Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 35B / iii / Solution

The exact energies and mixing coefficients containAs a function of complex $\lambda$, its nearest branch points to the origin occur when its argument vanishes:Their distance from the expansion point $\lambda =0$ determines the radius of convergence. ThereforeThe real-axis perturbation series converges for $|\lambda |<(E_{+}-E_{-})/2$.