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In a homogeneous overlapping two-component Bose-Einstein condensate, the long-wavelength energy change at fixed species numbers is one half the integral of the density perturbation quadratic form with matrix $\left(\begin{array}{cc}{U}_{11}& U_{12}\\ U_{12}& U_{22}\end{array}\right)$. Strict positive definiteness gives the displayed criterion. Strong positive cross-repulsion favors separation; sufficiently strong cross-attraction instead produces an unstable joint density mode. Equality is marginal at this order, not strict stability.

For an occupied uniform two-component Bose-Einstein condensate, the gradient and trapping terms vanish. Dividing each stationary equation by its nonzero field givesThe same relations follow by differentiating the uniform interaction-energy density $e=U_{11}{n}_1^2/2+U_{22}{n}_2^2/2+U_{12}{n}_1{n}_2$ with respect to each number density. When $D=U_{11}{U}_{22}-U_{12}^2\ne 0$, the inverse relations areThese describe a genuinely overlapping mixture only when both densities are nonnegative. An unoccupied component cannot be divided out of its stationary field equation; it requires the appropriate constrained equilibrium condition instead.

For a miscible two-component Bose-Einstein condensate with $V_i=A_i{r}^2$, $D=U_{11}{U}_{22}-U_{12}^2>0$, invert the local interaction matrix. The displayed result and $n_2=[U_{11}({\mu }_2-A_2{r}^2)-U_{12}({\mu }_1-A_1{r}^2)]/D$ hold in the overlap region. Factoring by $R_i^2={\mu }_i/A_i$ puts the ratio $\lambda =A_2/A_1$ in the first profile and ${\lambda }^{-1}$ in the second. Outside overlap, a negative formal density must be replaced by the appropriate single-component solution, not retained as a physical density.