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We develop an alternative description to solve the problem of theground-state energy of the Lieb-Liniger model that describes one-dimensionalbosons with contact repulsion. For this integrable model we express the Liebintegral equation in the representation of Chebyshev polynomials. The latterform is convenient to efficiently obtain very precise numerical results in thesingular limit of weak interaction. Such highly precise data enable us to usethe integer relation algorithm to discover the analytical form of thecoefficients in the expansion of the ground-state energy for small values ofthe interaction parameter. We obtained the first nine terms of the expansionusing quite moderate numerical efforts. The detailed knowledge of behavior ofthe ground-state energy on the interaction immediately leads to exactperturbative results for the excitation spectrum.

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