numerical result for $\xi < 0.3$. For larger values of $\xi$, the exact formula is close to the 3D expression.

![The ratio $ s^\mathrm{1d}(\xi)= e_\mathrm{LHY}^\mathrm{1d}(\xi)/e_\mathrm{LHY}^\mathrm{3d}(\xi)$ as the function of $\xi$ given by the thick black line. The additional thin meshed curve is the same ratio but using the approximate formula for $e_\mathrm{LHY}^\mathrm{1d}(\xi)$ given by Eq. (\[e1d\]).[]{data-label="fig2"}](fig2.pdf){width="49.00000%"}

*Droplets at 3D-1D crossover.*— We now analyze the quasi-1D regime defined in the limit $\xi \ll 1$. Including only the first term of Eq. (\[e1d\]), we find from Eq. (\[pzero\]) that at the equilibrium $\xi_0 = (2/9)\beta^2 = (128/9\pi^2)a^4/(\delta a^2 L^2)$. The corresponding droplet density is $$\label{xi01}
  n = \frac{32 }{9\pi } \frac{a^3}{\delta a^2 L^4},$$ where we assumed for simplicity $g_{11}=g_{22}= 4\pi \hbar^2 a/m$. To find the condition for the validity of this formula, we compare it with the density of the droplet using the full $e_\mathrm{LHY}^\mathrm{1d}(\xi)$ from Eq. (\[e1d\]). The relative density differs from the one given by Eq. (\[xi01\]) by $20\%$ for $\xi$ approximately equal to $0.0004$. Thus, for $\xi \lesssim 0.0004$, the formula for the density of the quasi-1D droplet is valid. However such small value of $\xi$ is probably out reach for current experiments. As in the 2D-3D crossover we 