UNFOLDING THE QUARTIC OSCILLATOR

The ''exact WKB method'' is applied to the general quartic oscillator, yielding rigorous results on the ramification properties of the energ y levels when the coefficients of the fourth degree polynomial are var ied in the complex domain. Simple though exact ''model forms'' are giv en for the avoided crossing phenomenon, easily interpreted in terms of complex branch points in the ''asymmetry parameter.'' In the almost s ymmetrical situation, this gives a generalization of the Zinn-Justin q uantization condition. The analogous ''model quantization condition'' near unstable equilibrium is thoroughly analysed in the symmetrical ca se, yielding complete confirmation of the branch point structure disco vered by Bender and Wu. The numerical results of this analysis are in excellent agreement with those computed by Shanley, overtaking the mos t optimistic expectations of the realm of validity of semiclassical mo dels. (C) 1997 Academic Press.