Point bosons in a one-dimensional box: the ground state, excitations and thermodynamics

We determine the ground-state energy and the effective dispersion law for a one-dimensional system of point bosons under zero boundary conditions. The ground-state energy is close to the value for a periodic system. But the dispersion law is essentially different from that for a periodic system, if the coupling is weak (weak interaction or high concentration) or intermediate. We propose also a new method for construction of the thermodynamics for a gas of point bosons. It turns out that the difference in the dispersion laws of systems with periodic and zero boundary conditions does not lead to a difference in the thermodynamic quantities. In addition, under zero boundary conditions, the microscopic sound velocity does not coincide with the macroscopic one. This means that either the method of determination of $k$ in the dispersion law $E(k)$ is unsuitable or the low-energy excitations are not phonons.