Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. III. Application to the One-Dimensional Model

We continue the study of the integrodifferential equation proposed previously for the evaluation of the ground-state energy of an imperfect Bose gas. We apply it here to the one-dimensional delta-function gas where the exact result is known for all values of the coupling constant $\gamma$. The results are: (i) For small $\gamma$, the equation gives the correct first two terms in an asymptotic series; (ii) a numerical solution of the equation shows that the maximum relative error occurs for $\gamma =\infty$ in which case it is 19%; (iii) for $\gamma =\infty$ we are able to compare the exact two-particle distribution function with that given by the equation. The agreement is quite good.