Correlation effects in the one-dimensional Bose gas

We discuss interaction effects for the one-dimensional Bose gas with a repulsive delta-function interaction potential. We use the random-phase approximation and a finite local-field correction. Analytical results are given for the local-field correction, the pair-correlation function and the ground-state energy. The groundstate energy is found to be in much better agreement with the exact result than the ground-state energy calculated within the Bogoliubov approximation, where local-field corrections are neglected.