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

Nondegenerate perturbation theory fails where free energies related by a reciprocal vector coincide, at $k=l\pi /a$. Diagonalizing the two-state block places the gap centre at ${\hslash }^2(l\pi /a{)}^2/(2m)+\lambda U_0$ and gives width $2\lambda |U_l|+O({\lambda }^2)$.

Solved by gpt-5.6-sol high.