  We study the ground state entanglement, energy and fidelities of a
two-electron system bounded by a core-shell potential, where the core width is
varied continuously until it eventually vanishes. This simple system displays a
rich and complex behavior: as the core width is varied, this system is
characterized by two peculiar transitions where, for different reasons, it
displays characteristics similar to a few-particle quantum phase transition.
The first occurrence corresponds to something akin to a second order quantum
phase transition, while the second transition is marked by a discontinuity,
with respect to the driving parameter, in the first derivatives of quantities
like energy and entanglement. The study of this system allows to shed light on
the sudden variation of entanglement and energy observed in S. Abdullah et al.
[Phys. Rev. B 80, 235302 (2009)]. We also compare the core-shell system with a
system where a core well is absent: this shows that, even when extremely
narrow, the core well has a relevant `pinning' effect. Interestingly, depending
on the potential symmetry, the pinning of the wavefunction may either halve or
double the system entanglement (with respect to the no-core-well system) when
the ground state is already bounded to the outer (shell) well. In the process
we discuss the system fidelity and show the usefulness of considering the
particle density fidelity as opposed to the more commonly used -- but much more
difficult to access -- wavefunction fidelity. In particular we demonstrate that
-- for ground-states with nodeless spatial wavefunctions -- the particle
density fidelity is zero if and only if the wavefunction fidelity is zero.
