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) = x(k+1) + c(1)x(k-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. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper demonstrates the different applications of Arnold's one-dimensional polynomial potentials in classical catastrophe theory and quantum mechanics, particularly within specific dynamical regimes. By relaxing constraints and utilizing perturbative methods, the characteristics of these potentials are investigated.