generalized eigenfunction

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
$x \mapsto e^ ,$
is an eigenfunction of the differential operator
$-i\frac$
on the real line , but isn't square-integrable for the usual ( Lebesgue) measure on . 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.

Definition

A rigged Hilbert space is a pair with a Hilbert space, a dense subspace, such that is given a topological vector space structure for which the inclusion map
$i : \Phi \to H,$
is continuous. Identifying with its dual space , the adjoint to is the map
$i^* : H = H^* \to \Phi^*.$

The duality pairing between and is then compatible with the inner product on , in the sense that:
$\langle u, v\rangle_ = (u, v)_H$
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 convention) or (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 is isomorphic to (via Riesz representation) if it happens that is a Hilbert space in its own right, this isomorphism is ''not'' the same as the composition of the inclusion with its adjoint
$i^* i: \Phi\subset H = H^* \to \Phi^*.$

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 , together with a subspace 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 is dense in for the Hilbert norm. We consider the inclusion of dual spaces in . The latter, dual to 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 of type
$\phi\mapsto\langle v,\phi\rangle$
for in are faithfully represented as distributions (because we assume dense).

Now by applying the Riesz representation theorem we can identify with . 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 is a nuclear space; this comment is an abstract expression of the idea that consists of test functions and 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
$\mathcal(\mathbb) \subset L^2 (\mathbb) \subset \mathcal'(\mathbb).$
Another example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on $\mathbb R^n$)
$H = L^2(\mathbb R^n),\ \Phi = H^s(\mathbb R^n),\ \Phi^* = H^(\mathbb R^n),$
where $s > 0$.

See also

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

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

*

{{Hilbert space

Generalized functions
Hilbert spaces
Spectral theory
Schwartz distributions