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 ,\phi)$ with $ \H$ a Hilbert space and $ \phi$ 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 ^* \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); $ 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, $ ISBN 3-540-64305-2$ .



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