\ll
\lambda_+$, and thus the density $n_- = (n_1 \sqrt{g_{22}} + n_2 \sqrt{g_{11}})/(\sqrt{g_{11} + g_{22}})$ corresponds to a soft mode, while $n_+ = (n_1 \sqrt{g_{11}} -
n_2 \sqrt{g_{22}})/(\sqrt{g_{11} + g_{22}})$, is the density of a hard mode. Deviation of the latter from zero is energetically very costly, so we assume that in the ground state the hard-mode density effectively vanishes, $n_+ = 0$. Consequently, the densities of both species are proportional to the density of the soft mode: $$\begin{aligned}
n_- =  n_1 \sqrt{(g_{11}+g_{22})/g_{22}} = n_2 \sqrt{(g_{11}+g_{22})/g_{11}}. \end{aligned}$$ To further specify our system we assume that it is confined in a box, and periodic boundary conditions are imposed. The standard LHY correction [@Petrov15] in 3D is evaluated under the assumption that all sides of the box have similar length.

To find the LHY correction for the tightly confined system we have to consider the case when a one side of the box is much smaller than the others, $L_z \ll L_x \simeq
L_y$ (3D-2D crossover), or much larger (3D-1D crossover) $L_x \simeq L_y \ll L_z $, than remaining sides. These two configurations are considered separately below. We denote the tight confinement extension by $L$ while a linear size of the box 