36
3 Efimov’s Universal Three-Body Effect
All the solutions of transcendental Eq. (3.24) are real except the one for which
s 0 ≈ ±1.00624 i, which is purely imaginary. As a result, in the case of resonantly
interacting particles, the channel n = 0 leads to an effective three-body attraction,
given by
V 0 (R) = −
s
2
0
+ 1/4
R 2
(3.26)
This result forms the basis of the Efimov physics.
3.4 Salient Feature of Efimov Effect
Fundamentally important findings of Efimov’s investigations can thus be summarized
as:
Efimov has shown that there exists an effective long-range attractive interaction of
kinetic origin produced by the resonance condition and is independent of the details
of the two-body potentials. Physically, this attractive interaction may be interpreted
as a mediated attraction between the two particles by the exchange of the third one
which is moving back and forth between the two. An important consequence of the
effective three-body potential 1/R
2 is that as long as |a| is sufficiently large compared
to the range r 0 , there is a sequence of three body bound states whose binding energies
form a geometric series lying in the interval between
2
/mr
2
0 and
2
/ma
2
. As |a| is
increased, new bound states appear in the spectrum at critical values of a determined
by a multiplicative factor exp(π/s 0 ), where in the case of identical bosons, s 0 is the
solution of the transcendental equation:
s 0 cosh
π s 0
2
=
8
√
3
sinh
π s 0
6
(3.27)
[Compare this equation with Eq. (3.24), where s 0 is taken as imaginary.] The
numerical value of s 0 is ≈ 1.00624 or exp(π/s 0 ) ≈ 22.7. The number of bound
states in the asymptotic limit, as |a|/r 0 → ∞, is
N →
s 0
π
ln(|a|/r 0 )
(3.28)
Figure 3.1 is a very nice representation of the underlying physics of the Efimov
effect and is guided by similar representations in the original works of Efimov [13].
There are infinitely many three-body bound states with an accumulation point at the
three-body threshold in the limit, and in the resonant limit, the ratio of the binding
energies of the successive bound states approaches a universal number near threshold,
i.e.,
3 Efimov’s Universal Three-Body Effect
All the solutions of transcendental Eq. (3.24) are real except the one for which
s 0 ≈ ±1.00624 i, which is purely imaginary. As a result, in the case of resonantly
interacting particles, the channel n = 0 leads to an effective three-body attraction,
given by
V 0 (R) = −
s
2
0
+ 1/4
R 2
(3.26)
This result forms the basis of the Efimov physics.
3.4 Salient Feature of Efimov Effect
Fundamentally important findings of Efimov’s investigations can thus be summarized
as:
Efimov has shown that there exists an effective long-range attractive interaction of
kinetic origin produced by the resonance condition and is independent of the details
of the two-body potentials. Physically, this attractive interaction may be interpreted
as a mediated attraction between the two particles by the exchange of the third one
which is moving back and forth between the two. An important consequence of the
effective three-body potential 1/R
2 is that as long as |a| is sufficiently large compared
to the range r 0 , there is a sequence of three body bound states whose binding energies
form a geometric series lying in the interval between
2
/mr
2
0 and
2
/ma
2
. As |a| is
increased, new bound states appear in the spectrum at critical values of a determined
by a multiplicative factor exp(π/s 0 ), where in the case of identical bosons, s 0 is the
solution of the transcendental equation:
s 0 cosh
π s 0
2
=
8
√
3
sinh
π s 0
6
(3.27)
[Compare this equation with Eq. (3.24), where s 0 is taken as imaginary.] The
numerical value of s 0 is ≈ 1.00624 or exp(π/s 0 ) ≈ 22.7. The number of bound
states in the asymptotic limit, as |a|/r 0 → ∞, is
N →
s 0
π
ln(|a|/r 0 )
(3.28)
Figure 3.1 is a very nice representation of the underlying physics of the Efimov
effect and is guided by similar representations in the original works of Efimov [13].
There are infinitely many three-body bound states with an accumulation point at the
three-body threshold in the limit, and in the resonant limit, the ratio of the binding
energies of the successive bound states approaches a universal number near threshold,
i.e.,