482
9 The Unification of Structure and Reaction Frameworks
C. ˆ
Q projects a given state on the subspace of core states, by suppressing all
components of the P space. ˆ
P projects a given state out of the Q space. Thus,
ˆ
P projects a given state in the physical valence space of the cluster orbital shell
model, which has no core components.
The Hamiltonian matrix in the cluster orbital shell model representation is
complex symmetric due to the use of the Berggren basis. However, it is necessary
for a Hamiltonian to be real symmetric when represented by real-energy basis
states, so that many-body bound states have real energies and resonance states
have positive widths. Thus, ˆ
Q and ˆ
P have to be Hermitian, and not complex
symmetric, as otherwise projecting Hamiltonians on the P space would lead
to complex energies for bound eigenstates and negative widths for resonance
eigenstates.
Exercise III.
A. Let us assume that the two conditions k
i k max and k
j k max are not verified
∀(i,j) ∈ {a, b, c, d}. Consequently, using the definitions of Sect. 9.2.6, one has
either k
i ≥ k
(H)
max or k
i ≤ k max − k
(H)
max ∀i ∈ {a, b, c, d}.
Let us firstly consider that k
i ≥ k
(H)
max ∀i ∈ {a, b, c, d}. It follows
from Eq. (9.96) that there is a state |i, with i ∈ {a, b, c, d}, with linear
momentum k i k max , i.e. so that k i < k
(H)
max (see Sect. 9.2.6). Therefore,
i i m i |k
i i m i B = 0 and + |k
i i m i B = 0 (see Sect. 9.2.6.2).
Symmetrically, let us consider that k
i ≤ k max − k
(H)
max ∀i
∈ {a, b, c, d}.
Similarly to the previous situation, it follows from Eq. (9.96) that one can find
a state |i, with i ∈ {a, b, c, d}, with linear momentum k i k max , i.e. so that
k i > k max − k
(H)
max (see Sect. 9.2.6). Hence, using the same arguments as before,
|i verifies i i m i |k
i i m i B = 0 and + |k
i i m i B = 0.
As a result, i |k
i i m i B = 0 for a given state i ∈ {a, b, c, d} for both
considered cases. The integrand of Eq. (9.106) is then negligible when: ∃(i,j) ∈
{a, b, c, d} so that k
i k max and k
j k max .
B. Let us expand
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B in plane waves:
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B
=
B
a a m a k
b b m b |k
a k
b
c k
d |k
c c m c k
d d m d B
× ×k
a k
b | ˆ
V res − ˆ
U 0 |k
c k
d dk
a dk
b dk
c dk
d
=
a m a b m b |Ω a Ω b c Ω d | c m c d m d
× ×k
a k
b | ˆ
V res − ˆ
U 0 |k
c k
d dΩ a dΩ b dΩ c dΩ d ,
(9.185)
9 The Unification of Structure and Reaction Frameworks
C. ˆ
Q projects a given state on the subspace of core states, by suppressing all
components of the P space. ˆ
P projects a given state out of the Q space. Thus,
ˆ
P projects a given state in the physical valence space of the cluster orbital shell
model, which has no core components.
The Hamiltonian matrix in the cluster orbital shell model representation is
complex symmetric due to the use of the Berggren basis. However, it is necessary
for a Hamiltonian to be real symmetric when represented by real-energy basis
states, so that many-body bound states have real energies and resonance states
have positive widths. Thus, ˆ
Q and ˆ
P have to be Hermitian, and not complex
symmetric, as otherwise projecting Hamiltonians on the P space would lead
to complex energies for bound eigenstates and negative widths for resonance
eigenstates.
Exercise III.
A. Let us assume that the two conditions k
i k max and k
j k max are not verified
∀(i,j) ∈ {a, b, c, d}. Consequently, using the definitions of Sect. 9.2.6, one has
either k
i ≥ k
(H)
max or k
i ≤ k max − k
(H)
max ∀i ∈ {a, b, c, d}.
Let us firstly consider that k
i ≥ k
(H)
max ∀i ∈ {a, b, c, d}. It follows
from Eq. (9.96) that there is a state |i, with i ∈ {a, b, c, d}, with linear
momentum k i k max , i.e. so that k i < k
(H)
max (see Sect. 9.2.6). Therefore,
i i m i |k
i i m i B = 0 and + |k
i i m i B = 0 (see Sect. 9.2.6.2).
Symmetrically, let us consider that k
i ≤ k max − k
(H)
max ∀i
∈ {a, b, c, d}.
Similarly to the previous situation, it follows from Eq. (9.96) that one can find
a state |i, with i ∈ {a, b, c, d}, with linear momentum k i k max , i.e. so that
k i > k max − k
(H)
max (see Sect. 9.2.6). Hence, using the same arguments as before,
|i verifies i i m i |k
i i m i B = 0 and + |k
i i m i B = 0.
As a result, i |k
i i m i B = 0 for a given state i ∈ {a, b, c, d} for both
considered cases. The integrand of Eq. (9.106) is then negligible when: ∃(i,j) ∈
{a, b, c, d} so that k
i k max and k
j k max .
B. Let us expand
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B in plane waves:
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B
=
B
a a m a k
b b m b |k
a k
b
c k
d |k
c c m c k
d d m d B
× ×k
a k
b | ˆ
V res − ˆ
U 0 |k
c k
d dk
a dk
b dk
c dk
d
=
a m a b m b |Ω a Ω b c Ω d | c m c d m d
× ×k
a k
b | ˆ
V res − ˆ
U 0 |k
c k
d dΩ a dΩ b dΩ c dΩ d ,
(9.185)