Arnold's potentials and quantum catastrophes

摘要

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) = x(k+1) + c(1)x(k-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 h(2)/(2 mu), 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 c(j). At the not too large N, a surprisingly user-friendly recipe for a systematic determination of parameters of the catastrophes is obtained and discussed. (C) 2019 Elsevier Inc. All rights reserved.