6.1 Effective Interaction in the Vicinity of the Particle-Emission Threshold
241
where w i = ˆ
H PQ |Φ i is the source term, and ω i
(+) (r) = G
(+)
P
ˆ
H PQ |Φ i is the
solution of an inhomogeneous radial equation with the outgoing boundary condition
[9, 10]:
E − ˆ
H PP
ω i
(+) (r) = w i (r) .
(6.3)
In the following, we shall assume that the Hamiltonian in P -space ˆ
H PP consists
of a kinetic energy operator and a finite-depth, square-well potential of radius R 0 .
The source term in Eq. (6.3) is modelled by w(r) ∝ r ν (ν ≥ + 1) for 0 ≤
r ≤ R 0 and w(r) = 0 for r > R 0 , where is the orbital angular momentum of a
particle. The assumed radial dependence of this localized source w(r) for r → 0 is
consistent with the radial dependence of single-particle wave functions which enter
in the microscopic calculation of this term [9, 10]. Under these assumptions, the
continuum-coupling correction to the eigenenergy of the closed quantum system at
the threshold (E = 0) can be calculated analytically as a function of the distance ε
of the eigenvalue of ˆ
H PP from the one-body continuum threshold:
E
(()
corr (ε) = −const|ε|
−1+
+ O(|ε|
0 ) .
(6.4)
The continuum-coupling correction at the threshold becomes singular for = 0, 1
states in the limit of ε → 0, independently of the nature of the considered S-matrix
pole. In the realistic case, the attenuation of this singular behavior is expected by
the residual nucleon–nucleon interaction and the channel–channel couplings. For
higher -values ( ≥ 2), a dependence of the continuum-coupling energy correction
on the position of S-matrix pole is not singular and therefore, certain aspects of
the continuum coupling can be imitated in standard shell model by introducing the
particle number dependence in monopole terms of the effective interaction.
6.1.1 Monitoring Branch Points and Avoided Crossings
in the Complex Plane
Monitoring effects of the configuration mixing in resonance wave function are an
important issue in the studies of open quantum systems. Branch points associated
with the reaction thresholds and avoided level crossings or exceptional points are
essential elements of the configuration mixing in open quantum systems. A useful
indicator of the effects caused by the continuum couplings is the phase rigidity [11–
13]:
r j =
˜
Ψ j |Ψ j
j |Ψ j
,
(6.5)
241
where w i = ˆ
H PQ |Φ i is the source term, and ω i
(+) (r) = G
(+)
P
ˆ
H PQ |Φ i is the
solution of an inhomogeneous radial equation with the outgoing boundary condition
[9, 10]:
E − ˆ
H PP
ω i
(+) (r) = w i (r) .
(6.3)
In the following, we shall assume that the Hamiltonian in P -space ˆ
H PP consists
of a kinetic energy operator and a finite-depth, square-well potential of radius R 0 .
The source term in Eq. (6.3) is modelled by w(r) ∝ r ν (ν ≥ + 1) for 0 ≤
r ≤ R 0 and w(r) = 0 for r > R 0 , where is the orbital angular momentum of a
particle. The assumed radial dependence of this localized source w(r) for r → 0 is
consistent with the radial dependence of single-particle wave functions which enter
in the microscopic calculation of this term [9, 10]. Under these assumptions, the
continuum-coupling correction to the eigenenergy of the closed quantum system at
the threshold (E = 0) can be calculated analytically as a function of the distance ε
of the eigenvalue of ˆ
H PP from the one-body continuum threshold:
E
(()
corr (ε) = −const|ε|
−1+
+ O(|ε|
0 ) .
(6.4)
The continuum-coupling correction at the threshold becomes singular for = 0, 1
states in the limit of ε → 0, independently of the nature of the considered S-matrix
pole. In the realistic case, the attenuation of this singular behavior is expected by
the residual nucleon–nucleon interaction and the channel–channel couplings. For
higher -values ( ≥ 2), a dependence of the continuum-coupling energy correction
on the position of S-matrix pole is not singular and therefore, certain aspects of
the continuum coupling can be imitated in standard shell model by introducing the
particle number dependence in monopole terms of the effective interaction.
6.1.1 Monitoring Branch Points and Avoided Crossings
in the Complex Plane
Monitoring effects of the configuration mixing in resonance wave function are an
important issue in the studies of open quantum systems. Branch points associated
with the reaction thresholds and avoided level crossings or exceptional points are
essential elements of the configuration mixing in open quantum systems. A useful
indicator of the effects caused by the continuum couplings is the phase rigidity [11–
13]:
r j =
˜
Ψ j |Ψ j
j |Ψ j
,
(6.5)