is accessible.

The conditions of the dimensional crossover may be reached by varying the trap geometries of the Bose-Bose mixture. Both strongly prolate and oblate shapes of BECs can be formed, and excitation energies in confined and extended direction(s) can be separated energetically, with a limited number of modes in confined direction(s) occupied at low temperatures in the weakly interacting limit. Such systems, with significantly varying spatial extensions in different directions, are in the region of dimensional crossover.

*Lee-Huang-Yang energy of a mixture in a box.*—The system we study is a two component mixture of interacting ultracold Bose gases in the ground state. The mean-field energy density is given by Eq. (\[MF\]). Following the analysis presented in [@Petrov15], we consider the case when both intraspecies interactions are repulsive, $g_{11}>0$, $g_{22}>0$, ($g_{11} \approx g_{22}$), while interspecies interaction is attractive, $g_{12}<0$. We also assume that the system is close to the region of collapse, and, thus, the parameter $\delta g = g_{12} + \sqrt{g_{11}g_{22}}$ is small, i.e., $|\delta g| \ll g_{11},g_{22}$. The diagonal form of the mean-field energy density reads: $$\begin{aligned}
\epsilon_\mathrm{MF} = \lambda_-n_-^2 + \lambda_+n_+^2.\,\end{aligned}$$ where the coefficients $\lambda_+ \simeq ({g_{11}+g_{22}})/2$, and $\lambda_- \simeq \delta g \sqrt{g_{11}g_{22}}/(g_{11}+g_{22})$. In our regime $|\lambda_-| 