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

$${\u27e8u,v\u27e9}_{\varphi \times {\varphi}^{*}}={(u,v)}_{\mathscr{H}}$$ |

whenever $u\in \varphi \subset \mathscr{H}$ and $v\in \mathscr{H}={\mathscr{H}}^{*}\subset {\varphi}^{*}$.

## 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 |