Estimates for eigenvalues of Sturm-Liouville operators with some PT-symmetric potentials

We provide some useful equations for calculating the periodic and antiperiodic eigenvalues of the one-dimensional Schrödinger operator $S(q)$ with a special potential that is a PT-symmetric trigonometric polynomial. We even give estimates to approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error estimation using Rouche's theorem.