Polynomial potentials and coupled quantum dots in two and three dimensions

Abstract

Non-separable $D-$dimensional partial differential Schrödinger equations are considered at $D=2$ and $D=3$, with the even-parity local potentials $V(x,y,\dots )$ 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 $\psi (x,y,\dots )$] is then found feasible in the dynamical regime simulating a coupled system of quantum dots in which the individual minima of $V(x,y,\dots )$ 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.

arXiv comment21 pages, 2 figures

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Cited by in corpus (4)

• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Displaced harmonic oscillator $V\sim \min [(x+d{)}^2,(x-d{)}^2]$ as a benchmark double-well quantum model
• Arnold's potentials and quantum catastrophes II
• Avoided level crossings in quasi-exact approach