rigged Hilbert space
In extensions of Quantum Mechanics [1, 2], 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).
Definition 0.1.
A rigged Hilbert space is a pair (ℋ,ϕ) with ℋ a Hilbert space and ϕ is a dense subspace with a topological vector space structure for which the inclusion map i is continuous. Between ℋ and its dual space ℋ* there is defined the adjoint map i*:ℋ*→ϕ* of the continuous inclusion map i. The duality pairing between ϕ and ϕ* also needs to be compatible with the inner product on ℋ:
⟨u,v⟩ϕ×ϕ*=(u,v)ℋ |
whenever u∈ϕ⊂ℋ and v∈ℋ=ℋ*⊂ϕ*.
References
- 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 |