Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State

A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, $\gamma$. When $\gamma$ is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of $\gamma$ and show that it is analytic for all $\gamma$, except $\gamma =0\mathrm{}$. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum—not one as is ordinarily supposed.