rigged Hilbert space | Physics Library

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 $(\H ,\phi)$ with $\H$ a Hilbert space and $\phi$ is a dense subspace with a topological vector space structure for which the inclusion map is continuous. Between $\H$ and its dual space $\H ^*$ there is defined the adjoint map $i^*: \H ^* \to \phi^*$ of the continuous inclusion map $i$

. The duality pairing between $\phi$ and $\phi^*$ also needs to be compatible with the inner product on $\H$ :

$\displaystyle \langle u, v\rangle_{\phi \times \phi^*} = (u, v)_{\H }$

whenever $u \in \phi \subset \H$ and $v \in \H = \H ^* \subset \phi^*$ .

Bibliography

1: R. de la Madrid, “The role of the rigged Hilbert space in Quantum Mechanics.”, Eur. J. Phys. 26, 287 (2005); .
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, .

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