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 $(\mathcal{H},\phi)$ with $\mathcal{H}$ a Hilbert space and $\phi$ is a dense subspace with a topological vector space structure for which the inclusion map $i$ is continuous. Between $\mathcal{H}$ and its dual space $\mathcal{H}^{*}$ there is defined the adjoint map $i^{*}:\mathcal{H}^{*}\to\phi^{*}$ of the continuous inclusion map $i$ . The duality pairing between $\phi$ and $\phi^{*}$ also needs to be compatible with the inner product on $\mathcal{H}$ :

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

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

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