Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schr\"odinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that...
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Published in | arXiv.org |
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Main Authors | , , |
Format | Paper Journal Article |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
27.10.2017
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Subjects | |
Online Access | Get full text |
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Summary: | We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schr\"odinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part \(S(\theta)\) does not satisfy any linear differential equation. In this case it satisfies a nonlinear ODE that has the Painlevé property and its solutions can be expressed in terms of the Painlevé transcendent \(P_6\). We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case. |
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Bibliography: | Volume 50, Number 49 |
ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.1706.08655 |