@#1}% \@mathmeasure6\displaystyle{#3\nolimits#2}% \dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@ \hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}% } \newenvironment{subarray}[1]{% \vcenter\bgroup \Let@ \restore@math@cr \default@tag \baselineskip\fontdimen10 \scriptfont\tw@ \advance\baselineskip\fontdimen12 \scriptfont\tw@ \lineskip\thr@@\fontdimen8 \scriptfont\thr@@ \lineskiplimit\lineskip \ialign\bgroup\ifx c#1\hfil\fi $\m@th\scriptstyle##$\hfil\crcr }{% \crcr\egroup\egroup } \newcommand{\substack}[1]{\subarray{c}#1\endsubarray} % definitions of dddot and ddddot(p. 225) \def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ \hbox{\normalfont ...}\vss}}}} \def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ \hbox{\normalfont....}\vss}}}} % definitions of leftroot, uproot (p.225) \begingroup \catcode`\"=12 \gdef\@@sqrt#1{\radical"270370 {#1}} \endgroup \def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha} \def\uproot{\@amsmath@err{\Invalid@@\uproot}\@eha} \newcount\uproot@ \newcount\leftroot@ \def\root{\relaxnext@ \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else \ifx\@let@token\leftroot\let\next@\nextiii@\else \let\next@\plainroot@\fi\fi\next@}% \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}% \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else \DN@.{\FN@\nextv@}\fi\next@.}% \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else \let\next@\plainroot@\fi\next@}% \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}% \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}% \def\nextvii@{\ifx\@let@token\@sptoken \DN@. {\FN@\nextviii@}\else \DN@.{\FN@\nextviii@}\fi\next@.}% \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else \let\next@\plainroot@\fi\next@}% \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}% \bgroup\uproot@\z@\leftroot@\z@\FN@\next@} \def\plainroot@#1\of#2{\setbox\rootbox\hbox{% $\m@th\scriptscriptstyle{#1}$}% \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}} {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup} \def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}% \dimen@\ht\z@\advance\dimen@-\dp\z@ \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}% \advance\dimen@ by1.667\wd\@ne \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox \mkern-10mu\mkern\leftroot@ mu\boxz@} % definition of \boxed (math in frame, no dollars) p 225 \def\boxed#1{\fbox{\m@th$\displaystyle#1$}} % definition of \smash for top and bottom parts of expression in braces (p.227) \renewcommand{\smash}[2][tb]{% \def\smash@{#1}% \ifmmode\@xp\mathpalette\@xp\mathsm@sh\else \@xp\makesm@sh\fi{#2}} % additional difinitions for arrows % zero mm wide #2 mm long vertical arrow shifted 1 mm to left, 1 mm up \newcommand{\varrow}[2]{% \unitlength=1mm \begin{picture}(0,6) % (0,0) % - for #1 to point arrow down,+ to point arrow up \end{picture}% \put(0,6){\vector(0,#1 3){#2}} % 6,- for down, 0, + for up } % 1 mm high #2 mm long horizontal arrow shifted 1 mm up (0,-1) \newcommand{\harrow}[2]{% \unitlength=1mm \begin{picture}(8,1)(0,-1) % % 1 mm distance between arrow and stackreled object over it (8,1) \put(0,0){\vector(#1 2,0){#2}} % - to point arrow left, + right \end{picture}% } \endinput %% end of setstack.sty %% Corrections history: %% ---------------0407231541763--