Critical parameters for non-hermitian Hamiltonians

•PT-symmetric Hamiltonians exhibit real eigenvalues.•Real eigenvalues appear when PT-symmetry is unbroken.•The range of real eigenvalues is determined by exceptional points or critical parameters.•The diagonalization method is suitable for the calculation of critical parameters.•Degenerate eigenvalues require a more general definition of unbroken symmetry.

We calculate accurate critical parameters for a class of non-hermitian Hamiltonians by means of the diagonalization method. We study three one-dimensional models and two perturbed rigid rotors with PT symmetry. One of the latter models illustrates the necessity of a more general condition for the appearance of real eigenvalues that we also discuss here.