This paper investigates the inverse spectral properties of a quadratic operator pencil with steplike almost-periodic potentials in space L₂ (R). The operator, which is generally non-self-adjoint, is analyzed under the conditions of specific potential forms. Key theorems are established to construct special solutions, examine spectral singularities, and demonstrate the absence of real eigenvalues. Additionally, the continuous spectrum is characterized, and the inverse spectral problem is formulated, providing a procedure to uniquely reconstruct the potentials from given spectral data. The findings contribute to the broader understanding of spectral theory in non-self-adjoint operator settings.
Efendiev et al. (Sun,) studied this question.