Supersymmetric model of a Bose-Einstein condensate in a $\mathcal{PT}$-symmetric double-delta trap

The most important properties of a Bose-Einstein condensate subject to balanced gain and loss can be modelled by a Gross-Pitaevskii equation with an external $\mathcal{PT}$-symmetric double-delta potential. We study its linear variant with a supersymmetric extension. It is shown that both in the $\mathcal{PT}$-symmetric as well as in the $\mathcal{PT}$-broken phase arbitrary stationary states can be removed in a supersymmetric partner potential without changing the energy eigenvalues of the other state. The characteristic structure of the singular delta potential in the supersymmetry formalism is discussed, and the applicability of the formalism to the nonlinear Gross-Pitaevskii equation is analysed. In the latter case the formalism could be used to remove $\mathcal{PT}$-broken states introducing an instability to the stationary $\mathcal{PT}$-symmetric states.