Solutions to Exercises
301
−0.09
−0.08
−0.07
0.8
0.81
0.82
Re E (MeV)
Im E (MeV)
Optimum value in GEM+CS
Pole of GSM
θ = 5°
θ = 25°
(θ = 13°)
GEM+CS
Fig. 6.22 Complex energies of the 6 He (2 + ) resonance state as a function of the rotation angle
θ for l max = 1. θ varies by 1 ◦ from one circle to another. The rotation angle θ varies from 5 ◦ to
25 ◦ . The notation “GEM + CS” stands for Gaussian Expansion Method + Complex Scaling, and
“GSM” for Gamow shell model (from Ref. [152])
so that the value θ = θ opt is used by default in Hamiltonian complex scaling
calculations. However, one can see in Fig. 6.22 that a sole variation of 1 ◦ of θ can
induce a change of about 10 keV. Consequently, the θ -dependence can play a role
in the energy differences noticed in the resonance eigenstates of 6 He and 6 Be when
performing diagonalizations with Gamow shell model and Hamiltonian complex
scaling frameworks.
Solutions to Exercises 1
Exercise I.
Commutators are straightforward to calculate from Eqs. (6.41), (6.43), (6.45),
(6.46), (6.50), and (6.52):
ˆ ˜
n q , B
†
η
=
2 ˜
b
†
q
√
w q
2 q − E η
ˆ ˜
n q , B
†
0
= 2
√
w q ˜
b
†
q
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.