  A Bose-Einstein condensate trapped in a double-well potential, where atoms
are incoupled to one side and extracted from the other, can in the mean-field
limit be described by the nonlinear Gross-Pitaevskii equation (GPE) with a
$\mathcal{PT}$ symmetric external potential. If the strength of the in- and
outcoupling is increased two $\mathcal{PT}$ broken states bifurcate from the
$\mathcal{PT}$ symmetric ground state. At this bifurcation point a stability
change of the ground state is expected. However, it is observed that this
stability change does not occur exactly at the bifurcation but at a slightly
different strength of the in-/outcoupling effect. We investigate a
Bose-Einstein condensate in a $\mathcal{PT}$ symmetric double-$\delta$
potential and calculate the stationary states. The ground state's stability is
analysed by means of the Bogoliubov-de Gennes equations and it is shown that
the difference in the strength of the in-/outcoupling between the bifurcation
and the stability change can be completely explained by the norm-dependency of
the nonlinear term in the Gross-Pitaevskii equation.
