nLab rigged Hilbert space

Idea

The notion of rigged Hilbert spaces serves to treat spectral theory of normal unbounded operators on a Hilbert space much as if it were about actual eigenvalues and eigenvectors. It may be used to make rigorous the idea of eigenstates for quantum observables as used in quantum mechanics.

Example

Let $H$ be the Hilbert space $L^2(\mathbb{R}, d x)$ consisting of square integrable functions $f$ with respect to Lebesgue measure. There is an unbounded self-adjoint operator

$$\array{ m_x \,\colon\, & H &\longrightarrow& H \\ & f &\mapsto& x \cdot f \mathrlap{\,,} }$$

where

$$(x \cdot f)(y) \coloneqq y f(y) \,.$$

This operator is not defined on all of $H$, but it is defined on a dense subspace of $H$. For example, if $S$ is the Schwartz space consisting of smooth functions $f$ on $\mathbb{R}$ all of whose derivatives $f^{(n)}(x)$ decay rapidly at infinity (more rapidly than any negative power of $|x|$), then there is a dense inclusion map $i: S \to H$, and $m_x$ is defined globally on $S$.

Meanwhile the Schwartz space $S$ carries its own topology (as described in the article distribution), stronger than the topology it inherits from $H$, and the space of tempered distributions $S^*$ is defined to be the continuous dual of the topological vector space $S$. Since the continuous inclusion $i: S \to H$ is dense, it follows that any continuous functional

$$f: S \to \mathbb{C}$$

has at most one extension to a continuous functional $H \to \mathbb{C}$. In other words, the adjoint map

$$i^*: H^* \to S^*$$

is injective. In addition, the topology on $S$ is such that the operator $m_x: S \to S$ is continuous.

In this example, there is a dense inclusion $S \to S^*$ defined by the inner product pairing, and the operator $m_x$ extends uniquely to an operator $S^* \to S^*$, called $m_x$ by abuse of notation. Again, in this example, the operator $m_x: S^* \to S^*$ has an eigenvector $s_{\xi}$ for each $\xi \in \mathbb{R}$:

$$m_x(s_{\xi}) = \xi s_{\xi}$$

Ugh. Lousy start on something that would be nice to understand properly. Maybe an expert can help out. John, you there?

References

Among the original treatises on the theory of rigged Hilbert spaces is

• John Roberts, Rigged Hilbert spaces in quantum mechanics, Communications in Mathematical Physics, 3 (1966) 98–119. doi:10.1007/BF01645448

A unification of various inequivalent approaches is claimed to be achieved in

• M. Gadella, F. Gómez, A Unified Mathematical Formalism for the Dirac Formulation of Quantum Mechanics, Foundations of Physics 32 (2002) 815–869, doi:10.1023/A:1016069311589

See also

• S. Wickramasekara, A. Bohm, Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics (arXiv)

Rigged Hilbert spaces are also known as Gelfand triples, written about in his series on generalized functions

• Gel’fand, I. M., and Shilov, Georgiĭ Evgenʹevich. Generalized Functions. United States, American Mathematical Society, (2016)

See also

Discussion on MathOverflow

• Good references for Rigged Hilbert spaces?