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

with

a
Hilbert space
and

is a dense subspace with a topological
vector space
structure for which the inclusion map
is continuous. Between

and its
dual space

there is defined the adjoint map

of the continuous inclusion map

. The
duality
pairing between

and

also needs to be compatible with the
inner product
on

:
whenever

and

.
- 1
-
R. de la Madrid, ``The role of the rigged Hilbert space in Quantum Mechanics.'', Eur. J. Phys. 26, 287 (2005);
.
- 2
-
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,
.
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