Exact solutions of the Schrödinger equation with a complex periodic potential

The exact solutions of 1D Schrödinger equation subject to a complex periodic potential V(x)=-[iasin(bx)+c]2 (a, b, c∈ R) are found as a confluent Heun function (CHF) H_C(α, β, γ, δ, η; z) . The energy spectra which are solved exactly by calculating the Wronskian determinant are found as real except for complex values. It is found that the eigenvalues obtained by two constraints on the CHF are not reliable or complete any more since they are only one small part of those evaluated by the Wronskian determinant. The wave functions are illustrated when eigenvalues are substituted into the eigenfunctions. We also notice that the energy spectra remain invariant when one substitutes a→ - a or b→ - b or c→ - c due to the PT symmetry with the property V(x) = V(- x) ^∗ .