Analytical solution of the Schr\"{o}dinger equation with $1/r^3$ and attractive $1/r^2$ potentials: Universal three-body parameter of mixed-dimensional Efimov states

Abstract:We study the Schrödinger equation with $1/r^3$ and attractive $1/r^2$ potentials. Using the quantum defect theory, we obtain analytical solutions for both repulsive and attractive $1/r^3$ 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 $1/r^3$ and $1/r^2$ 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.

Quantum Gases (cond-mat.quant-gas); Quantum Physics (quant-ph)

arXiv:2601.19517 [cond-mat.quant-gas]

(or arXiv:2601.19517v2 [cond-mat.quant-gas] for this version)

https://doi.org/10.48550/arXiv.2601.19517 Focus to learn more arXiv-issued DOI via DataCite

Phys. Rev. Research, 8, 023101 (2026)

https://doi.org/10.1103/8x96-ygjt Focus to learn more DOI(s) linking to related resources

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