9.1 Nucleon–Nucleus Reactions
409
introduces the projectors “in” and “out” of the core
ˆ
Q =
s∈core
|s
(9.16)
ˆ
P = ˆ
1 − ˆ
Q ,
(9.17)
where |s is a one-body state occupied in the core.
Exercise II
One will study the use of projectors within the cluster orbital shell model.
A. Show that ˆ
Q and ˆ
P are orthogonal Hermitian projectors and state their
eigenvalues and eigenvectors.
B. Explain how to project an arbitrary operator on the P space with projectors ˆ
Q
and ˆ
P .
Show that the projection of a Hermitian operator on the P space remains
Hermitian.
C. State the physical significance of the use of Hermitian projectors ˆ
Q and ˆ
P
within the cluster orbital shell model.
Explain why considering complex-symmetric projectors and Hamiltonians is
not sufficient for practical applications.
The projectile Hamiltonian ˆ
H P (see Eq. (9.15)) must be replaced by ˆ
P ˆ
H P ˆ
P in
order to have valence states orthogonal to core states (see Exercise II). Thus, one
obtains from Eqs. (9.16, 9.17)
ˆ
H P → ˆ
P ˆ
H P ˆ
P = ˆ
H P − ˆ
Q ˆ
H P − ˆ
H P ˆ
Q + ˆ
Q ˆ
H P ˆ
Q .
(9.18)
The |s states entering the ˆ
Q projector in Eq. (9.16) belong to the core, so that
they are well bound. As a consequence, the ˆ
Q projector takes a form of the potential
localized in the nuclear zone when represented in coordinate space. Therefore,
ˆ
P ˆ
H P ˆ
P induces an additional nonlocal potential in ˆ
U basis of Eq. (9.15). With this
modification of ˆ
U basis , all previously derived equations remain the same, as long as
core states |u n j τ are suppressed in all expansions involving Berggren basis states.
Therefore, composite states in the Gamow coupled-channel formalism, consisting of
target many-body states, provided by the Gamow shell model, and of the one-body
state of the projectile, are orthogonal to the core occupied states.
9.1.1.2 Coupled-Channel Equations
As one has defined the channel basis and the matrix elements of the Hamiltonian in
a basis of channel many-body states, one can now determine the A-body scattering
states solutions of coupled-channel equations. Let us define the A-body scattering
409
introduces the projectors “in” and “out” of the core
ˆ
Q =
s∈core
|s
(9.16)
ˆ
P = ˆ
1 − ˆ
Q ,
(9.17)
where |s is a one-body state occupied in the core.
Exercise II
One will study the use of projectors within the cluster orbital shell model.
A. Show that ˆ
Q and ˆ
P are orthogonal Hermitian projectors and state their
eigenvalues and eigenvectors.
B. Explain how to project an arbitrary operator on the P space with projectors ˆ
Q
and ˆ
P .
Show that the projection of a Hermitian operator on the P space remains
Hermitian.
C. State the physical significance of the use of Hermitian projectors ˆ
Q and ˆ
P
within the cluster orbital shell model.
Explain why considering complex-symmetric projectors and Hamiltonians is
not sufficient for practical applications.
The projectile Hamiltonian ˆ
H P (see Eq. (9.15)) must be replaced by ˆ
P ˆ
H P ˆ
P in
order to have valence states orthogonal to core states (see Exercise II). Thus, one
obtains from Eqs. (9.16, 9.17)
ˆ
H P → ˆ
P ˆ
H P ˆ
P = ˆ
H P − ˆ
Q ˆ
H P − ˆ
H P ˆ
Q + ˆ
Q ˆ
H P ˆ
Q .
(9.18)
The |s states entering the ˆ
Q projector in Eq. (9.16) belong to the core, so that
they are well bound. As a consequence, the ˆ
Q projector takes a form of the potential
localized in the nuclear zone when represented in coordinate space. Therefore,
ˆ
P ˆ
H P ˆ
P induces an additional nonlocal potential in ˆ
U basis of Eq. (9.15). With this
modification of ˆ
U basis , all previously derived equations remain the same, as long as
core states |u n j τ are suppressed in all expansions involving Berggren basis states.
Therefore, composite states in the Gamow coupled-channel formalism, consisting of
target many-body states, provided by the Gamow shell model, and of the one-body
state of the projectile, are orthogonal to the core occupied states.
9.1.1.2 Coupled-Channel Equations
As one has defined the channel basis and the matrix elements of the Hamiltonian in
a basis of channel many-body states, one can now determine the A-body scattering
states solutions of coupled-channel equations. Let us define the A-body scattering