Higher order solutions of Lieb-Liniger integral equation

We study higher order solutions of Lieb-Liniger integral equation for a one-dimensional δ-function Bose gas. By use of the power series expansion method, the integral equation is solved and the correction terms which improve the Bogoliubov theory are calculated analytically in the weak coupling regime. Physical quantities such as the ground state energy and the chemical potential are represented by a dimensionless parameter γ=c/ρ, where c is the interaction strength and ρ is the number density of particles while the quasi-momentum distribution function is expressed in terms of a dimensionless parameter λ=c/K, where K is the cut-off momentum. -- Highlights: → Exact analysis of a one-dimensional delta-function Bose gas for weak coupling case. → The third order corrections are given by the Bethe ansatz method explicitly for the first time. → The Lieb-Liniger equation is solved for the quasi-momentum distribution function. → The ground state energy and the chemical potential are obtained. → Some difference between the previous result and ours is pointed out.