Free electron approximation near the FBZ's boundary

hokhani

Consider nearly free-electron approximation (here in one dimensional crystal). There is degeneracy at the border of Brillouin zone and one can solve the problem using first order degenerate perturbation theory at the border. But many of text books also use degenerate perturbation theory for the points near the border whereas there isn't any degeneracy there. What is the reason?