Arnold's potentials and quantum catastrophes

Abstract

In the well known Thom's classification, every classical catastrophe is assigned a Lyapunov function. In the one-dimensional case, due to V. I. Arnold, these functions have polynomial form $V_{(k)}(x)=x^{k+1}+c_1{x}^{k-1}+\dots$. A natural question is which features of the theory survive when such a function (say, with an even value of asymptotically dominant exponent $k+1$) is used as a confining potential in Schrödinger equation. A few answers are formulated. Firstly, it is clarified that due to the tunneling, one of the possible classes of the measurable quantum catastrophes may be sought in a phenomenon of relocalization of the dominant part of the quantum particle density between different minima. For the sake of definiteness we just consider the spatially even Arnold's potentials and in the limit of the thick barriers and deep valleys we arrive at a systematic classification of the corresponding relocalization catastrophes.

31 pages, 6 figures

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Cited by in corpus (9)

• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Excepional Points from Hamiltonians of hybrid physical systems: Squeezing and anti-Squeezing
• Paths of unitary access to exceptional points
• Displaced harmonic oscillator $V\sim \min [(x+d{)}^2,(x-d{)}^2]$ as a benchmark double-well quantum model
• Relocalization switch in a triple quantum dot molecule in 2D
• Polynomial potentials and coupled quantum dots in two and three dimensions
• Application of the cranking method to the semimicroscopic algebraic cluster model and nuclear molecules
• Arnold's potentials and quantum catastrophes II
• Avoided level crossings in quasi-exact approach