away from $\eta_c$ the core-shell parameter region, which lies in-between $M$ and $SB$, grows in size making the estimate inefficient. The other reason is that the mean-field solution $\varphi^{\sigma}_{_{MF}}$ of the $SB$ phase is rather an asymmetric Gaussian.

Finally, we discuss the limiting cases. When $\eta \rightarrow \infty$ ($a_{I}/a_{M} \rightarrow 0$) the impurity becomes highly localized at $z=0$. It will not be affected by the majority atoms. Meanwhile the majority species will be subject to an additional delta-potential at $z=0$ with potential strength $g_{_{MI}} N_I$. This analytically solvable one-body problem results in a Weber differential equation. Upon increasing the delta-potential pre-factor $g_{_{MI}} N_I$ the initially unperturbed Gaussian solution develops a cusp at the trap center, whose depth tends to zero as the pre-factor goes to infinity. When $\eta \rightarrow 0$ ($a_{I}/a_{M} \rightarrow \infty$), we can change our perspective by rescaling the Hamiltonian in impurity harmonic units and argue in a similar way as above.

In the following section we compare to the results obtained for the corresponding correlated many-body approach of ML-X.

ML-X: modifications of the phase diagram due to correlations and entanglement {#sec:few_body_MLX}
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For the total wave-function in we use $S=8$ species orbitals and $s_{\sigma}=8$ SPFs for each 