Unfolding the quartic oscillator

The “exact WKB method” is applied to the general quartic oscillator, yielding rigorous results on the ramification properties of the energy levels when the coefficients of the fourth degree polynomial are varied in the complex domain. Simple though exact “model forms” are given for the avoided crossing phenomenon, easily interpreted in terms of complex branch points in the “asymmetry parameter.” In the almost symmetrical situation, this gives a generalization of the Zinn–Justin quantization condition. The analogous “model quantization condition” near unstable equilibrium is thoroughly analysed in the symmetrical case, yielding complete confirmation of the branch point structure discovered by Bender and Wu. The numerical results of this analysis are in excellent agreement with those computed by Shanley, overtaking the most optimistic expectations of the realm of validity of semiclassical models