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 with a Hilbert space and is a dense subspace with a topological vector space structure for which the inclusion map {\mathbf } is continuous. Between and its dual space there is defined the adjoint map of the continuous inclusion map . The duality pairing between and also needs to be compatible with the inner product on : whenever and .

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[2] [3]

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  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); .
  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, .