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. "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics."

Motivation

A function such as $$x\mapsto e^{ix},$$ is an eigenfunction of the differential operator $$-i\frac{d}{dx}$$ 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 $$i:\mathrm{\Phi }\to H,$$ is continuous.Template:SfnTemplate:Sfn Identifying Template:Math with its dual space Template:Math, the adjoint to Template:Math is the map $$i^{*}:H=H^{*}\to {\mathrm{\Phi }}^{*}.$$

The duality pairing between Template:Math and Template:Math is then compatible with the inner product on Template:Math, in the sense that: $$⟨u,v{⟩}_{\mathrm{\Phi }\times {\mathrm{\Phi }}^{*}}=(u,v{)}_H$$ whenever $u\in \mathrm{\Phi }\subset H$ and $v\in H=H^{*}\subset {\mathrm{\Phi }}^{*}$. 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 $(\mathrm{\Phi },H,{\mathrm{\Phi }}^{*})$ is often named the Gelfand triple (after Israel Gelfand). $H$ 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 $$i^{*}i:\mathrm{\Phi }\subset H=H^{*}\to {\mathrm{\Phi }}^{*}.$$

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 Template:Math, together with a subspace Template:Math which carries a finer topology, that is one for which the natural inclusion $$\mathrm{\Phi }\subseteq H$$ 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 $$\varphi \mapsto ⟨v,\varphi ⟩$$ 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: $$\mathrm{\Phi }\subseteq H\subseteq {\mathrm{\Phi }}^{*}.$$

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 $\mathrm{\Phi }$ and its dual ${\mathrm{\Phi }}^{*}$ is the Schwartz space $𝒮(ℝ)$ and the space of tempered distributions ${𝒮}^{\prime }(ℝ)$, respectively, rigging the Hilbert space of square-integrable functions. As such, the rigged Hilbert space is given byTemplate:Sfn $$𝒮(ℝ)\subset L^2(ℝ)\subset {𝒮}^{\prime }(ℝ).$$ Another example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on ${ℝ}^n$) $$H=L^2({ℝ}^n),\mathrm{\Phi }=H^s({ℝ}^n),{\mathrm{\Phi }}^{*}=H^{-s}({ℝ}^n),$$ where $s>0$.

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