Polynomial potentials and coupled quantum dots in two and three dimensions

摘要

Polynomial potentials V(x) = x(4) + O(x(2)) and V(x) = x(6) + O(x(4)) were introduced, in the Thom's purely geometric classification of bifurcations, as the benchmark models of the so called cusp catastrophes and of the so called butterfly catastrophes, respectively. Due to their asymptotically confining property, these two potentials are exceptional, viz., able to serve similar purposes even after quantization, in the presence of tunneling. In this paper the idea is generalized to apply also to quantum systems in two and three dimensions. Two related technical obstacles are addressed, both connected with the non-separability of the underlying partial differential Schrodinger equations. The first one [viz., the necessity of a non-numerical localization of the extremes (i.e., of the minima and maxima) of V(x, y,...)] is resolved via an ad hoc reparametrization of the coupling constants. The second one [viz., the necessity of explicit construction of the low lying bound states.(x, y,...)] is circumvented by the restriction of attention to the dynamical regime in which the individual minima of V(x, y,...) are well separated, with the potential being locally approximated by the harmonic oscillator wells simulating a coupled system of quantum dots (a.k.a. an artificial molecule). Subsequently it is argued that the measurable characteristics (and, in particular, the topologically protected probability-density distributions) could bifurcate in specific evolution scenarios called relocalization catastrophes. (C) 2020 Elsevier Inc. All rights reserved.