the system with mean-field energy corresponding to repulsive, weakly attractive, or even effectively vanishing interactions. The last possibility was not discussed in the literature so far. It is a droplet which is formed only due to quantum fluctuations.

Finally, let us compare our results for quasi-2D regime with the results from Ref. [@Petrov16] for strictly 2D systems. In the latter case, the LHY energy and droplet densities are expressed in terms of 2D scattering length. Here, we consider the case $a \ll L$, i.e., the scattering has a 3D character. To compare the results, we have to express the 2D scattering length, $a_\mathrm{2d}$, by the 3D one in the case of the box geometry analyzed in our paper. The scattering process in quasi-2D, when the confinement in a tight direction is provided by a box of length $L$, is expressed by [@Zin18]: $$\label{a2d3d}
  a_\mathrm{2d} = 2L e^{ - \gamma - \frac{L}{2a} }.$$ An analogous formula, in a situation when the tight confinement is provided by a harmonic potential, is given in [@shlyapnikov]. Inserting the above relation into equations for 2D LHY energy density and droplet density of [@Petrov16] we recover our results, given by Eqs. (\[Ehom\]) and (\[x0\]), together with 