Chemistry - Rigged Hilbert space

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

Since a function such as

$x \mapsto e^{ix} \quad$ ,

which is in a obvious sense an eigenvector of the differential operator

$i\frac{d}{dx}$

on the real line R, is not square-integrable for the usual Borel measure on R, this requires some way of stepping outside the strict confines of the Hilbert space theory. This was supplied by the apparatus of Schwartz distributions, and a generalized eigenfunction theory was developed in the years after 1950.

The concept of rigged Hilbert space places this idea in 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

$\Phi \subseteq H$

is continuous. It is no loss to assume that Φ is dense in H for the Hilbert norm. We consider the inclusion of dual spaces H^* in Φ^*. 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 functionals on the subspace Φ of type

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

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^* with H. Therefore the definition of rigged Hilbert space is in terms of a sandwich:

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

Reference

Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, K. Maurin, Polish Scientic Publishers, Warsaw, 1968.

Categories: Hilbert space | Spectral theory