# Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S{sub 3} symmetry

## Abstract

We study the inequivalent quantizations of the N=3 Calogero model by separation of variables, in which the model decomposes into the angular and the radial parts. Our inequivalent quantizations respect the 'mirror-S{sub 3}' invariance (which realizes the symmetry under the cyclic permutations of the particles) and the scale invariance in the limit of vanishing harmonic potential. We find a two-parameter family of novel quantizations in the angular part and classify the eigenstates in terms of the irreducible representations of the S{sub 3} group. The scale invariance restricts the quantization in the radial part uniquely, except for the eigenstates coupled to the lowest two angular levels for which two types of boundary conditions are allowed independently from all upper levels. It is also found that the eigenvalues corresponding to the singlet representations of the S{sub 3} are universal (parameter-independent) in the family, whereas those corresponding to the doublets of the S{sub 3} are dependent on one of the parameters. These properties are shown to be a consequence of the spectral preserving SU(2) (or its subgroup U(1)) transformations allowed in the family of inequivalent quantizations.

- Authors:

- Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 (Japan)

- Publication Date:

- OSTI Identifier:
- 20768706

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 1; Other Information: DOI: 10.1063/1.2162821; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUNDARY CONDITIONS; EIGENSTATES; EIGENVALUES; GROUP THEORY; HARMONIC POTENTIAL; IRREDUCIBLE REPRESENTATIONS; QUANTIZATION; SCALE INVARIANCE; SU-2 GROUPS; SYMMETRY; TRANSFORMATIONS; U-1 GROUPS

### Citation Formats

```
Yonezawa, Nobuhiro, and Tsutsui, Izumi.
```*Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S{sub 3} symmetry*. United States: N. p., 2006.
Web. doi:10.1063/1.2162821.

```
Yonezawa, Nobuhiro, & Tsutsui, Izumi.
```*Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S{sub 3} symmetry*. United States. doi:10.1063/1.2162821.

```
Yonezawa, Nobuhiro, and Tsutsui, Izumi. Sun .
"Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S{sub 3} symmetry". United States.
doi:10.1063/1.2162821.
```

```
@article{osti_20768706,
```

title = {Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S{sub 3} symmetry},

author = {Yonezawa, Nobuhiro and Tsutsui, Izumi},

abstractNote = {We study the inequivalent quantizations of the N=3 Calogero model by separation of variables, in which the model decomposes into the angular and the radial parts. Our inequivalent quantizations respect the 'mirror-S{sub 3}' invariance (which realizes the symmetry under the cyclic permutations of the particles) and the scale invariance in the limit of vanishing harmonic potential. We find a two-parameter family of novel quantizations in the angular part and classify the eigenstates in terms of the irreducible representations of the S{sub 3} group. The scale invariance restricts the quantization in the radial part uniquely, except for the eigenstates coupled to the lowest two angular levels for which two types of boundary conditions are allowed independently from all upper levels. It is also found that the eigenvalues corresponding to the singlet representations of the S{sub 3} are universal (parameter-independent) in the family, whereas those corresponding to the doublets of the S{sub 3} are dependent on one of the parameters. These properties are shown to be a consequence of the spectral preserving SU(2) (or its subgroup U(1)) transformations allowed in the family of inequivalent quantizations.},

doi = {10.1063/1.2162821},

journal = {Journal of Mathematical Physics},

number = 1,

volume = 47,

place = {United States},

year = {Sun Jan 15 00:00:00 EST 2006},

month = {Sun Jan 15 00:00:00 EST 2006}

}