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We consider the linear and nonlinear Schr\"odinger equation for aBose-Einstein condensate in a harmonic trap with $\cal {PT}$-symmetricdouble-delta function loss and gain terms. We verify that the conditions forthe applicability of a recent proposition by Mityagin and Siegl on singularperturbations of harmonic oscillator type self-adjoint operators are fulfilled.In both the linear and nonlinear case we calculate numerically the shifts ofthe unperturbed levels with quantum numbers $n$ of up to 89 in dependence onthe strength of the non-Hermiticity and compare with rigorous estimates derivedby those authors. We confirm that the predicted $1/n^{1/2}$ estimate provides avalid upper bound on the the shrink rate of the numerical eigenvalues.Moreover, we find that a more recent estimate of $\log(n)/n^{3/2}$ is inexcellent agreement with the numerical results. With nonlinearity the shrinkrates are found to be smaller than without nonlinearity, and the rigorousestimates, derived only for the linear case, are no longer applicable.

A Bose-Einstein Condensate with $\mathcal{PT}$-Symmetric Double-Delta Function Loss and Gain in a Harmonic Trap: A Test of Rigorous Estimates