Three-body continuum states and Efimov physics in non-integer geometry

TL;DR

This paper investigates three-body continuum states and Efimov physics in non-integer dimensions, deriving universal analytic expressions for wave functions and scattering properties near the critical dimension where Efimov effects occur.

Contribution

It introduces a non-integer dimension approach to analyze three-body continuum states, providing universal analytic formulas applicable to all short-range potentials near the Efimov regime.

Findings

Analytic expressions for wave functions, scattering lengths, phase shifts, and cross sections.

Good agreement between analytic formulas and numerical calculations for three identical bosons.

Cross sections vanish at zero energy with dimension-dependent power laws.

Abstract

Continuum structures of three short-range interacting particles in a deformed external one-body field are investigated. We use the equivalent $d$-method employing non-integer dimension, $d$, in a spherical calculation with a dimension-dependent angular momentum barrier. We focus on dimensions close to the critical dimension, $d=d_E$, between two and three, defined by zero two-body energies, where the Efimov effect can occur. We design for this dimension region a schematic, long-distance realistic, square-well based, three-body spherical model, which is used to derive analytic expressions for the wave functions, scattering lengths, phase shifts, and elastic scattering cross sections. The procedure and the results are universal, valid for all short-range potentials, and for large scattering lengths. We discuss the properties and validity of the derived expressions by means of the simplest system of three identical bosons. The derived expressions are particularly useful for very small energies, where full numerical calculations are often not feasible. For energies where the numerical calculations can be performed, a good agreement with the analytic results is found. These model results may be tested by scattering experiments for three particles in an equivalent external deformed oscillator potential. The cross sections all vanish in the zero-energy limit for $d<3$ with definite $d$-dependent power of energy.