294
6 Physical Applications of the Gamow Shell Model
resonant spectrum built from bound and resonance states, whose wave functions are
integrable on the real r-axis and eigenenergies are independent of θ (see Exercise III
and Ref. [139]). Obviously, the scattering spectrum of ˆ
H θ is made of scattering
eigenstates of energy e −2iθ E, with E > 0.
Exercise III
One will demonstrate the basic properties of ˆ
U θ and ˆ
H θ (see Eqs. (6.66) and 6.67))
in the one-body case.
A. Show that ˆ
U
−1
θ
= ˆ
U
†
θ = ˆ
U −θ (see Eq. (6.66)). One recalls that complex
conjugation does not enter adjoint definition in our case. Demonstrate that the
application of ˆ
U θ operators of Eq. (6.66) in Eq. (6.67) is equivalent to have
r → r e iθ and ∇ r → ∇ r e −iθ in the matrix elements defining ˆ
H .
B. Let us consider the matrix element of ˆ
H between two initial and final
resonance states, denoted as |u i and |u f , respectively.
Write f | ˆ
H |u i as a coordinate space integral in the complex plane using
complex scaling of integral (see Eq. (3.56)) and ˆ
H (see Eq. (6.67)), respectively.
We will assume that one can integrate in the complex plane along the complex
path defined by z = r e iθ . Deduce that both methods used to calculate
f | ˆ
H |u i are equivalent. Explain the presence of e iθ/2 in Eq. (6.66).
C. Explain why ˆ
H and ˆ
H θ have the same pole spectrum. Show how to determine
the resonant energies of ˆ
H by diagonalization of ˆ
H θ in a basis of bound states
(see Eq. (6.67)).
The generalization of Hamiltonian complex scaling method to the many-body
case is formulated in Exercise IV. The complex-scaled many-body Hamiltonian,
consisting of a one-body part U and local two-body interaction V , can then be
derived, and it takes the form similar to the one-body complex-scaled Hamiltonian
(see Exercise IV):
ˆ
H θ =
A
i=1
e
−2iθ p 2
i
2m i
+ ˆ
U i (r i e
iθ )
+
A
i
V ij (r i e
iθ , r j e
iθ , ∇ r i e
−iθ , ∇ r j e
−iθ ) ,
(6.69)
where m i is the mass of the i-th nucleon and angular dependence in V ij is not
explicitly written as it is the same as in the absence of complex scaling.
An illustration of the position of eigenenergies, studied in Exercise IV, is depicted
in the three-body case in Fig. 6.20. Similarly to the one-body case, the fact that
the used interaction functionals must be dilation analytic is the main theoretical
restriction for the use of many-body Hamiltonian complex scaling. Note that the
two-body interaction arising from a harmonic oscillator basis expansion, studied in
Sect. 5.6, is dilation analytic.
6 Physical Applications of the Gamow Shell Model
resonant spectrum built from bound and resonance states, whose wave functions are
integrable on the real r-axis and eigenenergies are independent of θ (see Exercise III
and Ref. [139]). Obviously, the scattering spectrum of ˆ
H θ is made of scattering
eigenstates of energy e −2iθ E, with E > 0.
Exercise III
One will demonstrate the basic properties of ˆ
U θ and ˆ
H θ (see Eqs. (6.66) and 6.67))
in the one-body case.
A. Show that ˆ
U
−1
θ
= ˆ
U
†
θ = ˆ
U −θ (see Eq. (6.66)). One recalls that complex
conjugation does not enter adjoint definition in our case. Demonstrate that the
application of ˆ
U θ operators of Eq. (6.66) in Eq. (6.67) is equivalent to have
r → r e iθ and ∇ r → ∇ r e −iθ in the matrix elements defining ˆ
H .
B. Let us consider the matrix element of ˆ
H between two initial and final
resonance states, denoted as |u i and |u f , respectively.
Write f | ˆ
H |u i as a coordinate space integral in the complex plane using
complex scaling of integral (see Eq. (3.56)) and ˆ
H (see Eq. (6.67)), respectively.
We will assume that one can integrate in the complex plane along the complex
path defined by z = r e iθ . Deduce that both methods used to calculate
f | ˆ
H |u i are equivalent. Explain the presence of e iθ/2 in Eq. (6.66).
C. Explain why ˆ
H and ˆ
H θ have the same pole spectrum. Show how to determine
the resonant energies of ˆ
H by diagonalization of ˆ
H θ in a basis of bound states
(see Eq. (6.67)).
The generalization of Hamiltonian complex scaling method to the many-body
case is formulated in Exercise IV. The complex-scaled many-body Hamiltonian,
consisting of a one-body part U and local two-body interaction V , can then be
derived, and it takes the form similar to the one-body complex-scaled Hamiltonian
(see Exercise IV):
ˆ
H θ =
A
i=1
e
−2iθ p 2
i
2m i
+ ˆ
U i (r i e
iθ )
+
A
i
V ij (r i e
iθ , r j e
iθ , ∇ r i e
−iθ , ∇ r j e
−iθ ) ,
(6.69)
where m i is the mass of the i-th nucleon and angular dependence in V ij is not
explicitly written as it is the same as in the absence of complex scaling.
An illustration of the position of eigenenergies, studied in Exercise IV, is depicted
in the three-body case in Fig. 6.20. Similarly to the one-body case, the fact that
the used interaction functionals must be dilation analytic is the main theoretical
restriction for the use of many-body Hamiltonian complex scaling. Note that the
two-body interaction arising from a harmonic oscillator basis expansion, studied in
Sect. 5.6, is dilation analytic.