# Universality in a one-dimensional three-body system

## TL;DR

This paper investigates the universal behavior of three-body bound states in a one-dimensional heavy-heavy-light system, demonstrating that near zero two-body energy, the three-body states are independent of potential shape.

## Contribution

It provides a comparative analysis of zero-range and finite-range potentials, showing universality in three-body bound states near zero two-body energy in one dimension.

## Key findings

- Three-body bound states become universal near zero two-body energy.
- Finite-range and zero-range potentials yield similar three-body behaviors.
- Numerical and analytical methods confirm potential shape independence.

## Abstract

We study a heavy-heavy-light three-body system confined to one space dimension. Both binding energies and corresponding wave functions are obtained for (i) the zero-range, and (ii) two finite-range attractive heavy-light interaction potentials. In case of the zero-range potential, we apply the method of Skorniakov and Ter-Martirosian to explore the accuracy of the Born-Oppenheimer approach. For the finite-range potentials, we solve the Schr\"odinger equation numerically using a pseudospectral method. We demonstrate that when the two-body ground state energy approaches zero, the three-body bound states display a universal behavior, independent of the shape of the interaction potential.