Abstract
It is shown that there is a QSE solution of the simple model exactly at a band edge if and only if the one Floquet solution at that band edge vanishes at the surface. This condition is applied to the free electron model, the nearly free electron model, the Kronig-Penney delta well model, and the tight binding model. The solutions near a band edge are also studied, and found to have simple approximate forms, namely u0(x) cos pi nx/L or u0(x) sin pi nx/L according as there is or is not a solution exactly at the band edge. The importance of the solutions at and near the band edges for spectroscopic studies of the QSE is pointed out.