Arnold's potentials and quantum catastrophes II

TL;DR

This paper explores the analogy between classical catastrophes and quantum phenomena, demonstrating how avoided level crossings in Schrödinger equations relate to topological changes in quantum states, revealing new insights into quantum relocalization.

Contribution

It introduces a family of polynomial potentials with multiple barriers and valleys, linking classical bifurcations to quantum avoided level crossings and topological quantum catastrophes.

Findings

Bifurcations in classical equilibria correspond to avoided level crossings in quantum spectra.

Quantum relocalization catastrophes involve topological changes in probability densities.

Potential landscapes with multiple barriers exhibit rich quantum-classical bifurcation phenomena.

Abstract

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.