RiggedHilbertSpace 2009-03-02 14:33:08 2009-03-02 18:04:18 Definition adjoint map i* rigged Hilbert space % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm 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\newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} In extensions of Quantum Mechanics \cite{JPA96,RdM2k5}, the concept of rigged Hilbert spaces allows one ``to put together'' the discrete spectrum of eigenvalues corresponding to the bound states (eigenvectors) with the continuous spectrum (as , for example, in the case of the ionization of an atom or the photoelectric effect). \begin{definition} A {\em rigged Hilbert space} is a pair $(\H,\phi)$ with $\H$ a Hilbert space and $\phi$ is a dense subspace with a topological vector space structure for which the inclusion map {\bf $i$} is continuous. Between $\H$ and its dual space $\H^*$ there is defined the adjoint map $i^*: \H^* \to \phi^*$ of the continuous inclusion map $i$. The duality pairing between $\phi$ and $\phi^*$ also needs to be compatible with the inner product on $\H$: $$\langle u, v\rangle_{\phi \times \phi^*} = (u, v)_{\H}$$ whenever $u \in \phi \subset \H$ and $v \in \H = \H^* \subset \phi^*$. \end{definition} \begin{thebibliography}{9} \bibitem{RdM2k5} R. de la Madrid, ``The role of the rigged Hilbert space in Quantum Mechanics.'', Eur. J. Phys. 26, 287 (2005); $quant-ph/0502053$. \bibitem{JPA96} J-P. Antoine, ``Quantum Mechanics Beyond Hilbert Space'' (1996), appearing in {\em Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces}, Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag, $ISBN 3-540-64305-2$. \end{thebibliography}