rigged Hilbert space

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{{Short description|Construction linking the study of "bound" and continuous eigenvalues in functional analysis}}

In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.

Using this notion, a version of the spectral theorem for unbounded operators on Hilbert space can be formulated.{{eom|id=Rigged Hilbert space|first=R. A.|last=Minlos|authorlink=Robert Minlos}} "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics."{{cite book |last1=Krasnoholovets |first1=Volodymyr |last2=Columbus |first2=Frank H. |title=New Research in Quantum Physics |date=2004 |publisher=Nova Science Publishers |isbn=978-1-59454-001-1 |page=79 |language=en}}

Motivation

A function such as

$x \mapsto e^{ix} ,$

is an eigenfunction of the differential operator

$-i\frac{d}{dx}$

on the real line {{math|R}}, but isn't square-integrable for the usual (Lebesgue) measure on {{math|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 distributions, and a generalized eigenfunction theory was developed in the years after 1950.{{sfn|Gel'fand|Vilenkin|1964|pp=103-105}}

Definition

A rigged Hilbert space is a pair {{math|(H, Φ)}} with {{math|H}} a Hilbert space, {{math|Φ}} a dense subspace, such that {{math|Φ}} is given a topological vector space structure for which the inclusion map

$i : \Phi \to H,$

is continuous.{{sfn|de la Madrid Modino|2001|pp=66-67}}{{sfn|van der Laan|2019|pp=21-22}} Identifying {{math|H}} with its dual space {{math|H^*}}, the adjoint to {{math|i}} is the map

$i^* : H = H^* \to \Phi^*.$

The duality pairing between {{math|Φ}} and {{math|Φ^*}} is then compatible with the inner product on {{math|H}}, in the sense that:

$\langle u, v\rangle_{\Phi\times\Phi^*} = (u, v)_H$

whenever $u \in \Phi\subset H$ and $v \in H = H^* \subset \Phi^*$ . In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in {{math|u}} (math convention) or {{math|v}} (physics convention), and conjugate-linear (complex anti-linear) in the other variable.

The triple $(\Phi,\,\,H,\,\,\Phi^*)$ is often named the Gelfand triple (after Israel Gelfand). $H$ is referred to as a pivot space.

Note that even though {{math|Φ}} is isomorphic to {{math|Φ^*}} (via Riesz representation) if it happens that {{math|Φ}} is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion {{math|i}} with its adjoint {{math|i*}}

$i^* i: \Phi\subset H = H^* \to \Phi^*.$

=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 {{math|H}}, together with a subspace {{math|Φ}} which carries a finer topology, that is one for which the natural inclusion

$\Phi \subseteq H$

is continuous. It is no loss to assume that {{math|Φ}} is dense in {{math|H}} for the Hilbert norm. We consider the inclusion of dual spaces {{math|H^*}} in {{math|Φ^*}}. The latter, dual to {{math|Φ}} in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the linear functionals on the subspace {{math|Φ}} of type

$\phi\mapsto\langle v,\phi\rangle$

for {{math|v}} in {{math|H}} are faithfully represented as distributions (because we assume {{math|Φ}} dense).

Now by applying the Riesz representation theorem we can identify {{math|H^*}} with {{math|H}}. Therefore, the definition of rigged Hilbert space is in terms of a sandwich:

$\Phi \subseteq H \subseteq \Phi^*.$

The most significant examples are those for which {{math|Φ}} is a nuclear space; this comment is an abstract expression of the idea that {{math|Φ}} consists of test functions and {{math|Φ*}} of the corresponding distributions.

An example of a nuclear countably Hilbert space $\Phi$ and its dual $\Phi^*$ is the Schwartz space $\mathcal S(\mathbb R)$ and the space of tempered distributions $\mathcal S'(\mathbb R)$ , respectively, rigging the Hilbert space of square-integrable functions. As such, the rigged Hilbert space is given by{{sfn|de la Madrid Modino|2001|p=72}}

$\mathcal{S}(\mathbb{R}) \subset L^2 (\mathbb{R}) \subset \mathcal{S}'(\mathbb{R}).$

Another example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on $\mathbb R^n$ )

$H = L^2(\mathbb R^n),\ \Phi = H^s(\mathbb R^n),\ \Phi^* = H^{-s}(\mathbb R^n),$

where $s > 0$ .

Notes

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.)
J. Dieudonné, Éléments d'analyse VII (1978). (See paragraphs 23.8 and 23.32)
{{cite book | last=Gel'fand | first=I. M. |author-link1=Israel_Gelfand| last2=Vilenkin | first2=N. Ya | author-link2=Naum_Yakovlevich_Vilenkin|title=Generalized Functions: Applications of Harmonic Analysis | publisher=Elsevier Science | publication-place=Burlington | date=1964 | isbn=978-1-4832-2974-4 |doi=10.1016/c2013-0-12221-0}}
K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968.
{{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree= PhD |publisher= Universidad de Valladolid}}
de la Madrid Modino, R. "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); [https://arxiv.org/abs/quant-ph/0502053 quant-ph/0502053].
{{cite thesis |last=van der Laan |first=L. |date= July 2019 |title=Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables |url=https://fse.studenttheses.ub.rug.nl/19933/ |degree=BSc |location=Groningen |publisher=Rijksuniversiteit Groningen |access-date=11 January 2025}}

Category:Generalized functions

Category:Hilbert spaces

Category:Spectral theory

Category:Schwartz distributions