PlanetPhysics/Rigged Hilbert Space

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In extensions of quantum mechanics [1], 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).


A rigged Hilbert space is a pair ${\displaystyle (\mathbb {H} ,\phi )}$ with ${\displaystyle \mathbb {H} }$ a Hilbert space and ${\displaystyle \phi }$ is a dense subspace with a topological vector space structure for which the inclusion map {\mathbf ${\displaystyle i}$} is continuous. Between ${\displaystyle \mathbb {H} }$ and its dual space ${\displaystyle \mathbb {H} ^{*}}$ there is defined the adjoint map ${\displaystyle i^{*}:\mathbb {H} ^{*}\to \phi ^{*}}$ of the continuous inclusion map ${\displaystyle i}$. The duality pairing between ${\displaystyle \phi }$ and ${\displaystyle \phi ^{*}}$ also needs to be compatible with the inner product on ${\displaystyle \mathbb {H} }$: ${\displaystyle \langle u,v\rangle _{\phi \times \phi ^{*}}=(u,v)_{\mathbb {H} }}$ whenever ${\displaystyle u\in \phi \subset \mathbb {H} }$ and ${\displaystyle v\in \mathbb {H} =\mathbb {H} ^{*}\subset \phi ^{*}}$.

References

1. Cite error: Invalid <ref> tag; no text was provided for refs named RdM2k5,JPA96
2. R. de la Madrid, "The role of the rigged Hilbert space in Quantum Mechanics.", Eur. J. Phys. 26, 287 (2005); ${\displaystyle quant-ph/0502053}$.
3. J-P. Antoine, "Quantum Mechanics Beyond Hilbert Space" (1996), appearing in Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces , Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag, ${\displaystyle ISBN3-540-64305-2}$.