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Efimov effect in the Born-Oppenheimer picture

Hamidreza Saberbaghi

TL;DR

This work analyzes Efimov physics for a mass-imbalanced 2+1 system within the Born–Oppenheimer framework, modeling heavy-light resonant interactions as two-center point interactions and treating exchange symmetry and antisymmetry. The authors rigorously derive an effective potential for the light particle and obtain an effective Hamiltonian for heavy motion, proving that in the unitary limit the three-body spectrum depends only on the mass ratio and statistics, with no three-body parameter, and establishing conditions that exclude non-Efimov states. Away from unitarity they refine the understanding of near-threshold states, showing the weakest bound state has size near 2.8 times the two-body scattering length, and they sharpen Bargmann bounds for the number of bound states. The results highlight the subtle interplay between short-range details and universal Efimov scaling in a zero-range, two-center setting, and they address the limitations and validity of the Born–Oppenheimer approximation in such few-body problems.

Abstract

In this work, we study the Efimov effect in a mass-imbalanced system consisting of two heavy particles and one light particle within the Born-Oppenheimer approximation. The result obtained in R. Figari, H. Saberbaghi, and A. Teta, J. Phys. A: Math. Theor. 57(5), 2024, for zero angular momentum is recovered here as a special case, and the analysis is extended to all angular momenta. In this setting, the resonant heavy-light interactions are modeled as point interactions, under the assumption of either exchange symmetry or antisymmetry with respect to the positions of the delta centers. Within this framework, we prove that in the unitary limit the Efimov spectrum depends solely on the mass ratio and particle statistics; that is, the so-called three-body parameter is absent. We also provide a condition ensuring that the spectrum contains no nonEfimov eigenvalues. Furthermore, we study the system away from the unitary limit and show that the spatial size of the weakest bound state near the threshold is approximately 2.8 times the two-body scattering length, in contrast to the common assumption that these two scales coincide. Finally, we compute Bargmann bound for the number of bound states and obtain a sharper estimate than the results in the literature.

Efimov effect in the Born-Oppenheimer picture

TL;DR

This work analyzes Efimov physics for a mass-imbalanced 2+1 system within the Born–Oppenheimer framework, modeling heavy-light resonant interactions as two-center point interactions and treating exchange symmetry and antisymmetry. The authors rigorously derive an effective potential for the light particle and obtain an effective Hamiltonian for heavy motion, proving that in the unitary limit the three-body spectrum depends only on the mass ratio and statistics, with no three-body parameter, and establishing conditions that exclude non-Efimov states. Away from unitarity they refine the understanding of near-threshold states, showing the weakest bound state has size near 2.8 times the two-body scattering length, and they sharpen Bargmann bounds for the number of bound states. The results highlight the subtle interplay between short-range details and universal Efimov scaling in a zero-range, two-center setting, and they address the limitations and validity of the Born–Oppenheimer approximation in such few-body problems.

Abstract

In this work, we study the Efimov effect in a mass-imbalanced system consisting of two heavy particles and one light particle within the Born-Oppenheimer approximation. The result obtained in R. Figari, H. Saberbaghi, and A. Teta, J. Phys. A: Math. Theor. 57(5), 2024, for zero angular momentum is recovered here as a special case, and the analysis is extended to all angular momenta. In this setting, the resonant heavy-light interactions are modeled as point interactions, under the assumption of either exchange symmetry or antisymmetry with respect to the positions of the delta centers. Within this framework, we prove that in the unitary limit the Efimov spectrum depends solely on the mass ratio and particle statistics; that is, the so-called three-body parameter is absent. We also provide a condition ensuring that the spectrum contains no nonEfimov eigenvalues. Furthermore, we study the system away from the unitary limit and show that the spatial size of the weakest bound state near the threshold is approximately 2.8 times the two-body scattering length, in contrast to the common assumption that these two scales coincide. Finally, we compute Bargmann bound for the number of bound states and obtain a sharper estimate than the results in the literature.

Paper Structure

This paper contains 11 sections, 11 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Born-Oppenheimer approximation
  4. Main results and plan of the paper
  5. Hamiltonian of the light particle
  6. The spectrum of the effective Hamiltonian
  7. Efimov effect in the unitary limit
  8. Finite scattering length
  9. Conclusions
  10. Proof of Lemma \ref{['analyticity_lemma']}
  11. Proof of Proposition \ref{['theorem_Efimov_constant_potential']}

Key Result

Theorem 1.1

There exists an infinite sequence of negative eigenvalues $E^{\mathrm{eff}}_n$ of the effective Hamiltonian slow with the effective potential effective_pontential_unitary_limit, such that $E^{\mathrm{eff}}_n \rightarrow 0$ for $n \rightarrow \infty$. Moreover, where $\beta = \sqrt{-l(l+1) + \varepsilon^{-2} \,W(1)^2 -1/4}$, $\phi_{\beta}= \arg \Gamma (1 + i \beta)$ and $\;\zeta_n\rightarrow 0\,,

Figures (2)

  • Figure 1: The light blue line represents $-\lambda_1(r)$ and the red line represents $-\lambda_{\mathrm{local}} = \frac{- W^2(1) }{r^2}$ for $\alpha =0$. The two curves intersect for $r_{k} = \sqrt{2}(\frac{3\pi}{4} + k \pi),\,\,k-1 \in \mathbb{N}$ and apart from an exponentially decreasing oscillation they coincide for $r > r_{0}$.
  • Figure 2: Effective potentials shifted by the constant $\frac{(1-t_{\theta})^2}{2}$. The light blue line represents the potential with infinite scattering length $a_\theta(0)$. Decreasing $a_\theta(0)$ increases the depth of the short-range well.

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 11 more