242
6 Physical Applications of the Gamow Shell Model
given by the ratio of Berggren and Hermitian norms of an eigenfunction. The
indicator (6.5) can also be written in a form [13]:
r j = e
2iθ j
dr
|ReΨ j (r)| 2 − |ImΨ j (r)| 2
dr
|ReΨ j (r)| 2 + |ImΨ j (r)| 2
,
(6.6)
which better illustrates its physical meaning. In this expression, the angle θ j
arises from a transformation of Ψ j so that ReΨ j and ImΨ j are orthogonal and it
characterizes the degree to which the eigenfunction Ψ j is complex.
Phase rigidity varies between 1 and 0. It equals 1 for bound state eigenfunctions,
whereas at the coalescence point of wave functions (the exceptional point) it equals
0. For unbound states, the condition r j =1 means that the continuum couplings exert
a negligible effect on the internal structure of an eigenfunction, i.e. its Berggren and
Hermitian norms are identical. Abrupt variations of the phase rigidity in a certain
interval of excitation energies are indicative of the instability of wave functions.
Another indicator of such instabilities could be the continuum-coupling energy
correction E
(i)
corr (Eq. (6.2)) which was studied extensively in the shell model
embedded in the continuum [6, 7, 15, 16]. In Gamow shell model, one may use
as an indicator the value of energy correction due to coupling to the nonresonant
continuum background.
ΔE B−space = =Ψ i | ˆ
H |Ψ i − −Ψ 0;i | ˆ
H |Ψ 0;i ,
(6.7)
where Ψ 0;i is the pole space wave function corresponding to the Gamow shell model
eigenfunction Ψ i . This energy correction is a measure of continuum-induced instability of the pole space wave function and resulting rearrangement of occupancies
of the resonant and nonresonant shells in Gamow shell model eigenfunctions, in
particular, due to the proximity of the branch points associated with the particleemission thresholds and the double-poles of the S-matrix (the exceptional points).
Figure 6.1 shows the difference ΔE B−space between Gamow shell model
energies calculated in the full Berggren ensemble and in the pole space for the 0
+
1
ground state of 6 He, as a function of the one neutron separation energy S 1n from
5 He. In this calculation, the Hamiltonian consists of one-body potential generated by
4 He core and the Furutani–Horiuchi–Tamagaki two-body interaction [17,18]. S 1n of
5 He is modified by changing the depth of the Woods–Saxon potential. One can see
in Fig. 6.1 a non-analytic variation of ΔE B−space at around the threshold S 1n = 0.
Further discussion of near-threshold singularities will be done in Sects. 7.4.1, 7.4.2,
and 7.4.2.1.
6.1.2 Effect of Continuum Coupling on the Spin–Orbit Splitting
The coupling to the particle continuum in low- orbits may lead to the instability
of the Hartree–Fock particle vacuum and the coexistence of local minima corresponding to different distribution of S-matrix poles around the threshold. In this
6 Physical Applications of the Gamow Shell Model
given by the ratio of Berggren and Hermitian norms of an eigenfunction. The
indicator (6.5) can also be written in a form [13]:
r j = e
2iθ j
dr
|ReΨ j (r)| 2 − |ImΨ j (r)| 2
dr
|ReΨ j (r)| 2 + |ImΨ j (r)| 2
,
(6.6)
which better illustrates its physical meaning. In this expression, the angle θ j
arises from a transformation of Ψ j so that ReΨ j and ImΨ j are orthogonal and it
characterizes the degree to which the eigenfunction Ψ j is complex.
Phase rigidity varies between 1 and 0. It equals 1 for bound state eigenfunctions,
whereas at the coalescence point of wave functions (the exceptional point) it equals
0. For unbound states, the condition r j =1 means that the continuum couplings exert
a negligible effect on the internal structure of an eigenfunction, i.e. its Berggren and
Hermitian norms are identical. Abrupt variations of the phase rigidity in a certain
interval of excitation energies are indicative of the instability of wave functions.
Another indicator of such instabilities could be the continuum-coupling energy
correction E
(i)
corr (Eq. (6.2)) which was studied extensively in the shell model
embedded in the continuum [6, 7, 15, 16]. In Gamow shell model, one may use
as an indicator the value of energy correction due to coupling to the nonresonant
continuum background.
ΔE B−space = =Ψ i | ˆ
H |Ψ i − −Ψ 0;i | ˆ
H |Ψ 0;i ,
(6.7)
where Ψ 0;i is the pole space wave function corresponding to the Gamow shell model
eigenfunction Ψ i . This energy correction is a measure of continuum-induced instability of the pole space wave function and resulting rearrangement of occupancies
of the resonant and nonresonant shells in Gamow shell model eigenfunctions, in
particular, due to the proximity of the branch points associated with the particleemission thresholds and the double-poles of the S-matrix (the exceptional points).
Figure 6.1 shows the difference ΔE B−space between Gamow shell model
energies calculated in the full Berggren ensemble and in the pole space for the 0
+
1
ground state of 6 He, as a function of the one neutron separation energy S 1n from
5 He. In this calculation, the Hamiltonian consists of one-body potential generated by
4 He core and the Furutani–Horiuchi–Tamagaki two-body interaction [17,18]. S 1n of
5 He is modified by changing the depth of the Woods–Saxon potential. One can see
in Fig. 6.1 a non-analytic variation of ΔE B−space at around the threshold S 1n = 0.
Further discussion of near-threshold singularities will be done in Sects. 7.4.1, 7.4.2,
and 7.4.2.1.
6.1.2 Effect of Continuum Coupling on the Spin–Orbit Splitting
The coupling to the particle continuum in low- orbits may lead to the instability
of the Hartree–Fock particle vacuum and the coexistence of local minima corresponding to different distribution of S-matrix poles around the threshold. In this