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 $\rho (x)$ 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 $\gamma \ge 0$. The analysis shows that the ideal crystal corresponds to the quantum numbers (from Gaudin's equations) $n_1=\dots =n_N=N$ and to the coupling constant $\gamma \le 1$. We also find that the ground state of the system ($n_1=\dots =n_N=1$) corresponds to a liquid for any $\gamma$ and any $N\gg 1$. In this case, the wave function of the ground state is nodeless, and the wave function of the ideal crystal has nodes.

arXiv comment16 pages, 5 figures; v4: a few minor changes; accepted version

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• Symmetry properties of the ground state of the system of interacting spinless bosons