Polynomial potentials and coupled quantum dots in two and three dimensions
Abstract
Non-separable D-dimensional partial differential Schr\"odinger equations are considered at D=2 and D=3, with the even-parity local potentials V(x,y,…) which are polynomials of degree four (cusp catastrophe resembling case) and six (butterfly resembling case). Their extremes (i.e., minima and maxima) are assumed pronounced, localized via a suitable ad hoc parametrization of the coupling constants. A non-numerical approximate construction of the low lying bound states (x,y,…)] is then found feasible in the dynamical regime simulating a coupled system of quantum dots in which the individual minima of V(x,y,…) are well separated, with the potential being locally approximated by the harmonic oscillator wells. The measurable characteristics (and, in particular, the topologically protected probability-density distributions) are then found bifurcating in a specific evolution scenario called a relocalization quantum catastrophe.