102
3 Berggren Basis and Completeness Relations
B. Run the Gauss-Legendre integration code calculating complex scaled integrals
for several values of R and k. Typical values are 15 < R < 30 fm and 0.01 <
|k| < 0.1 fm −1 for bound and resonance states.
Check that complex scaling is numerically precise only for θ values around the
middle of an interval ]θ k : θ k + π[.
C. A fundamental application of complex scaling is to calculate the norm of
resonance states (see Eq. (3.57)). Explain why this norm is not equal to one
in general and how it is used to normalize resonance states.
D. One will now numerically calculate the norm of a one-body neutron resonance
eigenstate generated by a Woods-Saxon potential.
For this, one will choose a Woods-Saxon potential depth so that the considered
eigenstate varies from narrow resonance to broad resonance.
Run the one-particle code of radial wave functions in these conditions in order
to obtain the norm of one-body resonance states. Explain the dependence of
the norm of the calculated resonance eigenstate with the depth of the used
Woods-Saxon potential.
3.4
Matrix Elements Involving Scattering States
The calculation of matrix elements involving the scattering states is based on the
(3.57) and (3.58). The one-body matrix element can be written as:
F (k f ) =
R
0
u f (r)V (r)u i (r) dr + A f A i F ++ (k f )
+A f B i F +− (k f ) + B f A i F −+ (k f ) + B f B i F −− (k f ) , (3.63)
where
• u f (r) = A f u
+
f (r) + B f u
−
f (r)
• u i (r) = A i u
+
i (r) + B i u
−
i (r), where k i in F (k f ) is fixed and, in general, u i (r)
can be either bound, resonance, or scattering state
• F s f s i (k f ) =
+∞
0
u
s f
f (R + x) ˆ
O(R + x)u
s i
i (R + x)dx with s f , s i ∈ (+, −) .
This separation is necessary because the presence of incoming and outgoing waves
in the same integral does not allow one to find a unique path in the complex
plane along which the integrand decreases exponentially. Consequently, for each
F s f s i (k f ) one has to consider the domain of the complex plane where it converges,
and then one performs an analytical continuation with the appropriate angle θ s f s i .
Certain integrals cannot be regularized in the above sense. Those include
F +− (k f ) and F −+ (k f ) with u i (r) = u f (r). For ˆ
O(r) = 1, the integrand tends
toward a constant value at +∞, independently of the values of θ +− and θ −+ . This
3 Berggren Basis and Completeness Relations
B. Run the Gauss-Legendre integration code calculating complex scaled integrals
for several values of R and k. Typical values are 15 < R < 30 fm and 0.01 <
|k| < 0.1 fm −1 for bound and resonance states.
Check that complex scaling is numerically precise only for θ values around the
middle of an interval ]θ k : θ k + π[.
C. A fundamental application of complex scaling is to calculate the norm of
resonance states (see Eq. (3.57)). Explain why this norm is not equal to one
in general and how it is used to normalize resonance states.
D. One will now numerically calculate the norm of a one-body neutron resonance
eigenstate generated by a Woods-Saxon potential.
For this, one will choose a Woods-Saxon potential depth so that the considered
eigenstate varies from narrow resonance to broad resonance.
Run the one-particle code of radial wave functions in these conditions in order
to obtain the norm of one-body resonance states. Explain the dependence of
the norm of the calculated resonance eigenstate with the depth of the used
Woods-Saxon potential.
3.4
Matrix Elements Involving Scattering States
The calculation of matrix elements involving the scattering states is based on the
(3.57) and (3.58). The one-body matrix element can be written as:
F (k f ) =
R
0
u f (r)V (r)u i (r) dr + A f A i F ++ (k f )
+A f B i F +− (k f ) + B f A i F −+ (k f ) + B f B i F −− (k f ) , (3.63)
where
• u f (r) = A f u
+
f (r) + B f u
−
f (r)
• u i (r) = A i u
+
i (r) + B i u
−
i (r), where k i in F (k f ) is fixed and, in general, u i (r)
can be either bound, resonance, or scattering state
• F s f s i (k f ) =
+∞
0
u
s f
f (R + x) ˆ
O(R + x)u
s i
i (R + x)dx with s f , s i ∈ (+, −) .
This separation is necessary because the presence of incoming and outgoing waves
in the same integral does not allow one to find a unique path in the complex
plane along which the integrand decreases exponentially. Consequently, for each
F s f s i (k f ) one has to consider the domain of the complex plane where it converges,
and then one performs an analytical continuation with the appropriate angle θ s f s i .
Certain integrals cannot be regularized in the above sense. Those include
F +− (k f ) and F −+ (k f ) with u i (r) = u f (r). For ˆ
O(r) = 1, the integrand tends
toward a constant value at +∞, independently of the values of θ +− and θ −+ . This