Arnold's potentials and quantum catastrophes

TL;DR

This paper explores how classical Arnold potentials influence quantum catastrophes, focusing on relocalization phenomena in Schrödinger equations with polynomial confining potentials.

Contribution

It systematically classifies quantum relocalization catastrophes arising from Arnold's polynomial potentials in the Schrödinger equation.

Findings

Relocalization of quantum particle density between minima due to tunneling.

Classification of relocalization catastrophes for even Arnold potentials.

Insights into the survival of classical features in quantum systems with polynomial potentials.

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 $V_{(k)}(x)= x^{k+1} + c_1x^{k-1} + \ldots$. 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 Schr\"{o}dinger 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.