Avoided level crossings in polynomial potentials with N thick barriers

A family of one-dimensional Schrödinger equations is considered, with the multi-well polynomial potentials characterized by an N−plet of the high and thick barriers separating the (N+1)−plet of the deep confining valleys. It is shown how the approximate low-lying spectra can be constructed participating in the tunneling-controlled fine-tuned competition of these valleys for the groundor excited-state dominance and stability. These phenomena are interpreted as a quantum avoidedlevel-crossing analogue of the bifurcations of the long-time equilibria (also known, as Thom’s catastrophes) in classical dynamical systems.