TIGHT UPPER AND LOWER BOUNDS ON PERTURBATION COEFFICIENTS FOR ANHARMONIC OSCILLATORS

An extremely simple argument is given which provides tight upper and lower bounds on the growth of the perturbation coefficients in the expansion of the ground-state energy for the anharmonic oscillator [-d2dx2+x24+gx2K2K-E(g)]y(x)=0, y(±)=0. We prove that if E(g)∼12-(-g)nAn, then for large n, An grows like Cn[n(K-1)]!, where [K(K-1)]K-1≤C≤2K[K(K-1)]K-1. © 1979 The American Physical Society.