306
6 Physical Applications of the Gamow Shell Model
This implies that the complex scaled many-body Hamiltonian ˆ
U θ H ˆ
U
−1
θ is
also symmetric with respect to the coordinates of the A nucleons involved.
Therefore, the many-body operator ˆ
H θ has the same expression as in Eq. (6.67,
provided that the complex scaling operator of Eq. (6.77) is used.
Similarly to the one-body case (see Eq. (6.68)), the used interaction in ˆ
H θ
must be dilatation analytic with respect to all rotated nucleon coordinates r i e iθ .
B. The ˆ
H θ operator in the non-interacting case is evidently the sum of A one-body
Hamiltonians of Eq. (6.68):
ˆ
H θ =
A
i=1
e
−2iθ p 2
i
2m i
+ ˆ
U i (r i e
iθ )
.
(6.78)
The eigenstates of ˆ
H θ are clearly the Slater determinants |s 1 , . . . , s A generated
by the eigenstates of the one-body Hamiltonians of the sum of Eq. (6.78).
One has |s 1 , . . . , s A = |s
(res)
1
, . . . , s
(res)
a
, s
(scat)
a+1 , . . . , s
(scat)
A
, with a =
0, 1, 2, . . . , A, where the resonant or scattering character of occupied states
is explicitly written. Let us denote the eigenenergy of |s 1 , . . . , s A by E. From
the discussion following Eq. (6.68), one deduces that
E = E res (a) + e
−2iθ E scat (A − a) ,
where E res (a) is the sum of the resonant energies of |s
(res)
1
, . . . , |s
(res)
a
and
E scat (A − a) ≥ 0.
As the one-body states generated by a complex-scaled one-body Hamiltonian
build a completeness relation of Newton type, the Slater determinants generated
by ˆ
H θ of Eq. (6.78) form a many-body complete set of states in the Hilbert space.
C. The eigenenergies of many-body correlated resonant eigenstate are equal to
E res (A) by definition. The θ -independence of the many-body resonant eigenenergies of ˆ
H θ can be demonstrated as in C of Exercise III, because the used
argument in the one-body case clearly holds in the many-body case.
Let us now turn to the scattering eigenstates of the Hamiltonian ˆ
H θ where
nucleons interact. For this, one firstly constructs an approximate Hamiltonian
ˆ
H
(app)
θ
from ˆ
H θ , by suppressing many-body interactions in ˆ
H θ so that A − a
nucleons at least do not interact, with a = 0, 1, 2, . . . , A − 1. Consequently,
one can generate scattering eigenstates of ˆ
H
(app)
θ
of the form |Ψ (app) =
|Ψ
(a)
res ⊗ |Φ
(A−a)
scat , where |Ψ
(a)
res is a many-body resonant state of a nucleons
and |Φ
(A−a)
scat is a Slater determinant of A − a nucleons, where only one-body
scattering states are occupied. Thus, similarly to the non-interacting case, the
eigenenergy of |Ψ
(a)
res ⊗ |Φ
(A−a)
scat is equal to E = E res (a) + e −2iθ E scat (A − a),
where E res (a) is the eigenenergy of |Ψ
(a)
res and E scat (A − a) ≥ 0.
One can now study the general case of correlated scattering eigenstates of
ˆ
H θ . For this, one will suppose that there exists a scattering eigenstate |Ψ of ˆ
H θ
of energy E so that E = E res (a) + e −2iθ E scat (A − a) ∀a ∈ 0, 1, 2, . . . , A − 1.
6 Physical Applications of the Gamow Shell Model
This implies that the complex scaled many-body Hamiltonian ˆ
U θ H ˆ
U
−1
θ is
also symmetric with respect to the coordinates of the A nucleons involved.
Therefore, the many-body operator ˆ
H θ has the same expression as in Eq. (6.67,
provided that the complex scaling operator of Eq. (6.77) is used.
Similarly to the one-body case (see Eq. (6.68)), the used interaction in ˆ
H θ
must be dilatation analytic with respect to all rotated nucleon coordinates r i e iθ .
B. The ˆ
H θ operator in the non-interacting case is evidently the sum of A one-body
Hamiltonians of Eq. (6.68):
ˆ
H θ =
A
i=1
e
−2iθ p 2
i
2m i
+ ˆ
U i (r i e
iθ )
.
(6.78)
The eigenstates of ˆ
H θ are clearly the Slater determinants |s 1 , . . . , s A generated
by the eigenstates of the one-body Hamiltonians of the sum of Eq. (6.78).
One has |s 1 , . . . , s A = |s
(res)
1
, . . . , s
(res)
a
, s
(scat)
a+1 , . . . , s
(scat)
A
, with a =
0, 1, 2, . . . , A, where the resonant or scattering character of occupied states
is explicitly written. Let us denote the eigenenergy of |s 1 , . . . , s A by E. From
the discussion following Eq. (6.68), one deduces that
E = E res (a) + e
−2iθ E scat (A − a) ,
where E res (a) is the sum of the resonant energies of |s
(res)
1
, . . . , |s
(res)
a
and
E scat (A − a) ≥ 0.
As the one-body states generated by a complex-scaled one-body Hamiltonian
build a completeness relation of Newton type, the Slater determinants generated
by ˆ
H θ of Eq. (6.78) form a many-body complete set of states in the Hilbert space.
C. The eigenenergies of many-body correlated resonant eigenstate are equal to
E res (A) by definition. The θ -independence of the many-body resonant eigenenergies of ˆ
H θ can be demonstrated as in C of Exercise III, because the used
argument in the one-body case clearly holds in the many-body case.
Let us now turn to the scattering eigenstates of the Hamiltonian ˆ
H θ where
nucleons interact. For this, one firstly constructs an approximate Hamiltonian
ˆ
H
(app)
θ
from ˆ
H θ , by suppressing many-body interactions in ˆ
H θ so that A − a
nucleons at least do not interact, with a = 0, 1, 2, . . . , A − 1. Consequently,
one can generate scattering eigenstates of ˆ
H
(app)
θ
of the form |Ψ (app) =
|Ψ
(a)
res ⊗ |Φ
(A−a)
scat , where |Ψ
(a)
res is a many-body resonant state of a nucleons
and |Φ
(A−a)
scat is a Slater determinant of A − a nucleons, where only one-body
scattering states are occupied. Thus, similarly to the non-interacting case, the
eigenenergy of |Ψ
(a)
res ⊗ |Φ
(A−a)
scat is equal to E = E res (a) + e −2iθ E scat (A − a),
where E res (a) is the eigenenergy of |Ψ
(a)
res and E scat (A − a) ≥ 0.
One can now study the general case of correlated scattering eigenstates of
ˆ
H θ . For this, one will suppose that there exists a scattering eigenstate |Ψ of ˆ
H θ
of energy E so that E = E res (a) + e −2iθ E scat (A − a) ∀a ∈ 0, 1, 2, . . . , A − 1.