On the Perturbation Theory of the Anharmonic Oscillator at Large Orders

Summary Using the Lipatov approach, we study the large- n behaviour of the n -th coefficient E n in the perturbation expansion for the ground-state energy E ( g ) of the anharmonic oscillator. Our calculations do not rely on the analyticity properties of E ( g ) in the coupling constant g . We compute the 1/ n correction to the leading term and recover the result of Bender and Wu. A general « renormalization » procedure is conjectured, which has been checked up to the order 1/ n 3 .