Two-body continuum states in non-integer geometry

TL;DR

This paper explores two-body continuum states in a non-integer dimensional space using an equivalent $d$-method, deriving analytic expressions for scattering properties and demonstrating the method's effectiveness for low-energy elastic cross sections.

Contribution

It introduces a novel application of the $d$-method in non-integer dimensions to analyze two-body scattering, providing explicit formulas and showing equivalence with traditional 3D results.

Findings

Derived analytic expressions for scattering lengths and phase shifts.

Showed phase shifts are identical in $d$-method and 3D space.

Validated the $d$-method for calculating low-energy cross sections.

Abstract

Wave functions, phase shifts and corresponding elastic cross sections are investigated for two short-range interacting particles in a deformed external oscillator field. For this we use the equivalent $d$-method employing a non-integer dimension $d$. Using a square-well potential, we derive analytic expressions for scattering lengths and phase shifts. In particular, we consider the dimension, $d_E$, for infinite scattering length, where the Efimov effect occurs by addition of a third particle. We give explicitly the equivalent continuum wave functions in $d$ and ordinary three dimensional (3D) space, and show that the phase shifts are the same in both methods. Consequently the $d$-method can be used to obtain low-energy two-body elastic cross sections in an external field.