# Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion

## TL;DR

This paper introduces a Chebyshev polynomial-based method to accurately compute the ground-state energy of the Lieb-Liniger model at weak interactions, revealing new analytical expansion coefficients.

## Contribution

It presents a novel numerical approach using Chebyshev polynomials and integer relation algorithms to derive analytical energy expansions in an integrable quantum model.

## Key findings

- First nine terms of the energy expansion obtained
- High-precision numerical data achieved in the weak interaction limit
- Exact perturbative results for excitation spectrum derived

## Abstract

We develop an alternative description to solve the problem of the ground-state energy of the Lieb-Liniger model that describes one-dimensional bosons with contact repulsion. For this integrable model we express the Lieb integral equation in the representation of Chebyshev polynomials. The latter form is convenient to efficiently obtain very precise numerical results in the singular limit of weak interaction. Such highly precise data enable us to use the integer relation algorithm to discover the analytical form of the coefficients in the expansion of the ground-state energy for small values of the interaction parameter. We obtained the first nine terms of the expansion using quite moderate numerical efforts. The detailed knowledge of behavior of the ground-state energy on the interaction immediately leads to exact perturbative results for the excitation spectrum.