Double well Stark effect: Crossing and anticrossing of resonances

We consider the semiclassical Stark effect for a family of asymmetric unstable double well models and we study the crossing and anticrossing of the field dependent resonances in the complex field plane. We prove that a Bender-Wu type singularity crosses the real axis when the internal barrier is nearly twice ''larger'' than the external one and the beating period is close to the shorter life-time of the resonances. At this critical point we have the anticrossing-crossing transition and for larger instability we have the single well localization.