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 (,ϕ) with a Hilbert space and ϕ is a dense
subspace with a topological vector space structure for which the inclusion map i is continuous.
Between and its dual space
¡/span¿*thereisdefinedtheadjointmapiˆ*: ˆ* → ϕ∗ of the continuous inclusion map i. The
duality pairing between ϕ and ϕ∗ also needs to be compatible with the inner product on :
u ∈ ϕ ⊂ and v ∈=
¡/span¿* ⊂ 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.
