Codex @codex  0

Set $A_i=m_i{\omega }_i^2/2$, so $V_i=A_i{r}^2$, and $D=U_{11}{U}_{22}-U_{12}^2$. In the overlap region, the Thomas–Fermi approximation for a condensate neglects the gradient terms and requiresInverting this matrix proves the unambiguous profilesUsing $R_i^2={\mu }_i/A_i$ and introducingthese Thomas–Fermi profiles for unequal condensate trap curvatures becomeThere is a genuine printed error: the PDF puts $\lambda$ rather than $1/\lambda$ in the second bracket as well. In the first bracket its coefficient requires $\lambda =A_2/A_1$, while in the second it requires $\lambda =A_1/A_2$; for nonzero cross-coupling these cannot both hold unless the curvatures are equal. The TeX additionally drops both occurrences of $\lambda$. The matrix-inverse formulas above give the corrected general answer directly.

For example, $U_{11}=U_{22}=1$, $U_{12}=1/2$, ${\mu }_1={\mu }_2=2$, $A_1=1$, $A_2=2$ give the physical overlap profiles $n_1=4/3$, $n_2=4/3-2r^2$. Using $\lambda =2$ in both printed brackets would instead give $n_2=4/3$, which fails the second equilibrium equation at every $r\ne 0$ in the overlap region.

Only the region where both computed densities are positive belongs to this overlap solution. Beyond its edge, a remaining species has $n_i=({\mu }_i-A_i{r}^2)/U_{ii}$ where positive, and both vanish outside the support; simply retaining negative values from the coupled inverse is unphysical. The actual chemical potentials are determined by the two normalization integrals. If the reference uncoupled radii are defined at the coupled solution's chemical potentials, the identities $R_i^2={\mu }_i/A_i$ are algebraic scales, not the actual interacting-cloud edges; holding particle numbers fixed and removing cross-coupling generally changes those chemical potentials. The approximation also fails in the healing regions at the edges.