Thomas–Fermi profiles for unequal condensate trap curvatures ($n_1=\frac{U_{22}({\mu }_1-A_1{r}^2)-U_{12}({\mu }_2-A_2{r}^2)}{D}$)

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.