292
6 Physical Applications of the Gamow Shell Model
Alternatively, there exists a many-body formalism allowing to calculate the
many-body resonances without recurring to the use of the Berggren basis. This
formalism is based on the use of complex-scaled Hamiltonians [139, 140]. In the
Hamiltonian complex scaling approach, the Hamiltonian is firstly transformed using
a complex rotation similar to that described in Sect. 3.3 [139,140]. The fundamental
advantage of this transformation is that resonance states become integrable on the
real r-axis, so that they can be expanded with a basis of bound states, as in standard
shell model.
The fact that resonance many-body states, of unbound character, can be represented with bound states expansions, relies on Aguilar-Blaslev-Combes theorem,
or shortly the ABC theorem [139, 140]. The ABC theorem states that resonance
energies are independent of the complex scaling angle θ used in the transformation
of the considered Hamiltonian, provided that θ is sufficiently large. Consequently,
Hamiltonian complex scaling, which is based on the use of the ABC theorem,
has been developed in order to calculate resonance energies of unbound quantum
systems using the apparatus of bound state expansions. In fact, Hamiltonian
complex scaling has been widely applied for many years to calculate the unbound
spectrum of atomic systems and molecules [141–144] and light nuclei [145–154].
In the following, the ABC theorem is briefly studied in the one-body and manybody cases. From a practical point of view, it is of interest to compare the results
obtained with the Gamow shell model and from the diagonalization of complexscaled Hamiltonians. For that purpose, the Gamow shell model and Hamiltonian
complex scaling method will be considered in a simple three-particle system. Due
to the small model space dimensions involved, the un-truncated calculations can be
performed to guarantee the convergence with respect to basis size at a numerical
level. The overall numerical properties of Hamiltonian complex scaling in practical
calculations will then be stated and compared to those of the Gamow shell model
(see Chap. 5).
6.6.1 Complex-Scaled Hamiltonians and Their Eigenstates
The Hamiltonian complex scaling method is defined from the ˆ
U θ operator, which
applies a complex rotation to radial one-body wave functions [139]:
ˆ
U θ |u = e
iθ/2
u(r e iθ )
r
,
(6.66)
where θ is the rotation angle and |u is a radial one-body state. Note that, similarly to
Sect. 5.1.1, no complex conjugation enters the matrix elements of Eq. (6.66) and that
angular matrix elements do not change when complex scaling is applied. This is the
case because Hamiltonian complex scaling is defined from the analytic continuation
of the real-axis dilation r → re θ [139].
6 Physical Applications of the Gamow Shell Model
Alternatively, there exists a many-body formalism allowing to calculate the
many-body resonances without recurring to the use of the Berggren basis. This
formalism is based on the use of complex-scaled Hamiltonians [139, 140]. In the
Hamiltonian complex scaling approach, the Hamiltonian is firstly transformed using
a complex rotation similar to that described in Sect. 3.3 [139,140]. The fundamental
advantage of this transformation is that resonance states become integrable on the
real r-axis, so that they can be expanded with a basis of bound states, as in standard
shell model.
The fact that resonance many-body states, of unbound character, can be represented with bound states expansions, relies on Aguilar-Blaslev-Combes theorem,
or shortly the ABC theorem [139, 140]. The ABC theorem states that resonance
energies are independent of the complex scaling angle θ used in the transformation
of the considered Hamiltonian, provided that θ is sufficiently large. Consequently,
Hamiltonian complex scaling, which is based on the use of the ABC theorem,
has been developed in order to calculate resonance energies of unbound quantum
systems using the apparatus of bound state expansions. In fact, Hamiltonian
complex scaling has been widely applied for many years to calculate the unbound
spectrum of atomic systems and molecules [141–144] and light nuclei [145–154].
In the following, the ABC theorem is briefly studied in the one-body and manybody cases. From a practical point of view, it is of interest to compare the results
obtained with the Gamow shell model and from the diagonalization of complexscaled Hamiltonians. For that purpose, the Gamow shell model and Hamiltonian
complex scaling method will be considered in a simple three-particle system. Due
to the small model space dimensions involved, the un-truncated calculations can be
performed to guarantee the convergence with respect to basis size at a numerical
level. The overall numerical properties of Hamiltonian complex scaling in practical
calculations will then be stated and compared to those of the Gamow shell model
(see Chap. 5).
6.6.1 Complex-Scaled Hamiltonians and Their Eigenstates
The Hamiltonian complex scaling method is defined from the ˆ
U θ operator, which
applies a complex rotation to radial one-body wave functions [139]:
ˆ
U θ |u = e
iθ/2
u(r e iθ )
r
,
(6.66)
where θ is the rotation angle and |u is a radial one-body state. Note that, similarly to
Sect. 5.1.1, no complex conjugation enters the matrix elements of Eq. (6.66) and that
angular matrix elements do not change when complex scaling is applied. This is the
case because Hamiltonian complex scaling is defined from the analytic continuation
of the real-axis dilation r → re θ [139].