Solutions of the Lieb-Liniger integral equation

Proposed is a method of solving the Lieb-Liniger integral equation for a one-dimensional delta-function bose gas at zero temperature. The integral equation reduces to a set of algebraic equations for the coefficients of the series expansion. For strong coupling case, a few coefficients are enough to have a good approximation. On the contrary, all the coefficients are in the same order for weak coupling case. The expansion lambda = c/K, where c is the coupling constant and K is the cutoff momentum, is a formal one in the sense that the coefficients may include the divergent sums. However, the physical quantities such as the ground state energy are shown to be finite in gamma = c/rho, where rho is the number density, which agree with the results of the Bogoliubov theory.