Bound States of a Non‐Hermitian System With a QES Sine Potential via AIM

ABSTRACT In this study, the eigenfunctions (wavefunctions) of a non‐Hermitian system defined by a complex periodic sine potential are obtained in the form of a hypergeometric series, while its real eigenvalues are derived in closed form. To this end, the one‐dimensional Schrödinger equation is formulated for a PT‐symmetric system of this type, and asymptotic iteration method is employed as the mathematical tool. Owing to the method's applicability to quasi‐exact, numerical, or perturbative regimes, the obtained real analytical eigenvalues are shown to be in good agreement with the literature. Accordingly, a specific condition on the potential parameters, determined by means of the method, indicates that the wavefunction can be expressed in the form of a hypergeometric series. Furthermore, the method demonstrates that the analytical eigenvalues (and the corresponding wavefunction) remain valid, particularly at high‐energy levels, even when the potential parameters deviate from this specific condition. Additionally, the obtained wavefunctions are shown to be consistent with Bloch's theorem in solid‐state physics.