Abstract: Quasi-exactly solvable (QES) systems are a class of quantum mechanical models in which part of the energy spectrum and eigenstates can be solved analytically. This paper reviews the definition and development of the QES problem and its importance in physics, centering on the QES periodic potential. By analyzing the energy band structure of the Kronig-Penney model, the Lamé potential, the PT-symmetric periodic potential, and the Mathieu potential, the key role played by these models in the description of the electron behavior in periodic lattices is revealed. In this paper, the energy band properties of the QES periodic potentials and their analytic solutions are systematically explored by combining the theoretical tools of Lie algebra, polynomial invariant subspaces, and supersymmetric quantum mechanics. The results show that different QES periodic potentials exhibit rich energy band structures, which can be used to modulate the bandgap of crystalline materials. The QES system is of great significance in solving complex quantum systems and provides theoretical guidance and prospective ideas for studying novel materials and Non-Hermitian systems.

Key words: quasi-exactly solvable, periodic potential, energy band structure