2 \xi q^2}$ and $
A_{\bf q}=q^2 + \xi$. The ratio $\xi$ of the sum of mean field energies of both component to the excitation energy in the tight direction is the crucial parameter characterizing the system. We note that Eq. (\[LHY\_2d\]) applies not only to a system at 3D-2D crossover, but also in the case of strongly oblate geometry, where the characteristic spacing of kinetic momenta in the tighter direction is much larger than spacing in the perpendicular directions. Then, the densely spaced momenta in the perpendicular direction can be considered as continuous.

![ The ratio $ s^\mathrm{2d}(\xi)= e_\mathrm{LHY}^\mathrm{2d}(\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 $e_\mathrm{LHY}^\mathrm{2d}(\xi)$ given by Eq. (\[e2d\]). The red dashed horizontal line is the asymptotic 3D result.[]{data-label="fig1"}](fig1.pdf "fig:"){width="49.00000%"}\

For small $\xi$, the result can be obtained analytically (see the Supplementary Material). The formula, derived for $\xi \ll 1$, is the following: $$\label{e2d}
e_\mathrm{LHY}^\mathrm{2d}(\xi) = \frac{\pi}{4} \xi^2 \left(   \log(\xi) + \log(2\pi^2) + \frac{1}{2}  +  \frac{\pi^2 \xi}{3} \right).$$

We compare this expansion to the direct numerical evaluation of Eq. (\[LHY\_mix\]). In Fig. \[fig1\], we plot the ratio $ 