Rigged Hilbert space

Template:Multiple issues Template:Short description 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.<ref>Template:Eom</ref> "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics."<ref>Template:Cite book</ref>

Motivation

A function such as <math display="block"> x \mapsto e^{ix} , </math> is an eigenfunction of the differential operator <math display="block">-i\frac{d}{dx}</math> on the real line Template:Math, but isn't square-integrable for the usual (Lebesgue) measure on Template:Math. 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.Template:Sfn

Definition

A rigged Hilbert space is a pair Template:Math with Template:Math a Hilbert space, Template:Math a dense subspace, such that Template:Math is given a topological vector space structure for which the inclusion map <math display="block">i : \Phi \to H,</math> is continuous.Template:SfnTemplate:Sfn Identifying Template:Math with its dual space Template:Math, the adjoint to Template:Math is the map <math display="block">i^* : H = H^* \to \Phi^*.</math>

The duality pairing between Template:Math and Template:Math is then compatible with the inner product on Template:Math, in the sense that: <math display="block">\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>. In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in Template:Math (math convention) or Template:Math (physics convention), and conjugate-linear (complex anti-linear) in the other variable.

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

Note that even though Template:Math is isomorphic to Template:Math (via Riesz representation) if it happens that Template:Math is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion Template:Math with its adjoint Template:Math <math display="block">i^* i: \Phi\subset H = H^* \to \Phi^*.</math>

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 Template:Math, together with a subspace Template:Math which carries a finer topology, that is one for which the natural inclusion <math display="block"> \Phi \subseteq H </math> is continuous. It is no loss to assume that Template:Math is dense in Template:Math for the Hilbert norm. We consider the inclusion of dual spaces Template:Math in Template:Math. The latter, dual to Template:Math 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 Template:Math of type <math display="block">\phi\mapsto\langle v,\phi\rangle</math> for Template:Math in Template:Math are faithfully represented as distributions (because we assume Template:Math dense).

Now by applying the Riesz representation theorem we can identify Template:Math with Template:Math. Therefore, the definition of rigged Hilbert space is in terms of a sandwich: <math display="block">\Phi \subseteq H \subseteq \Phi^*. </math>

The most significant examples are those for which Template:Math is a nuclear space; this comment is an abstract expression of the idea that Template:Math consists of test functions and Template:Math of the corresponding distributions.

An example of a nuclear countably Hilbert space <math>\Phi</math> and its dual <math>\Phi^*</math> is the Schwartz space <math>\mathcal S(\mathbb R)</math> and the space of tempered distributions <math>\mathcal S'(\mathbb R)</math>, respectively, rigging the Hilbert space of square-integrable functions. As such, the rigged Hilbert space is given byTemplate:Sfn <math display="block">\mathcal{S}(\mathbb{R}) \subset L^2 (\mathbb{R}) \subset \mathcal{S}'(\mathbb{R}).</math> Another example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on <math>\mathbb R^n</math>) <math display="block">H = L^2(\mathbb R^n),\ \Phi = H^s(\mathbb R^n),\ \Phi^* = H^{-s}(\mathbb R^n),</math> where <math>s > 0</math>.

See also

• Fourier inversion theorem
• Fourier transform § Tempered distributions
• Self-adjoint operator § Spectral theorem

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, Template:Isbn. (Provides a survey overview.)
• J. Dieudonné, Éléments d'analyse VII (1978). (See paragraphs 23.8 and 23.32)
• Template:Cite book
• K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968.
• Template:Cite thesis
• de la Madrid Modino, R. "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); quant-ph/0502053.
• Template:Cite thesis