Solutions to Exercises
305
Let us now evaluate f | ˆ
H |u i with complex scaling of Hamiltonian, i.e. by
using Eqs. (6.66) and (6.67):
f | ˆ
H |u i = =u
(θ)
f | ˆ
H θ |u
(θ)
i
=
+∞
0
e
iθ/2 u f (r e
iθ ) H (r e
iθ , ∇ r e
−iθ ) e
iθ/2 u i (r e
iθ ) dr
= e
iθ
+∞
0
u f (r e
iθ ) H (r e
iθ , ∇ r e
−iθ ) u i (r e
iθ ) dr , (6.76)
where |u
(θ)
i = ˆ
U θ |u i and |u
(θ)
f = ˆ
U θ |u f . The integrals converge because
u i (r e iθ ) and u f (r e iθ ) exponentially vanish for r → +∞. As f | ˆ
H |u i =
(θ)
f | ˆ
H θ |u
(θ)
i , the application of complex scaling of integral with unbound
resonance states or complex scaling of Hamiltonian with rotated resonance states
are then equivalent. Previous results imply that the e iθ/2 factor in Eq. (6.66)
arises from the Jacobian of the change of variable r → r e iθ in complex
integration.
C. Let us consider a resonant state |Ψ of ˆ
H , so that ˆ
H |Ψ = E |Ψ , with E the
energy of |Ψ . Thus, from Eq. (6.67), one has ˆ
H θ |Ψ θ = E |Ψ θ , with |Ψ θ =
ˆ
U θ |Ψ . Consequently, ˆ
H θ has the same resonant eigenenergies as ˆ
H , which are
the poles of the S-matrix (see Sect. 2.6.6).
As the resonance eigenstates |Ψ θ of ˆ
H θ are integrable on the real r-axis
(see Eq. (6.66)) for a sufficiently large angle θ in [0 : π/4], they belong to the
Hilbert space. Consequently, for θ sufficiently large, they can be obtained by
diagonalizing ˆ
H θ with a complete basis in the Hilbert space, hence with a basis
of bound states.
Exercise IV.
A. One aims at defining a generalization of the one-body ˆ
U θ operator of Eq. (6.66)
in the A-body case. For, let us consider a Slater determinant of the form
|s 1 , . . . , s A , with |s i i ∈ [1 : A], a one-body state. One can then define an
A-body operator ˆ
U θ from the tensor product of A one-body complex scaling
operators of Eq. (6.66):
ˆ
U θ |u 1 , . . . , u A =
A
i=1
U
(i)
θ |u i ,
(6.77)
where |u i is the radial ket state associated with |s i , and U
(i)
θ is the complex
scaling operator associated with the i-th nucleon. The θ angle must be the
same for all nucleon coordinates so that the ˆ
U θ operator of Eq. (6.77) acts
symmetrically on all nucleon states.
305
Let us now evaluate f | ˆ
H |u i with complex scaling of Hamiltonian, i.e. by
using Eqs. (6.66) and (6.67):
f | ˆ
H |u i = =u
(θ)
f | ˆ
H θ |u
(θ)
i
=
+∞
0
e
iθ/2 u f (r e
iθ ) H (r e
iθ , ∇ r e
−iθ ) e
iθ/2 u i (r e
iθ ) dr
= e
iθ
+∞
0
u f (r e
iθ ) H (r e
iθ , ∇ r e
−iθ ) u i (r e
iθ ) dr , (6.76)
where |u
(θ)
i = ˆ
U θ |u i and |u
(θ)
f = ˆ
U θ |u f . The integrals converge because
u i (r e iθ ) and u f (r e iθ ) exponentially vanish for r → +∞. As f | ˆ
H |u i =
(θ)
f | ˆ
H θ |u
(θ)
i , the application of complex scaling of integral with unbound
resonance states or complex scaling of Hamiltonian with rotated resonance states
are then equivalent. Previous results imply that the e iθ/2 factor in Eq. (6.66)
arises from the Jacobian of the change of variable r → r e iθ in complex
integration.
C. Let us consider a resonant state |Ψ of ˆ
H , so that ˆ
H |Ψ = E |Ψ , with E the
energy of |Ψ . Thus, from Eq. (6.67), one has ˆ
H θ |Ψ θ = E |Ψ θ , with |Ψ θ =
ˆ
U θ |Ψ . Consequently, ˆ
H θ has the same resonant eigenenergies as ˆ
H , which are
the poles of the S-matrix (see Sect. 2.6.6).
As the resonance eigenstates |Ψ θ of ˆ
H θ are integrable on the real r-axis
(see Eq. (6.66)) for a sufficiently large angle θ in [0 : π/4], they belong to the
Hilbert space. Consequently, for θ sufficiently large, they can be obtained by
diagonalizing ˆ
H θ with a complete basis in the Hilbert space, hence with a basis
of bound states.
Exercise IV.
A. One aims at defining a generalization of the one-body ˆ
U θ operator of Eq. (6.66)
in the A-body case. For, let us consider a Slater determinant of the form
|s 1 , . . . , s A , with |s i i ∈ [1 : A], a one-body state. One can then define an
A-body operator ˆ
U θ from the tensor product of A one-body complex scaling
operators of Eq. (6.66):
ˆ
U θ |u 1 , . . . , u A =
A
i=1
U
(i)
θ |u i ,
(6.77)
where |u i is the radial ket state associated with |s i , and U
(i)
θ is the complex
scaling operator associated with the i-th nucleon. The θ angle must be the
same for all nucleon coordinates so that the ˆ
U θ operator of Eq. (6.77) acts
symmetrically on all nucleon states.