Computing Periodic and Antiperiodic Eigenvalues with a PT-Symmetric Optical Potential

We give estimates for the eigenvalues of nonself-adjoint Sturm–Liouville operators with periodic and antiperiodic boundary conditions for the special potential $4\cos^{2}x+4iV\sin2x$ that is a PT-symmetric optical potential, especially when $|\sqrt{1-4V^{2}}|<3$ or equally $0\leq V<\sqrt{10}/2$. We provide some useful equations for calculating the periodic and antiperiodic eigenvalues. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.