186
5 Formulation and Implementation of the Gamow Shell Model
purely formal and one would have no guarantee that they will provide with the
proper eigenvectors of the considered Hamiltonian. Consequently, one has to leave
the Hilbert space, where only square integrable states can be considered, and replace
it by a larger space able to deal with resonance and scattering states. This is possible
within the so-called rigged Hilbert space, at the price, however, of introducing a
more complicated abstract framework involving different spaces for the states and
for the linear functionals acting on them.
5.1.1 Rigged Hilbert Space Setting of the Gamow Shell Model
The Dirac notation, consisting in the standard bra and ket notation, was devised in
the context of bound states. Its usefulness and mathematical rigor is based on the use
of the Riesz representation theorem of the Hilbert space. Indeed, the latter theorem
implies that the Hilbert space and its dual space are anti-isomorphic, so that the
action of the dual of a state vector |A, i.e., the bra state on another ket state
vector |B, provides with the same result as the scalar product between the two latter
state vectors: = =A|B
This notation greatly simplifies equations, as all coordinate and momentum
space dependences, as well as multidimensional integration, are embedded in an
algebraic formalism of operators and vectors. It is not possible to use it rigorously if
one includes scattering states, as they cannot be normalized except with a Dirac
delta. Nevertheless, they can be formally handled with bra and ket states, as
they correspond to well-defined wave functions. However, a rigorous mathematical
framework with which bound, resonance, and scattering states can all be represented
with the Dirac notation demands the introduction of rigged Hilbert spaces.
The rigged Hilbert space, or Gel’fand triple, is defined from three space
inclusions [2–4]:
Φ ⊂ H ⊂ Φ
× .
(5.1)
Φ in this relation is the subspace of test functions, dense in the Hilbert space H ,
whose elements can be seen as bounded operators bearing a fast decrease on the
real axis. Φ × is the space of anti-linear functionals over Φ, comprising in particular
distributions, where all bound, resonance, and scattering states will be defined.
Finally, Φ × is thus the rigged Hilbert space of interest.
The linear functionals over Φ are contained in another rigged Hilbert space,
denoted as Φ
:
Φ ⊂ H ⊂ Φ
,
(5.2)
so that unbound bras and kets belong to Φ
and Φ × , respectively.
The unbound states in Φ × will enter scalar products only with test functions of
Φ, so that they all are well defined even though Φ × contains non-integrable states.
One may note that the Hilbert space H does not have to be included, as bound,
5 Formulation and Implementation of the Gamow Shell Model
purely formal and one would have no guarantee that they will provide with the
proper eigenvectors of the considered Hamiltonian. Consequently, one has to leave
the Hilbert space, where only square integrable states can be considered, and replace
it by a larger space able to deal with resonance and scattering states. This is possible
within the so-called rigged Hilbert space, at the price, however, of introducing a
more complicated abstract framework involving different spaces for the states and
for the linear functionals acting on them.
5.1.1 Rigged Hilbert Space Setting of the Gamow Shell Model
The Dirac notation, consisting in the standard bra and ket notation, was devised in
the context of bound states. Its usefulness and mathematical rigor is based on the use
of the Riesz representation theorem of the Hilbert space. Indeed, the latter theorem
implies that the Hilbert space and its dual space are anti-isomorphic, so that the
action of the dual of a state vector |A, i.e., the bra state on another ket state
vector |B, provides with the same result as the scalar product between the two latter
state vectors: = =A|B
This notation greatly simplifies equations, as all coordinate and momentum
space dependences, as well as multidimensional integration, are embedded in an
algebraic formalism of operators and vectors. It is not possible to use it rigorously if
one includes scattering states, as they cannot be normalized except with a Dirac
delta. Nevertheless, they can be formally handled with bra and ket states, as
they correspond to well-defined wave functions. However, a rigorous mathematical
framework with which bound, resonance, and scattering states can all be represented
with the Dirac notation demands the introduction of rigged Hilbert spaces.
The rigged Hilbert space, or Gel’fand triple, is defined from three space
inclusions [2–4]:
Φ ⊂ H ⊂ Φ
× .
(5.1)
Φ in this relation is the subspace of test functions, dense in the Hilbert space H ,
whose elements can be seen as bounded operators bearing a fast decrease on the
real axis. Φ × is the space of anti-linear functionals over Φ, comprising in particular
distributions, where all bound, resonance, and scattering states will be defined.
Finally, Φ × is thus the rigged Hilbert space of interest.
The linear functionals over Φ are contained in another rigged Hilbert space,
denoted as Φ
:
Φ ⊂ H ⊂ Φ
,
(5.2)
so that unbound bras and kets belong to Φ
and Φ × , respectively.
The unbound states in Φ × will enter scalar products only with test functions of
Φ, so that they all are well defined even though Φ × contains non-integrable states.
One may note that the Hilbert space H does not have to be included, as bound,