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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Efimov's Effect: A New Pathology of Three-Particle Systems. II

R. D. Amado, Julian V. Noble

Published Apr 15, 1972• 134 citations
Publisher

Abstract

By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length $a$ becomes large. [The number of bound states is then roughly $(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)$, where $\ensuremath{\Lambda}$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.

DOI
10.1103/physrevd.5.1992

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By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length aaa becomes large. [The number of bound states is then roughly (1\ensuremathπ)ln(\ensuremathΛ∣a∣)(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)(\ensuremathπ1​)ln(\ensuremathΛ∣a∣), where \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.
Introducing a suitable cutoff parameter r0{r}_{0}r0​ for the effective potential in the three-body subsystem, we show that the number of four-body bound states is roughly $k\mathrm{ln}[\ensuremath{\epsilon}a({r}_{0})]$, where $\ensuremath{\epsilon}$ is a momentum cutoff and $\frac{2}{3\ensuremath{\pi}}\ensuremath{\le}k\ensuremath{\le}\frac{1}{3}$, if the scattering length a(r0)a({r}_{0})a(r0​) for a two-body bound state and a third particle tends to infinity. In a numerical calculation we found that the value ${r}_{0}\ensuremath{\approx}1.82$ fm, which provides a sufficiently resonant effective interaction in the three-body subsystem, is independent of the applied two-body potentials within the numerical accuracy.NUCLEAR REACTIONS Efimov effect.
The possibility of a zero-energy Efimov effect is investigated in a model consisting of three identical heavy particles and a lighter one, when the light-heavy interaction leads to a zero-energy bound state for the two-heavy---one-light subsystem. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable sss-wave potential of Yamaguchi form. The heavy-heavy interaction is short range and, if attractive, is to be taken weak enough to support no two- or three-heavy-particle bound states. The relevant parameter is the potential strength $\ensuremath{\lambda}$ of the light-heavy interaction which for $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ supports a single zero-energy light-heavy bound state. If the first zero-energy bound state of the two-heavy---one-light subsystem occurs for $\ensuremath{\lambda}={\ensuremath{\lambda}}^{\ensuremath{'}}$ such that $|{\ensuremath{\lambda}}^{\ensuremath{'}}|<|{\ensuremath{\lambda}}_{c}|$, there are no Efimov four-body bound states. On the contrary, if the chosen heavy-heavy potential is repulsive enough to prevent the existence of three-body bound states for $|\ensuremath{\lambda}|<|{\ensuremath{\lambda}}_{c}|$, then as $\ensuremath{\lambda}\ensuremath{\rightarrow}{\ensuremath{\lambda}}_{c}$ a few four-body Efimov states may emerge but their number remains finite. These states disappear for $|\ensuremath{\lambda}|\ensuremath{\gtrsim}|{\ensuremath{\lambda}}_{c}|$ as the three-body cut overrides them.
Two particles that are just shy of binding may develop an infinite number of shallow bound states when a third particle is added. This counterintuitive effect was first predicted by Efimov for identical bosons interacting with a short-range pairwise potential. The Efimov effect persists for nonidentical particles if at least two of the three bonds are almost bound. The Efimov effect has recently been verified experimentally using ultracold atoms. We explain the origin of this effect using elementary quantum mechanics and summarize the experimental evidence for it.
Abstract We consider a system of three identical bosons in R3\mathbb {R}^3R3 R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius a>0 a > 0a>0 a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues EnE_nEn​ E n accumulating at zero and fulfilling the asymptotic geometrical law   En+1/En  →  e−2πs0    for   n→+∞\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty En+1​/En​→e−s0​2π​forn→+∞ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where s0≈1.00624s_0\approx 1.00624s0​≈1.00624 s 0 ≈ 1.00624 .