40
2 The Discrete Spectrum and the Continuum
vanishes identically (see Eq. (2.107)). N > ν and N ≤ B Λ thus provide with all
other types of states.
Equation (2.114) implies that k lies on the negative imaginary axis if 0 ≤ Δ <
(N + 1/2)
2 and if k is closest to the real axis by using the “+” sign. Hence, an
antibound state occurs therein. There is another antibound state for “−” sign, but it
is usually far from the real axis as
√
Δ cannot partially cancel with N + 1/2.
The last remaining case of complex states of negative energy and positive width
occurs for N > ν, N ≤ B Λ , and Δ < 0 either using the “+” or “−” sign in
Eqs. (2.121) and (2.122). Even though these states do not represent unbound states,
they can have a physical importance and they can influence cross sections if their
linear momentum is sufficiently close to the real k-axis (see Sect. 4.2).
One may notice that all states bearing 2n + + 1 = N are degenerate, as for
the harmonic oscillator potential. This is due to the SU(3) symmetry obeyed by the
Pöschl-Teller-Ginocchio potential. Moreover, one can see in Eqs. (2.117), (2.114),
and (2.115) that Λ =
√
2/(1 − a) is a critical value for resonances. Indeed, on the
one hand, no resonance can exist if Λ ≤
√
2/(1 − a) as B Λ = +∞ in this case. On
the other hand, resonance states exist when Λ >
√
2/(1 − a). This is consistent with
the fact that the Pöschl-Teller-Ginocchio potential can properly describe a nucleus
if Λ is sufficiently large. This also points out to the importance of flat bottom of a
potential for a purpose of generating the resonances.
2.4.4 Modified Pöschl-Teller-Ginocchio Potential
The Pöschl-Teller-Ginocchio potential cannot provide physical values of widths
due to the exponential decrease of its centrifugal part. In the following, one will
discuss a modification of the Pöschl-Teller-Ginocchio potential which yields a
proper centrifugal+Coulomb asymptote. Let us consider the potential:
V PTG–mod (r) = V PTG (r) + E b , 0 < r < r as
(2.123)
V PTG–mod (r) =
v c
r
, r > r as
(2.124)
where the barrier energy E b is fixed and the radius defining the asymptoting region
r as is chosen so that V PTG–mod (r) is continuous in r = r as . In order to determine the
parameters entering V PTG–mod (r), one considers a Woods-Saxon potential with the
centrifugal part included,
+ 1)
r 2
+
v c
r
+ V WS (r)
and one finds its maximal value in the asymptotic zone, which is the barrier energy
E b . Parameters of the Pöschl-Teller-Ginocchio potential V PTG (r) of Eq. (2.123) (see
Eq. (2.88)–(2.92)) are defined so that the Pöschl-Teller-Ginocchio potential closely
resembles the Woods-Saxon potential V WS (r).
2 The Discrete Spectrum and the Continuum
vanishes identically (see Eq. (2.107)). N > ν and N ≤ B Λ thus provide with all
other types of states.
Equation (2.114) implies that k lies on the negative imaginary axis if 0 ≤ Δ <
(N + 1/2)
2 and if k is closest to the real axis by using the “+” sign. Hence, an
antibound state occurs therein. There is another antibound state for “−” sign, but it
is usually far from the real axis as
√
Δ cannot partially cancel with N + 1/2.
The last remaining case of complex states of negative energy and positive width
occurs for N > ν, N ≤ B Λ , and Δ < 0 either using the “+” or “−” sign in
Eqs. (2.121) and (2.122). Even though these states do not represent unbound states,
they can have a physical importance and they can influence cross sections if their
linear momentum is sufficiently close to the real k-axis (see Sect. 4.2).
One may notice that all states bearing 2n + + 1 = N are degenerate, as for
the harmonic oscillator potential. This is due to the SU(3) symmetry obeyed by the
Pöschl-Teller-Ginocchio potential. Moreover, one can see in Eqs. (2.117), (2.114),
and (2.115) that Λ =
√
2/(1 − a) is a critical value for resonances. Indeed, on the
one hand, no resonance can exist if Λ ≤
√
2/(1 − a) as B Λ = +∞ in this case. On
the other hand, resonance states exist when Λ >
√
2/(1 − a). This is consistent with
the fact that the Pöschl-Teller-Ginocchio potential can properly describe a nucleus
if Λ is sufficiently large. This also points out to the importance of flat bottom of a
potential for a purpose of generating the resonances.
2.4.4 Modified Pöschl-Teller-Ginocchio Potential
The Pöschl-Teller-Ginocchio potential cannot provide physical values of widths
due to the exponential decrease of its centrifugal part. In the following, one will
discuss a modification of the Pöschl-Teller-Ginocchio potential which yields a
proper centrifugal+Coulomb asymptote. Let us consider the potential:
V PTG–mod (r) = V PTG (r) + E b , 0 < r < r as
(2.123)
V PTG–mod (r) =
v c
r
, r > r as
(2.124)
where the barrier energy E b is fixed and the radius defining the asymptoting region
r as is chosen so that V PTG–mod (r) is continuous in r = r as . In order to determine the
parameters entering V PTG–mod (r), one considers a Woods-Saxon potential with the
centrifugal part included,
+ 1)
r 2
+
v c
r
+ V WS (r)
and one finds its maximal value in the asymptotic zone, which is the barrier energy
E b . Parameters of the Pöschl-Teller-Ginocchio potential V PTG (r) of Eq. (2.123) (see
Eq. (2.88)–(2.92)) are defined so that the Pöschl-Teller-Ginocchio potential closely
resembles the Woods-Saxon potential V WS (r).