74
2 The Discrete Spectrum and the Continuum
B. Phase shifts vary smoothly using V PTG (r), due to the absence of centrifugal
or Coulomb barrier. This implies that no narrow resonance can exist, as
narrow resonances always generate important variations of phase shifts near the
resonance energy. Conversely, the numerical examples considered present a narrow resonance in calculations using V PTG–mod (r) and Woods-Saxon potentials,
which is visible on phase shifts by their rapid change therein.
C. The second-order approximation is, in principle, valid when |δk| | |k| in
Eq. (2.127), that is, when one considers narrow resonances. Indeed, numerical
calculations show that widths issued from the second-order approximation
whose value is slightly smaller or close to 100 keV are almost exact, whereas
their precision deteriorates after 1 MeV. However, very small widths are not
well reproduced in absolute value, as the width provided by the second-order
approximation is typically too large. Indeed, even if the width arising from the
second-order approximation is small, it varies by several orders of magnitude
when Γ E. The current formula of Sect. 2.6.8 is, in fact, needed in this case,
as it is very precise when widths are much smaller than 1 keV.
D. As the Woods-Saxon and modified Pöschl-Teller-Ginocchio potentials are not
exactly equal, their phase shifts and resonance energies slightly differ. The quick
change of phase shifts is directly related to the small value of the width of the
narrow resonance.
Exercise XI. To show this property, let us multiply the Schrödinger equation (2.2)
by u ∗ (k, r) and integrate from 0 to +∞:
+∞
0
u
(k, r)u
∗ (k, r) dr =
+∞
0
+ 1)
r 2
+ v l (r) − k
2
u(k, r)u
∗ (k, r) dr
⇒ k
2
+∞
0
|u(k, r)|
2 dr =
+∞
0
|u
(k, r)|
2 dr
+
+∞
0
+ 1)
r 2
+ v l (r)
|u(k, r)|
2 dr . (2.218)
In order to obtain Eq. (2.218), one effected an integration by parts:
+∞
0
u
(k, r)u
∗ (k, r) dr =
u
(k, r)u
∗ (k, r)
r=+∞
r=0
−
+∞
0
|u
(k, r)|
2 dr .
(2.219)
The first term on the right-hand-side vanishes because u(k, 0) = 0 and u(k, r) → 0
for r → +∞. It is then clear from Eq. (2.218) that k 2 has to be real if u(k, r) is a
bound state.
2 The Discrete Spectrum and the Continuum
B. Phase shifts vary smoothly using V PTG (r), due to the absence of centrifugal
or Coulomb barrier. This implies that no narrow resonance can exist, as
narrow resonances always generate important variations of phase shifts near the
resonance energy. Conversely, the numerical examples considered present a narrow resonance in calculations using V PTG–mod (r) and Woods-Saxon potentials,
which is visible on phase shifts by their rapid change therein.
C. The second-order approximation is, in principle, valid when |δk| | |k| in
Eq. (2.127), that is, when one considers narrow resonances. Indeed, numerical
calculations show that widths issued from the second-order approximation
whose value is slightly smaller or close to 100 keV are almost exact, whereas
their precision deteriorates after 1 MeV. However, very small widths are not
well reproduced in absolute value, as the width provided by the second-order
approximation is typically too large. Indeed, even if the width arising from the
second-order approximation is small, it varies by several orders of magnitude
when Γ E. The current formula of Sect. 2.6.8 is, in fact, needed in this case,
as it is very precise when widths are much smaller than 1 keV.
D. As the Woods-Saxon and modified Pöschl-Teller-Ginocchio potentials are not
exactly equal, their phase shifts and resonance energies slightly differ. The quick
change of phase shifts is directly related to the small value of the width of the
narrow resonance.
Exercise XI. To show this property, let us multiply the Schrödinger equation (2.2)
by u ∗ (k, r) and integrate from 0 to +∞:
+∞
0
u
(k, r)u
∗ (k, r) dr =
+∞
0
+ 1)
r 2
+ v l (r) − k
2
u(k, r)u
∗ (k, r) dr
⇒ k
2
+∞
0
|u(k, r)|
2 dr =
+∞
0
|u
(k, r)|
2 dr
+
+∞
0
+ 1)
r 2
+ v l (r)
|u(k, r)|
2 dr . (2.218)
In order to obtain Eq. (2.218), one effected an integration by parts:
+∞
0
u
(k, r)u
∗ (k, r) dr =
u
(k, r)u
∗ (k, r)
r=+∞
r=0
−
+∞
0
|u
(k, r)|
2 dr .
(2.219)
The first term on the right-hand-side vanishes because u(k, 0) = 0 and u(k, r) → 0
for r → +∞. It is then clear from Eq. (2.218) that k 2 has to be real if u(k, r) is a
bound state.