Scarcity of real discrete eigenvalues in non-analytic complex ΡΤ-symmetric potentials

We find that a non-differentiability occurring whether in real or imaginary part of a complex ΡΤ-symmetric potential causes a scarcity of the real discrete eigenvalues despite the real part alone possessing an infinite spectrum. We demonstrate this by perturbing the real potentials x2 and mean x by imaginary ΡΤ-symmetric potentials ix mean x and ix, respectively. (author)