rigged Hilbert space

In extensionsPlanetmathPlanetmath of Quantum Mechanics [1, 2], the concept of rigged Hilbert spacesPlanetmathPlanetmath allows one “to put together” the discrete spectrum of eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath corresponding to the bound states (eigenvectorsMathworldPlanetmathPlanetmathPlanetmath) with the continuous spectrum (as , for example, in the case of the ionization of an atom or the photoelectric effect).

Definition 0.1.

A rigged Hilbert space is a pair (,ϕ) with a Hilbert spaceMathworldPlanetmath and ϕ is a dense subspace with a topological vector spaceMathworldPlanetmath structureMathworldPlanetmath for which the inclusion mapMathworldPlanetmath i is continuousPlanetmathPlanetmath. Between and its dual spaceMathworldPlanetmathPlanetmathPlanetmath * there is defined the adjoint map i*:*ϕ* of the continuous inclusion map i. The duality pairing between ϕ and ϕ* also needs to be compatibleMathworldPlanetmath with the inner productMathworldPlanetmath on :


whenever uϕ and v=*ϕ*.


  • 1 R. de la Madrid, “The role of the rigged Hilbert space in Quantum Mechanics.”, Eur. J. Phys. 26, 287 (2005); quant-ph/0502053.
  • 2 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, ISBN3-540-64305-2.
Title rigged Hilbert space
Canonical name RiggedHilbertSpace
Date of creation 2013-03-22 19:22:48
Last modified on 2013-03-22 19:22:48
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 6
Author bci1 (20947)
Entry type Definition
Classification msc 81Q20
Synonym Gelfand triple
Defines dual Hilbert space
Defines adjoint map
Defines eigen spectrum