Abstract 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 ψ ( 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.

中文翻译：

---

二维和三维的多项式势和耦合量子点

摘要 多项式势 V ( x ) = x 4 + O ( x 2 ) 和 V ( x ) = x 6 + O ( x 4 ) 在 Thom 对分岔的纯几何分类中被引入，作为所谓的基准模型。尖头灾难和所谓的蝴蝶灾难，分别。由于它们的渐近限制特性，这两个势能是例外的，即，即使在量子化之后，在存在隧道效应的情况下也能达到类似的目的。在这篇论文中，这个想法被推广到也适用于二维和三维的量子系统。解决了两个相关的技术障碍，两者都与基础偏微分薛定谔方程的不可分离性有关。第一个 [即，必须对 V ( x , y , … ) ] 是通过耦合常数的特别重新参数化来解决的。第二个[即，显式构造低束缚态 ψ ( x , y , … ) 的必要性] 通过将注意力限制在 V ( x , y , y , … ) 被很好地分离，电势由模拟量子点耦合系统（又名人工分子）的谐振子阱局部逼近。随后有人认为，可测量的特征（特别是受拓扑保护的概率密度分布）可能会在称为重定位灾难的特定进化场景中分叉。显式构造低束缚态 ψ ( x , y , … ) ] 的必要性通过将注意力限制在 V ( x , y , … ) 的个体最小值被很好地分离的动力学状态的限制来规避，电势由模拟量子点耦合系统（又名人工分子）的谐振子阱局部逼近。随后有人认为，可测量的特征（特别是受拓扑保护的概率密度分布）可能会在称为重定位灾难的特定进化场景中分叉。显式构造低束缚态 ψ ( x , y , … ) ] 的必要性通过将注意力限制在 V ( x , y , … ) 的个体最小值被很好地分离的动力学状态的限制来规避，模拟量子点（又名人工分子）耦合系统的谐振子阱局部近似电位。随后有人认为，可测量的特征（特别是受拓扑保护的概率密度分布）可能会在称为重定位灾难的特定进化场景中分叉。电势由模拟量子点耦合系统（又名人工分子）的谐振子阱局部逼近。随后有人认为，可测量的特征（特别是受拓扑保护的概率密度分布）可能会在称为重定位灾难的特定进化场景中分叉。电势由模拟量子点耦合系统（又名人工分子）的谐振子阱局部逼近。随后有人认为，可测量的特征（特别是受拓扑保护的概率密度分布）可能会在称为重定位灾难的特定进化场景中分叉。