# Nondegenerate superintegrable systems in n-dimensional Euclidean spaces

## Abstract

We analyze the concept of a nondegenerate superintegrable system in n-dimensional Euclidean space. Attached to this idea is the notion that every such system affords a separation of variables in one of the various types of generic elliptical coordinates that are possible in complex Euclidean space. An analysis of how these coordinates are arrived at in terms of their expression in terms of Cartesian coordinates is presented in detail. The use of well-defined limiting processes illustrates just how all these systems can be obtained from the most general nondegenerate superintegrable system in n-dimensional Euclidean space. Two examples help with the understanding of how the general results are obtained.

- Authors:

- University of Waikato, Department of Mathematics and Statistics (New Zealand)
- The University of New South Wales, School of Mathematics (Australia), E-mail: j.kress@unsw.edu.au
- University of Minnesota, School of Mathematics (United States)
- Joint Institute of Nuclear Research, Laboratory of Theoretical Physics (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21075916

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Atomic Nuclei; Journal Volume: 70; Journal Issue: 3; Other Information: DOI: 10.1134/S1063778807030143; Copyright (c) 2007 Nauka/Interperiodica; Article Copyright (c) 2007 Pleiades Publishing, Ltd; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CARTESIAN COORDINATES; EUCLIDEAN SPACE; MANY-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Kalnins, E. G., Kress, J. M., Miller, W., and Pogosyan, G. S.
```*Nondegenerate superintegrable systems in n-dimensional Euclidean spaces*. United States: N. p., 2007.
Web. doi:10.1134/S1063778807030143.

```
Kalnins, E. G., Kress, J. M., Miller, W., & Pogosyan, G. S.
```*Nondegenerate superintegrable systems in n-dimensional Euclidean spaces*. United States. doi:10.1134/S1063778807030143.

```
Kalnins, E. G., Kress, J. M., Miller, W., and Pogosyan, G. S. Thu .
"Nondegenerate superintegrable systems in n-dimensional Euclidean spaces". United States.
doi:10.1134/S1063778807030143.
```

```
@article{osti_21075916,
```

title = {Nondegenerate superintegrable systems in n-dimensional Euclidean spaces},

author = {Kalnins, E. G. and Kress, J. M. and Miller, W. and Pogosyan, G. S.},

abstractNote = {We analyze the concept of a nondegenerate superintegrable system in n-dimensional Euclidean space. Attached to this idea is the notion that every such system affords a separation of variables in one of the various types of generic elliptical coordinates that are possible in complex Euclidean space. An analysis of how these coordinates are arrived at in terms of their expression in terms of Cartesian coordinates is presented in detail. The use of well-defined limiting processes illustrates just how all these systems can be obtained from the most general nondegenerate superintegrable system in n-dimensional Euclidean space. Two examples help with the understanding of how the general results are obtained.},

doi = {10.1134/S1063778807030143},

journal = {Physics of Atomic Nuclei},

number = 3,

volume = 70,

place = {United States},

year = {Thu Mar 15 00:00:00 EDT 2007},

month = {Thu Mar 15 00:00:00 EDT 2007}

}

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