Arnold’s potentials and quantum catastrophes

Abstract

In the Thom’s approach to the classification of instabilities in one-dimensional classical systems every equilibrium is assigned a local minimum in one of the Arnold’s benchmark potentials V(k)(x)=xk+1+c1xk−1+…. We claim that in quantum theory, due to the tunneling, the genuine catastrophes (in fact, abrupt “relocalizations” caused by a minor change of parameters) can occur when the number N of the sufficiently high barriers in the Arnold’s potential becomes larger than one. A systematic classification of the catastrophes is then offered using the variable mass term ħ2∕(2μ), odd exponents k=2N+1 and symmetry assumption V(k)(x)=V(k)(−x). The goal is achieved via a symbolic-manipulation-based explicit reparametrization of the couplings cj. At the not too large N, a surprisingly user-friendly recipe for a systematic determination of parameters of the catastrophes is obtained and discussed.