270
6 Physical Applications of the Gamow Shell Model
channel equations:
−
¯
h 2
2m
d 2
dρ 2 −
(K + 3/2)(K + 5/2)
ρ 2
− ˜
E
ψ
J π
γ K (ρ)
+
K γ
ˆ
V
J π
K γ ,Kγ (ρ)ψ
J π
γ K (ρ) +
K γ
+∞
0
ˆ
W K γ ,Kγ (ρ, ρ
)ψ
Lπ
γ K (ρ
)dρ
= 0 ,
(6.27)
where
ˆ
V
Lπ
K γ ,Kγ (ρ) = =Y
J M
γ K |
3
i>j =1
ˆ
V ij (r ij )|Y
J M
γ K
ˆ
W K γ ,Kγ (ρ, ρ
) = =Y
J M
γ K |Λ
c
|ϕ
j c m c
j c m c | Y
J M
γ K ,
(6.28)
and where ˆ
W K γ ,Kγ (ρ, ρ ) is the nonlocal potential generated by the Pauli projection operator in Eq. (6.21). The radial matrix elements of the Hamiltonian
in the Berggren basis are: n (k n , ρ)|V (ρ)|B m (k m , ρ). The radial integral in
these matrix elements can oscillate at large distances, especially for Coulomb and
centrifugal potentials, so that it is numerically unstable. The nuclear potential in
a hyperspherical harmonics expansion has also a long-range behavior in O(1/ρ 3 )
[107], so that it is difficult to handle numerically as well. Hence, one has to find an
integral path where wavefunctions vanish in the asymptotic region. For potentials
that decrease as O(1/ρ 2 ) (the centrifugal potential) or faster (the nuclear potential),
one can use the exterior complex scaling [108], where integrals are calculated along
a complex radial path:
n | ˆ
V (ρ)|B m =
R
0
B n (ρ) ˆ
V (ρ)B m (ρ)dρ
+
+∞
0
B n (R + ρe
iθ ) ˆ
V (R + ρe
iθ )B m (R + ρe
iθ )dρ .
(6.29)
In the above equation, R is a radius taken sufficiently large to bypass all singularities, and θ is a rotation angle chosen so that the integral converges. θ can be positive
or negative, according to the different behavior of outgoing (H + ) and incoming
(H − ) wave functions. More details can be found in Ref. [109]. As the Coulomb
potential is infinite-range, its diagonal matrix elements involving Berggren basis
scattering states diverge even using exterior complex scaling. A practical solution
of this problem is the off-diagonal method [110]. Basically, a small offset ±δk is
added to the linear momenta k n and k m of involved scattering wave functions, so
6 Physical Applications of the Gamow Shell Model
channel equations:
−
¯
h 2
2m
d 2
dρ 2 −
(K + 3/2)(K + 5/2)
ρ 2
− ˜
E
ψ
J π
γ K (ρ)
+
K γ
ˆ
V
J π
K γ ,Kγ (ρ)ψ
J π
γ K (ρ) +
K γ
+∞
0
ˆ
W K γ ,Kγ (ρ, ρ
)ψ
Lπ
γ K (ρ
)dρ
= 0 ,
(6.27)
where
ˆ
V
Lπ
K γ ,Kγ (ρ) = =Y
J M
γ K |
3
i>j =1
ˆ
V ij (r ij )|Y
J M
γ K
ˆ
W K γ ,Kγ (ρ, ρ
) = =Y
J M
γ K |Λ
c
|ϕ
j c m c
j c m c | Y
J M
γ K ,
(6.28)
and where ˆ
W K γ ,Kγ (ρ, ρ ) is the nonlocal potential generated by the Pauli projection operator in Eq. (6.21). The radial matrix elements of the Hamiltonian
in the Berggren basis are: n (k n , ρ)|V (ρ)|B m (k m , ρ). The radial integral in
these matrix elements can oscillate at large distances, especially for Coulomb and
centrifugal potentials, so that it is numerically unstable. The nuclear potential in
a hyperspherical harmonics expansion has also a long-range behavior in O(1/ρ 3 )
[107], so that it is difficult to handle numerically as well. Hence, one has to find an
integral path where wavefunctions vanish in the asymptotic region. For potentials
that decrease as O(1/ρ 2 ) (the centrifugal potential) or faster (the nuclear potential),
one can use the exterior complex scaling [108], where integrals are calculated along
a complex radial path:
n | ˆ
V (ρ)|B m =
R
0
B n (ρ) ˆ
V (ρ)B m (ρ)dρ
+
+∞
0
B n (R + ρe
iθ ) ˆ
V (R + ρe
iθ )B m (R + ρe
iθ )dρ .
(6.29)
In the above equation, R is a radius taken sufficiently large to bypass all singularities, and θ is a rotation angle chosen so that the integral converges. θ can be positive
or negative, according to the different behavior of outgoing (H + ) and incoming
(H − ) wave functions. More details can be found in Ref. [109]. As the Coulomb
potential is infinite-range, its diagonal matrix elements involving Berggren basis
scattering states diverge even using exterior complex scaling. A practical solution
of this problem is the off-diagonal method [110]. Basically, a small offset ±δk is
added to the linear momenta k n and k m of involved scattering wave functions, so