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Analytical solution of the Schrödinger equation with 1/r3 and attractive 1/r2 potentials: Universal three-body parameter of mixed-dimensional Efimov states

Yuki Ohishi1,*,†, Kazuki Oi1,2,*,‡, and Shimpei Endo1,3,§

  • *These authors contributed equally to this work.
  • †Contact author: y.ohishi@uec.ac.jp
  • ‡Contact author: kazuki.oi.t8@dc.tohoku.ac.jp
  • §Contact author: shimpei.endo@uec.ac.jp
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Phys. Rev. Research 8, 023101 – Published 30 April, 2026

DOI: https://doi.org/10.1103/8x96-ygjt

Abstract

We study the Schrödinger equation with 1/r3 and attractive 1/r2 potentials. Using the quantum defect theory, we obtain analytical solutions for both repulsive and attractive 1/r3 interactions. The obtained discrete-scale-invariant energies and wave functions, validated by excellent agreement with numerical results, provide a natural framework for describing the universality of Efimov states in mixed dimension. Specifically, we consider a three-body system consisting of two heavy particles with large dipole moments confined to a quasi-one-dimensional geometry and resonantly interacting with an unconfined light particle. With the Born-Oppenheimer approximation, this system is effectively reduced to the Schrödinger equation with 1/r3 and 1/r2 potentials, and manifests the Efimov effect. Our analytical solution suggests that, for repulsive dipole interactions, the three-body parameter of the mixed-dimensional Efimov states is universally set by the dipolar length scale, whereas for attractive interactions it explicitly depends on the short-range phase. We also investigate the effects of finite transverse confinement and find that our analytical results are useful for describing the Efimov states composed of two polar molecules and a light atom.

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Phys. Rev. Research 8, 023101– Published 30 April, 2026
  • Received 26 January 2026
  • Accepted 2 April 2026
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