profiles being Gaussians of different widths we observe the appearance of a small barrier in $V_M^{eff}$ at $g_{_{MI}}=0.2$ (Fig. \[fig:dw\_picture.eps\] a2). This barrier grows with $g_{_{MI}}$ and at $g_{_{MI}}=0.4$ (Fig. \[fig:dw\_picture.eps\] a3) it becomes comparable to the ground state energy of the effective potential, while the one-body density $\rho_{_{MF}}^M$ turns flat at the trap origin. Once the ground state energy drops below the barrier height, two density humps appear and core-shell $MIM$ is established (Fig. \[fig:dw\_picture.eps\] a4). Meanwhile, the effective potential of the impurity $V_I^{eff}$ does not show significant deviations from the harmonic case (second column). Especially, the induced part $V_I^{ind}$, being initially also a Gaussian, is not capable to produce a barrier at the trap center. Similar statements can be made for columns 3 and 4, corresponding to $a_I/a_M=1.1$. The only difference is that $V_I^{eff}$ develops a barrier instead, whereas $V_M^{eff}$ shows only a slight variation, which finally leads to the core-shell $IMI$ phase.

![image](dw_picture.eps){width="1.0\linewidth"}

Motivated by the above observation we define an alternative phase classification from an energetical point of view: $$\label{eq:classification_energy}
    \begin{cases} 
    \text{M} & \mbox{:} \
    E_{_{0,\sigma}}^{MF}-V_{\sigma}^{eff}(0) > 0 \\
    \text{IMI} & \mbox{:} \
    E_{_{0,M}}^{MF}-V_{M}^{eff}(0) > 0 
    \ \land \
    E_{_{0,I}}^{MF}-V_{I}^{eff}(0) < 0  \\
    \text{MIM} & \mbox{:} 