9.2 Generalization to Many-Nucleon Projectiles
441
One will now solve Eqs. (9.122–9.124) by constructing target–projectile channels that verify coupled-channel equations. In the continuum discretized coupledchannel method [58–60, 62], the scattering states |Φ P are dealt with bin states,
obtained by integrating scattering states on finite energy interval, so as to obtain
a basis of discrete bound states. However, this method does not allow to calculate
complex-energy resonance states. Thus, let us consider the Berggren basis associated to the intrinsic Hamiltonian ˆ
H P of the projectile:
ˆ
H P |Φ P = E P |Φ P .
(9.125)
|Φ P is straightforward to calculate as H P only depends on the relative space
coordinate r (see Eq. (9.124)). We can now expand the solution of Eq. (9.122) in
a coupled-channel expansion of intrinsic and center-of-mass projectile states:
|Ψ =
n
|Φ
(n)
P |Ψ
(n)
J
,
(9.126)
where J is the total angular momentum of the system, n represents a given target–
projectile channel, |Φ
(n)
P is a Berggren basis eigenstate of H P , and |Ψ (n) is a
center-of-mass projectile state to be determined.
By using the coupled-channel representation of ˆ
V T (r, R) and expanding |Ψ (n)
in partial waves (see Eq. (9.126)), one can show that |Ψ (n) verifies the equations:
Ψ
(n) (R) =
L
χ
(L)
n (R)Y L (Ω R )
×
¯
h
2
2M CM
−
d 2
dR 2 +
L(L + 1)
R 2
+ V
(T)
nn (R) − E n
χ
(L)
n (R)
+
n =n
V
(T)
nn (R)χ
(L )
n (R) = 0 ,
(9.127)
where
V
(T)
nn (R) = =
Φ
(n )
P Y L
J
| ˆ
V T |
Φ
(n)
P Y L
J
.
(9.128)
In this equation, L is the orbital angular momentum of the system, n represents the
different quantum numbers associated to channels, and E n = E − E
(n)
P , with E
(n)
P
the energy of state |Φ
(n)
P (see Eq. (9.125)).
Equation (9.128) has the same form as Eq. (9.149), so that it can be solved with
the Green’s function method described in Sect. 9.3.3.2. As this method replaces the
direct integration of coupled-channel equations by a linear system, it should be
efficient, insofar as the number of channels in Faddeev equations is usually very
large.
441
One will now solve Eqs. (9.122–9.124) by constructing target–projectile channels that verify coupled-channel equations. In the continuum discretized coupledchannel method [58–60, 62], the scattering states |Φ P are dealt with bin states,
obtained by integrating scattering states on finite energy interval, so as to obtain
a basis of discrete bound states. However, this method does not allow to calculate
complex-energy resonance states. Thus, let us consider the Berggren basis associated to the intrinsic Hamiltonian ˆ
H P of the projectile:
ˆ
H P |Φ P = E P |Φ P .
(9.125)
|Φ P is straightforward to calculate as H P only depends on the relative space
coordinate r (see Eq. (9.124)). We can now expand the solution of Eq. (9.122) in
a coupled-channel expansion of intrinsic and center-of-mass projectile states:
|Ψ =
n
|Φ
(n)
P |Ψ
(n)
J
,
(9.126)
where J is the total angular momentum of the system, n represents a given target–
projectile channel, |Φ
(n)
P is a Berggren basis eigenstate of H P , and |Ψ (n) is a
center-of-mass projectile state to be determined.
By using the coupled-channel representation of ˆ
V T (r, R) and expanding |Ψ (n)
in partial waves (see Eq. (9.126)), one can show that |Ψ (n) verifies the equations:
Ψ
(n) (R) =
L
χ
(L)
n (R)Y L (Ω R )
×
¯
h
2
2M CM
−
d 2
dR 2 +
L(L + 1)
R 2
+ V
(T)
nn (R) − E n
χ
(L)
n (R)
+
n =n
V
(T)
nn (R)χ
(L )
n (R) = 0 ,
(9.127)
where
V
(T)
nn (R) = =
Φ
(n )
P Y L
J
| ˆ
V T |
Φ
(n)
P Y L
J
.
(9.128)
In this equation, L is the orbital angular momentum of the system, n represents the
different quantum numbers associated to channels, and E n = E − E
(n)
P , with E
(n)
P
the energy of state |Φ
(n)
P (see Eq. (9.125)).
Equation (9.128) has the same form as Eq. (9.149), so that it can be solved with
the Green’s function method described in Sect. 9.3.3.2. As this method replaces the
direct integration of coupled-channel equations by a linear system, it should be
efficient, insofar as the number of channels in Faddeev equations is usually very
large.