Talk:PlanetPhysics/Rigged Hilbert Space

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\begin{document}

 In extensions of \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} \cite{RdM2k5,JPA96}, the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of rigged Hilbert spaces allows one ``to put together'' the discrete \htmladdnormallink{spectrum}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} 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 \htmladdnormallink{photoelectric effect}{http://planetphysics.us/encyclopedia/PhotoelectricEffectIntroduction.html}).

\begin{definition}
A {\em rigged Hilbert space} is a pair $(\H,\phi)$ with $\H$ a \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} and $\phi$ is a dense subspace with a \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} structure for which the inclusion map {\bf $i$} is continuous. Between $\H$ and its \htmladdnormallink{dual space}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} $\H^*$ there is defined the adjoint map $i^*: \H^* \to \phi^*$ of the continuous inclusion map $i$. The \htmladdnormallink{duality}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} pairing between $\phi$ and $\phi^*$ also needs to be compatible with the \htmladdnormallink{inner product}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} 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} 

\end{document}