rigged Hilbert space

In extensions of Quantum Mechanics [1, 2], the concept of rigged Hilbert spaces allows one “to put together” the discrete spectrum of eigenvalues corresponding to the bound states (eigenvectors) with the continuous spectrum (as , for example, in the case of the ionization of an atom or the photoelectric effect).

Definition 0.1.

A rigged Hilbert space is a pair $(\mathscr{H},\varphi )$ with $\mathscr{H}$ a Hilbert space and $\varphi$ is a dense subspace with a topological vector space structure for which the inclusion map $i$ is continuous. Between $\mathscr{H}$ and its dual space ${\mathscr{H}}^{*}$ there is defined the adjoint map $i^{*}:{\mathscr{H}}^{*}\to {\varphi }^{*}$ of the continuous inclusion map $i$. The duality pairing between $\varphi$ and ${\varphi }^{*}$ also needs to be compatible with the inner product on$\mathscr{H}$:

$${⟨u,v⟩}_{\varphi \times {\varphi }^{*}}={(u,v)}_{\mathscr{H}}$$

whenever$u\in \varphi \subset \mathscr{H}$ and $v\in \mathscr{H}={\mathscr{H}}^{*}\subset {\varphi }^{*}$.