PlanetPhysics/Rigged Hilbert Space

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In extensions of [[../QuantumParadox/|quantum mechanics]] , the [[../PreciseIdea/|concept]] of rigged Hilbert spaces allows one "to put together" the discrete [[../CohomologyTheoryOnCWComplexes/|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 [[../PhotoelectricEffectIntroduction/|photoelectric effect]]).

A rigged Hilbert space is a pair $(ℍ,\varphi )$ with $ℍ$ a [[../NormInducedByInnerProduct/|Hilbert space]] and $\varphi$ is a dense subspace with a [[../CoIntersections/|topological]] [[../NormInducedByInnerProduct/|vector space]] structure for which the inclusion map {\mathbf $i$} is continuous. Between $ℍ$ and its [[../DualityAndTriality/|dual space]] ${ℍ}^{*}$ there is defined the adjoint map $i^{*}:{ℍ}^{*}\to {\varphi }^{*}$ of the continuous inclusion map $i$. The [[../TrivialGroupoid/|duality]] pairing between $\varphi$ and ${\varphi }^{*}$ also needs to be compatible with the [[../NormInducedByInnerProduct/|inner product]] on $ℍ$: $⟨u,v{⟩}_{\varphi \times {\varphi }^{*}}=(u,v{)}_{ℍ}$ whenever $u\in \varphi \subset ℍ$ and $v\in ℍ={ℍ}^{*}\subset {\varphi }^{*}$.