Arnold's potentials and quantum catastrophes II

The well known phenomenon of avoided level crossing (ALC) can be perceived as a quantum analogue of the Thom's catastrophes in classical dynamical systems. One-dimensional Schr\"{o}dinger equation is chosen for illustration. In constructive manner, a family of confining polynomial potentials is considered, characterized by the presence of an \(N-\)plet of high barriers separating the \((N+1)-\)plet of deep valleys. The bifurcations of the long-time classical equilibria are shown paralleled by the ALCs in the quantum low-lying spectra. Every tunneling-controlled fine-tuned switch of dominance between the valleys is finally interpreted as a change of the topological structure of the probability density representing a genuine quantum relocalization catastrophe.