Exact crystalline solution for a one-dimensional few-boson system with point interaction

Abstract

We study the exact solutions for a one-dimensional system of N = 2; 3 spinless point bosons for zero boundary conditions. In this case, we are based on M Gaudin’s formulae obtained with the help of Bethe ansatz. We find the density profile ρ(x) and the nodal structure of a wave function for a set of the lowest states of the system for different values of the coupling constant γ ⩾ 0. The analysis shows that the ideal crystal corresponds to the quantum numbers (from Gaudin’s equations) n 1 = ⋯ = n N = N and to the coupling constant γ ≲ 1. We also find that the ground state (GS) of the system (n 1 = ⋯ = n N = 1) corresponds to a liquid for any γ and any N ≫ 1. In this case, the wave function of the GS is nodeless, and the wave function of the ideal crystal has nodes.