PlanetPhysics/Rigged Hilbert Space

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