One spatial dimensional finite volume three-body interaction for a short-range potential

TL;DR

This paper derives finite volume quantization conditions for three-body scattering in one dimension with short-range interactions, providing analytic solutions and formulas similar to L"uscher's method, extending previous theoretical results.

Contribution

It presents the first analytic solutions of Faddeev's equations for three particles in one dimension and derives L"uscher-like quantization conditions for short-range interactions.

Findings

Analytic solutions for three-body scattering in 1D.

Finite volume quantization conditions matching free space solutions.

Consistency with previous Yang's results.

Abstract

In this work, we use McGuire's model to describe scattering of three spinless identical particles in one spatial dimension, we first present analytic solutions of Faddeev's equation for scattering of three spinless particles in free space. The three particles interaction in finite volume is derived subsequently, and the quantization conditions by matching wave functions in free space and finite volume are presented in terms of two-body scattering phase shifts. The quantization conditions obtained in this work for short range interaction are L\"uscher's formula like and consistent with Yang's results in \cite{Yang:1967bm}.