140
3 Berggren Basis and Completeness Relations
of validity of a complex-valued function. As a consequence, no θ dependence
can appear as the analytic continuation of a complex function is unique.
Exercise VIII.
A. The Gauss-Legendre quadrature can be applied only in finite intervals. Thus,
it can be used if x ∈ [0 : R −1/4 ], but not if x ≥ 0. In order to show that
the proposed change of variable is efficient, let us consider x ∈ [0 : R −1/4 ].
|k|x −4 quickly increases when x → 0 + , so that the integral converges rapidly
even if k is small. Indeed, if sin(θ − θ k ) = 0.1, |k| = 10 −5 fm −1 , x = 0.01,
exp(−|k|x −4 sin(θ − θ k )) x −5 = 10 10 exp(−100) < 10 −33 . Consequently, the
integrand goes quickly to zero when x → 0 in all situations of practical interest.
Thus, the proposed change of variable to calculate integrals initially defined on
[0 : +∞) is efficient.
B. Even though theoretically the integral converges for θ ∈]θ k : θ k + π[,
nevertheless it converges very slowly for θ ∼ θ k or θ ∼ θ k +π, so that the GaussLegendre integration cannot be precise therein. Conversely, exp(−|k|x −4 sin(θ −
θ k )) is smallest for θ in the middle of ]θ k : θ k + π[, so that numerical integration
is precise therein.
C. When one numerically calculates a resonance state, it is firstly necessary to fix
its integration constant arbitrarily, as the norm of the resonance eigenstate is
unknown. For example, one typically demands that u(k, r) ∼ r +1 if r → 0
(see Eq. (2.6)). Therefore, the norm of u(k, r), calculated with complex scaling,
will not be equal to one in general. u(k, r) will be normalized to one after it is
divided by the square root of the norm calculated with complex scaling.
D. Resonances states have a small width when their energy above the lowest decay
threshold is close to zero. The particle-emission width quickly increases along
with energy. As a consequence, the norm of resonance states has a very small
imaginary part at small energies, which increases with increasing width of the
Woods-Saxon unbound eigenstate.
Exercise IX.
(i) The 2p 3/2 single-particle resonance (E = 3.287 MeV, Γ =931 keV) in
a Woods-Saxon potential of the depth V 0 = 65 MeV is expanded in the
basis generated by the Woods-Saxon potential of the depth V
(B)
0
= 70 MeV,
where the 2p 3/2 single-particle resonance has an energy E = 1.905 MeV and
width Γ =61.89 keV. After diagonalization in the discretized basis (3.87),
one obtains E = 3.287 MeV and Γ =931 keV for the 2p 3/2 single-particle
resonance, that is, the discretization error is negligible.
As both states are resonant, the squared amplitude of the 2p 3/2 basis state is
close to one. Nevertheless, the contribution from the nonresonant continuum is
essential. It is due to the fact that the resonant state in the basis is very narrow,
whereas the expanded resonant state is fairly broad. It is interesting to notice
3 Berggren Basis and Completeness Relations
of validity of a complex-valued function. As a consequence, no θ dependence
can appear as the analytic continuation of a complex function is unique.
Exercise VIII.
A. The Gauss-Legendre quadrature can be applied only in finite intervals. Thus,
it can be used if x ∈ [0 : R −1/4 ], but not if x ≥ 0. In order to show that
the proposed change of variable is efficient, let us consider x ∈ [0 : R −1/4 ].
|k|x −4 quickly increases when x → 0 + , so that the integral converges rapidly
even if k is small. Indeed, if sin(θ − θ k ) = 0.1, |k| = 10 −5 fm −1 , x = 0.01,
exp(−|k|x −4 sin(θ − θ k )) x −5 = 10 10 exp(−100) < 10 −33 . Consequently, the
integrand goes quickly to zero when x → 0 in all situations of practical interest.
Thus, the proposed change of variable to calculate integrals initially defined on
[0 : +∞) is efficient.
B. Even though theoretically the integral converges for θ ∈]θ k : θ k + π[,
nevertheless it converges very slowly for θ ∼ θ k or θ ∼ θ k +π, so that the GaussLegendre integration cannot be precise therein. Conversely, exp(−|k|x −4 sin(θ −
θ k )) is smallest for θ in the middle of ]θ k : θ k + π[, so that numerical integration
is precise therein.
C. When one numerically calculates a resonance state, it is firstly necessary to fix
its integration constant arbitrarily, as the norm of the resonance eigenstate is
unknown. For example, one typically demands that u(k, r) ∼ r +1 if r → 0
(see Eq. (2.6)). Therefore, the norm of u(k, r), calculated with complex scaling,
will not be equal to one in general. u(k, r) will be normalized to one after it is
divided by the square root of the norm calculated with complex scaling.
D. Resonances states have a small width when their energy above the lowest decay
threshold is close to zero. The particle-emission width quickly increases along
with energy. As a consequence, the norm of resonance states has a very small
imaginary part at small energies, which increases with increasing width of the
Woods-Saxon unbound eigenstate.
Exercise IX.
(i) The 2p 3/2 single-particle resonance (E = 3.287 MeV, Γ =931 keV) in
a Woods-Saxon potential of the depth V 0 = 65 MeV is expanded in the
basis generated by the Woods-Saxon potential of the depth V
(B)
0
= 70 MeV,
where the 2p 3/2 single-particle resonance has an energy E = 1.905 MeV and
width Γ =61.89 keV. After diagonalization in the discretized basis (3.87),
one obtains E = 3.287 MeV and Γ =931 keV for the 2p 3/2 single-particle
resonance, that is, the discretization error is negligible.
As both states are resonant, the squared amplitude of the 2p 3/2 basis state is
close to one. Nevertheless, the contribution from the nonresonant continuum is
essential. It is due to the fact that the resonant state in the basis is very narrow,
whereas the expanded resonant state is fairly broad. It is interesting to notice