a single component Bose gas with effective $(gn)_\mathrm{eff} = g_{11}n_1+g_{22}n_2$.

The summation over discrete momentum states is essential to account for a tight confinement. If we substituted the summation over momenta with the integral, i.e., $1/V
\sum_{\bf k} \rightarrow \int {\rm d} {\bf k}/(2 \pi)^{3}$, we would recover the limit of an infinite box and the LHY energy of a Bose-Bose mixture in 3D space [@Petrov15].

*LHY energy at the 3D-2D crossover.*—Our main goal here is to find the LHY energy for a system confined in one spatial direction. In such a situation the $z$-axis is a tight direction, i.e., $L=L_z$. Assuming that $L_x=L_y \rightarrow \infty$, in Eq. (\[LHY\_mix\]) one has to substitute $\frac{1}{V}\sum_{{\bf k}}\rightarrow
\frac{1}{(2\pi)^2} \int {\rm d}^2 k_{\perp} \frac{1}{L} \sum_{k_z} $, and the LHY energy in quasi-2D takes the form: $$\label{LHY_2d}
e_\mathrm{LHY}^\mathrm{2d}(\xi)  = \lim_{r \rightarrow 0} \frac{\partial }{\partial r} \left(\! r   \frac{1}{2}\sum_{q_z} \int {\rm d}^2 q_{\perp} 
\, e^{i{\bf q} {{\bf r}}} \left( \varepsilon_{\bf q} - A_{\bf q} \right) \!\right),$$ where $\xi=(g_{11} n_1 + g_{22} n_2)/\varepsilon_0$, ${\bf q} = ({\bf q}_\perp,q_z)$ and $q_z$, ${\bf q}_\perp$ are the integer dimensionless momenta: $q_z=(L/2\pi)k_z$, and ${\bf q}_\perp=(L/2\pi) {\bf k}_{\perp}$. Bogoliubov’s energies expressed in the units of $\varepsilon_0$ are: $\varepsilon_{\bf q}= \sqrt{q^4 + 