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 $$v \in \mathbb{H} = \mathbb{H}^* \subset \phi^*$$.