5.1 Mathematical Foundation of the Gamow Shell Model
191
Φ| ˆ
A|Φ (see Eq. (5.13)):
Φ| ˆ
A|Φ = A( ˜
E) A(E) − i
Γ
2
A
(E) ,
(5.15)
so that A r A(E) and A i −(Γ /2) A (E). Moreover, Δ t (A) ∝ Γ at first-order in
Γ if Γ E (see Eq. (5.10)). Consequently, at leading order, A r can be interpreted
as the average value of measurements made at times t T , whereas A i can be seen
as its weighted dispersion rate over time:
A r = A(t)
(5.16)
A i = − ¯
h
√
3
A (E)
|A(E)|
Δ t (A) .
(5.17)
Consequently, the interpretation of complex observables in a quasi-stationary formalism is formally clear. Indeed, both A r and A i can be in principle experimentally
measured. However, as the values of cross sections involving unbound states are
much smaller than those involving stable nuclei, a statistical analysis of observables
related to resonance states seems very difficult to obtain. Moreover, electromagnetic
and beta decays typically have time scales which are much longer than those of
particle emission, so that they are very unlikely to precede particle decay. Thus, it
seems that Eqs. (5.16) and (5.17) are primarily of theoretical interest, and that their
experimental study would demand very precise experimental instruments, which are
not available at present.
5.1.3 Many-Body Berggren Completeness Relation
The discretized basis (3.82) can be a starting point for establishing the completeness
relation in the many-body case, in a full analogy with the standard shell model in
a complete discrete basis, e.g., the harmonic oscillator basis. Indeed, the Berggren
completeness relation is discretized with the Gauss-Legendre quadrature, so that it
is formally identical to a discrete complete set of states (see Sect. 3.5). One can then
derive the many-body completeness relation:
n
|Ψ n
Ψ n | | ˆ
1 .
(5.18)
The N-body Slater determinants |Ψ n have the form |φ 1 . . . φ N , where |φ k
are resonant (bound and decaying) and scattering (contour) single-particle states.
The approximate equality in (5.18) is an obvious consequence of the continuum
discretization, similarly as in Eq. (3.82), so that the exact completeness relation is
restored when the number of discretized basis scattering states becomes infinite.
Like in the case of single-particle Gamow states, the normalization of the Gamow
191
Φ| ˆ
A|Φ (see Eq. (5.13)):
Φ| ˆ
A|Φ = A( ˜
E) A(E) − i
Γ
2
A
(E) ,
(5.15)
so that A r A(E) and A i −(Γ /2) A (E). Moreover, Δ t (A) ∝ Γ at first-order in
Γ if Γ E (see Eq. (5.10)). Consequently, at leading order, A r can be interpreted
as the average value of measurements made at times t T , whereas A i can be seen
as its weighted dispersion rate over time:
A r = A(t)
(5.16)
A i = − ¯
h
√
3
A (E)
|A(E)|
Δ t (A) .
(5.17)
Consequently, the interpretation of complex observables in a quasi-stationary formalism is formally clear. Indeed, both A r and A i can be in principle experimentally
measured. However, as the values of cross sections involving unbound states are
much smaller than those involving stable nuclei, a statistical analysis of observables
related to resonance states seems very difficult to obtain. Moreover, electromagnetic
and beta decays typically have time scales which are much longer than those of
particle emission, so that they are very unlikely to precede particle decay. Thus, it
seems that Eqs. (5.16) and (5.17) are primarily of theoretical interest, and that their
experimental study would demand very precise experimental instruments, which are
not available at present.
5.1.3 Many-Body Berggren Completeness Relation
The discretized basis (3.82) can be a starting point for establishing the completeness
relation in the many-body case, in a full analogy with the standard shell model in
a complete discrete basis, e.g., the harmonic oscillator basis. Indeed, the Berggren
completeness relation is discretized with the Gauss-Legendre quadrature, so that it
is formally identical to a discrete complete set of states (see Sect. 3.5). One can then
derive the many-body completeness relation:
n
|Ψ n
Ψ n | | ˆ
1 .
(5.18)
The N-body Slater determinants |Ψ n have the form |φ 1 . . . φ N , where |φ k
are resonant (bound and decaying) and scattering (contour) single-particle states.
The approximate equality in (5.18) is an obvious consequence of the continuum
discretization, similarly as in Eq. (3.82), so that the exact completeness relation is
restored when the number of discretized basis scattering states becomes infinite.
Like in the case of single-particle Gamow states, the normalization of the Gamow