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

We consider the linear and nonlinear Schr\"odinger equation for a Bose-Einstein condensate in a harmonic trap with $\cal {PT}$-symmetric double-delta function loss and gain terms. We verify that the conditions for the applicability of a recent proposition by Mityagin and Siegl on singular perturbations of harmonic oscillator type self-adjoint operators are fulfilled. In both the linear and nonlinear case we calculate numerically the shifts of the unperturbed levels with quantum numbers $n$ of up to 89 in dependence on the strength of the non-Hermiticity and compare with rigorous estimates derived by those authors. We confirm that the predicted $1/n^{1/2}$ estimate provides a valid 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 in excellent agreement with the numerical results. With nonlinearity the shrink rates are found to be smaller than without nonlinearity, and the rigorous estimates, derived only for the linear case, are no longer applicable.