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

TL;DR

This paper derives analytical solutions for the Schrödinger equation with specific potentials relevant to mixed-dimensional Efimov states, revealing universal and interaction-dependent aspects of the three-body parameter.

Contribution

It provides the first analytical framework for understanding Efimov states with $1/r^3$ and $1/r^2$ potentials in mixed dimensions, linking the three-body parameter to physical scales.

Findings

Universal three-body parameter for repulsive dipoles set by dipolar length

Explicit dependence of the three-body parameter on short-range phase for attractive interactions

Analytical solutions agree well with numerical results and describe Efimov states in mixed dimensions

Abstract

We study the Schr\"{o}dinger 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 Schr\"{o}dinger 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.