|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], a '''rigged Hilbert space''' ('''Gelfand triple''', '''nested Hilbert space''', '''equipped Hilbert space''') is a construction designed to link the [[distribution (mathematics)|distribution]] and [[square-integrable]] aspects of [[functional analysis]]. Such spaces were introduced to study [[spectral theory]] in the broad sense.{{vague|It does something that isn't everything|date=January 2012}} They can bring together the '[[bound state]]' ([[eigenvector]]) and '[[Decomposition of spectrum (functional analysis)|continuous spectrum]]', in one place.
| | Hiya. The author's name is Eusebio but rather he never really adored that name. The hobby for him and his kids is to drive and he's found itself doing it for several years. Auditing is how he supports your man's family. Massachusetts is where that he and his wife live. He's been working in his website for several time now. Check it out here: http://prometeu.net<br><br>my weblog - [http://prometeu.net clash of clans hack] |
| | |
| ==Motivation==
| |
| | |
| A function such as the canonical homomorphism of the real line into the complex plane
| |
| | |
| :<math> x \mapsto e^{ix} , </math>
| |
| | |
| is an [[eigenvector]] of the [[differential operator]] | |
| | |
| :<math>-i\frac{d}{dx}</math>
| |
| | |
| on the [[real line]] '''R''', but isn't [[square-integrable]] for the usual [[Borel measure]] on '''R'''. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the [[Hilbert space]] theory. This was supplied by the apparatus of [[Schwartz distribution]]s, and a ''generalized eigenfunction'' theory was developed in the years after 1950.
| |
| | |
| ==Functional analysis approach==
| |
| | |
| The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a [[Hilbert space]] ''H'', together with a subspace Φ which carries a [[finer topology]], that is one for which the natural inclusion | |
| | |
| :<math> \Phi \subseteq H </math>
| |
| | |
| is continuous. It is [[Without loss of generality|no loss]] to assume that Φ is [[Dense_set|dense]] in ''H'' for the Hilbert norm. We consider the inclusion of [[dual space]]s ''H''<sup>*</sup> in Φ<sup>*</sup>. The latter, dual to Φ in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the [[linear functional]]s on the subspace Φ of type | |
| | |
| :<math>\phi\mapsto\langle v,\phi\rangle</math>
| |
| | |
| for ''v'' in ''H'' are faithfully represented as distributions (because we assume Φ dense).
| |
| | |
| Now by applying the [[Riesz representation theorem]] we can identify ''H''<sup>*</sup> with ''H''. Therefore the definition of ''rigged Hilbert space'' is in terms of a sandwich:
| |
| | |
| :<math>\Phi \subseteq H \subseteq \Phi^*. </math>
| |
| | |
| The most significant examples are those for which Φ is a [[nuclear space]]; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding [[distribution (mathematics)|distributions]].
| |
| | |
| ==Formal definition (Gelfand triple)==
| |
| | |
| A '''rigged Hilbert space''' is a pair (''H'',Φ) with ''H'' a Hilbert space, Φ a dense subspace, such that Φ is given a [[topological vector space]] structure for which the [[inclusion map]] ''i'' is continuous.
| |
| | |
| Identifying ''H'' with its dual space ''H<sup>*</sup>'', the adjoint to ''i'' is the map
| |
| | |
| :<math>i^*:H=H^*\to\Phi^*.</math> | |
| | |
| The duality pairing between Φ and Φ<sup>*</sup> has to be compatible with the inner product on ''H'', in the sense that:
| |
| | |
| :<math>\langle u, v\rangle_{\Phi\times\Phi^*} = (u, v)_H</math>
| |
| | |
| whenever <math>u\in\Phi\subset H</math> and <math>v \in H=H^* \subset \Phi^*</math>.
| |
| | |
| The specific triple <math> (\Phi,\,\,H,\,\,\Phi^*)</math> is often named the "Gelfand triple" (after the mathematician [[Israel Gelfand]]).
| |
| | |
| Note that even though Φ is isomorphic to Φ<sup>*</sup> if Φ is a Hilbert space in its own right, this isomorphism is ''not'' the same as the composition of the inclusion ''i'' with its adjoint ''i''*
| |
| | |
| :<math>i^* i:\Phi\subset H=H^*\to\Phi^*.</math>
| |
| | |
| ==References==
| |
| | |
| * 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. ''(Provides a survey overview.)''
| |
| * [[Jean Dieudonné]], ''Éléments d'analyse'' VII (1978). ''(See paragraphs 23.8 and 23.32)''
| |
| * [[Israel Gelfand|I. M. Gelfand]] and N. J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
| |
| * R. de la Madrid, "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); [http://arxiv.org/abs/quant-ph/0502053 quant-ph/0502053].
| |
| * K. Maurin, ''Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups'', Polish Scientific Publishers, Warsaw, 1968.
| |
| *{{eom|id=Rigged_Hilbert_space|first=R.A.|last= Minlos}}
| |
| | |
| [[Category:Hilbert space]]
| |
| [[Category:Spectral theory]]
| |
| [[Category:Generalized functions]]
| |
Hiya. The author's name is Eusebio but rather he never really adored that name. The hobby for him and his kids is to drive and he's found itself doing it for several years. Auditing is how he supports your man's family. Massachusetts is where that he and his wife live. He's been working in his website for several time now. Check it out here: http://prometeu.net
my weblog - clash of clans hack