Equation of state for the one-dimensional attractive $\delta$-potential Bose gas in the weak-coupling regime

Approximated formulas for real quasimomentum and the associated energy spectrum are presented for one-dimensional Bose gas with weak attractive contact interactions. On the basis of the energy spectrum, we obtain the equation of state in the high-temperature region, which is found to be the van der Waals equation without volume correction.

Figure 1

Pressure of one-dimensional Cs gas as a function of the one-dimensional scattering length $a_{1\mathrm{D}}$, evaluated for $N=25$, $L=30\mu \mathrm{m}$, $T=0.16\mathit{nK}$. For lower values of $a_{1\mathrm{D}}$, we find that $P$ may become negative, which occurs before reaching the shaded range $a_{1\mathrm{D}}<2\times 10\mu \mathrm{m}$ where the weak-coupling condition (5) is no longer valid.