3.5 Predictions on the Properties of Three-Nucleon System
39
In the Sect. 6 on halo nuclei, we encountered precisely such a situation where we
found two Efimov states in the case of halo nucleus
20 C, and while studying n −
19 C
scattering, we found these Efimov states moved over to produce a resonance just
near threshold. We shall discuss these details in Chap. 6.
3.5.2 Appearance of Efimov Effect in Three-Body Separable
Potential Approach
To conclude this study, it would be interesting to record how the effect of long-range
three-body potential of the type 1/R
2 as originally found by Efimov appears in the
three-body problem with separable potentials. It should be noted that to analyze the
Efimov effect in the limit when two-body scattering length is large as compared to
the short-range potential (corresponding to large values of β in the present case), the
kernel of the three-body Schrodinger equation is basically governed by the function
H(p), defined by Eq. (2.73), where
H ( p) = [λ
−1
− h( p)]
−1
=
π
2
β(β + α) 2 −
π
2
β[β +
α 2 − 3k 2 /4 + 3 p 2 /4] 2
−1
So as long as the two-body potential is sufficiently strong to produce a three-body
bound state with k
2
< 0, H(p) will be a regular function of p for real p. However,
as k
2
→ 0, three-body binding energy approaches that of the two-particle system
represented by the parameter α
2 . In such a case,
H ( p) =
π
2
β(β + α) 2 −
π
2
β[β +
α 2 + 3 p 2 /4] 2
−1
⇒ p→0
4αβ(α + β)
3
3π 2 p 2
Thus, H(p) becomes singular near p = 0. Further, in the limiting case, where in
addition α = 0, corresponding to infinite scattering length, we find
H ( p) =
π
2
β 3 −
π
2
β[β +
√
3 p/2] 2
−1
⇒ p→0
β
4
√
3π 2 p
.
This 1/p singular behavior in momentum space corresponds to the singularity of
the type 1/R
2 in configuration space leading to the Efimov effect [31]. To see that
for α = 0, the scattering length does become infinite, we have
H ( p) = λ[1 − λh( p)]
−1
⇒ p→0 λ[1 − λ
π
2
β 3 ]
−1
39
In the Sect. 6 on halo nuclei, we encountered precisely such a situation where we
found two Efimov states in the case of halo nucleus
20 C, and while studying n −
19 C
scattering, we found these Efimov states moved over to produce a resonance just
near threshold. We shall discuss these details in Chap. 6.
3.5.2 Appearance of Efimov Effect in Three-Body Separable
Potential Approach
To conclude this study, it would be interesting to record how the effect of long-range
three-body potential of the type 1/R
2 as originally found by Efimov appears in the
three-body problem with separable potentials. It should be noted that to analyze the
Efimov effect in the limit when two-body scattering length is large as compared to
the short-range potential (corresponding to large values of β in the present case), the
kernel of the three-body Schrodinger equation is basically governed by the function
H(p), defined by Eq. (2.73), where
H ( p) = [λ
−1
− h( p)]
−1
=
π
2
β(β + α) 2 −
π
2
β[β +
α 2 − 3k 2 /4 + 3 p 2 /4] 2
−1
So as long as the two-body potential is sufficiently strong to produce a three-body
bound state with k
2
< 0, H(p) will be a regular function of p for real p. However,
as k
2
→ 0, three-body binding energy approaches that of the two-particle system
represented by the parameter α
2 . In such a case,
H ( p) =
π
2
β(β + α) 2 −
π
2
β[β +
α 2 + 3 p 2 /4] 2
−1
⇒ p→0
4αβ(α + β)
3
3π 2 p 2
Thus, H(p) becomes singular near p = 0. Further, in the limiting case, where in
addition α = 0, corresponding to infinite scattering length, we find
H ( p) =
π
2
β 3 −
π
2
β[β +
√
3 p/2] 2
−1
⇒ p→0
β
4
√
3π 2 p
.
This 1/p singular behavior in momentum space corresponds to the singularity of
the type 1/R
2 in configuration space leading to the Efimov effect [31]. To see that
for α = 0, the scattering length does become infinite, we have
H ( p) = λ[1 − λh( p)]
−1
⇒ p→0 λ[1 − λ
π
2
β 3 ]
−1