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Efimov spectrum in the Born-Oppenheimer picture of a 2+1 system with zero-range heavy-light interactions

Hamidreza Saberbaghi*

  • *Contact author: hamidreza.saberbaghi@uninsubria.it
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Phys. Rev. A 113, 063320 – Published 22 June, 2026

DOI: https://doi.org/10.1103/kxd7-t5rc

Abstract

We study the Born–Oppenheimer approximation of a mass-imbalanced three-body system made of two heavy particles of mass M and one light particle of mass m for arbitrary angular momentum. In this system, heavy-light pairs interact via a zero-range force. We construct the light-particle Hamiltonian using self-adjoint extensions of the two-center point interaction and show that the corresponding effective potential is regular at the coincidence point of the heavy particles. Consequently, this model presents an alternative method to finite-range, cutoff, or short-distance heavy-heavy regularizations: the necessary three-body input is encoded in the self-adjoint realization of the light-particle Hamiltonian, while the heavy-light interactions remain point-like. In the unitary limit, after fixing the characteristic length scale, we derive an explicit Efimov spectrum. Our results recover the zero-angular-momentum case of [J. Phys. A: Math. Theor. 57, 055303 (2024)] and provide a sufficient condition ensuring the absence of non-Efimov bound states. Away from unitarity, we show that the spatial size of the shallowest trimer near the threshold is approximately 2.8 times the heavy-light scattering length, in contrast with the common assumption that these two length scales coincide. We also derive a Bargmann-type bound on the number of three-body bound states and obtain an estimate sharper than previous results. Finally, we illustrate the method with numerical results for selected alkali mixtures.

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Phys. Rev. A 113, 063320– Published 22 June, 2026
  • Received 1 December 2025
  • Revised 13 March 2026
  • Accepted 3 June 2026
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