Arnold’s potentials and quantum catastrophes II

In paper I (Znojil, 2020) it has been demonstrated that besides the well known use of the Arnold’s one-dimensional polynomial potentials V(k)(x)=xk+1+c1xk−1+… in the classical Thom’s catastrophe theory, some of these potentials (viz., the confining ones, with k=2N+1) could also play an analogous role of genuine benchmark models in quantum mechanics, especially in the dynamical regime in which N+1 valleys are separated by N barriers. For technical reasons, just the ground states in the spatially symmetric subset of V(k)(x)=V(k)(−x) have been considered. In the present paper II we will show that and how both of these constraints can be relaxed. Thus, even the knowledge of the trivial leading-order form of the excited states will be shown sufficient to provide a new, truly rich level-avoiding spectral pattern. Secondly, the fully general asymmetric-potential scenarios will be shown tractable perturbatively. •Avoided level crossings interpreted as quantum catastrophes.•Density distribution topology phase transitions described.•Asymmetric multi-well Arnold’s potentials also considered.•Description extended to cover excited states.•Perturbation approximation treatment outlined.