Spectral singularities in PT?>-symmetric Bose–Einstein condensates

We consider the model of a PT?>-symmetric Bose–Einstein condensate in a delta-function double-well potential. We demonstrate that analytic continuation of the primarily non-analytic term |ψ|2ψ—occurring in the underlying Gross–Pitaevskii equation—yields new branch points where three levels coalesce. We show numerically that the new branch points exhibit the behaviour of exceptional points of second and third order. A matrix model which confirms the numerical findings in analytic terms is given.