Nondegenerate perturbation theory fails where free energies related by a reciprocal vector coincide, at $k=lπ/a$. Diagonalizing the two-state block places the gap centre at $\hbar^2(lπ/a)^2/(2m)+λU_0$ and gives width $2λ|U_l|+O(λ^2)$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 34B / b / Solution

 Solution (source code)