A generalized Lieb-Liniger model

TL;DR

This paper extends the Lieb-Liniger model to include a three-function potential with mixed attractive and repulsive parts, providing an approximate solution for dilute gases and identifying stability regimes.

Contribution

It introduces a generalized Lieb-Liniger model with a three-function potential and derives an effective model for low-energy dilute gases, expanding the understanding of bosonic interactions.

Findings

Reduced to an effective Lieb-Liniger model at low energies

Identified stability regimes for the generalized model

Provided an approximate solution for dilute gases

Abstract

In 1963, Lieb and Liniger solved exactly a one dimensional model of bosons interacting by a repulsive \delta-potential and calculated the ground state in the thermodynamic limit. In the present work, we extend this model to a potential of three \delta-functions, one of them is repulsive and the other two are attractive, modeling some aspects of the interaction between atoms, and present an approximate solution for a dilute gas. In this limit, for low energy states, the results are found to be reduced to the ones of an effective Lieb Liniger model with an effective \delta-function of strength $c_{eff}$ and the regime of stability is identified. This may shed light on some aspects of interacting bosons.