The ₂ Harmonic Oscillator with Reflections and Superintegrability
The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group ₂, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2023 |
| Main Author: | |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211918 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The ₂ Harmonic Oscillator with Reflections and Superintegrability. Charles F. Dunkl. SIGMA 19 (2023), 025, 18 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862692694858399744 |
|---|---|
| author | Dunkl, Charles F. |
| author_facet | Dunkl, Charles F. |
| citation_txt | The ₂ Harmonic Oscillator with Reflections and Superintegrability. Charles F. Dunkl. SIGMA 19 (2023), 025, 18 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group ₂, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated with finite reflection groups and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement.
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| first_indexed | 2026-03-17T23:35:14Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211918 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T23:35:14Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dunkl, Charles F. 2026-01-16T11:19:30Z 2023 The ₂ Harmonic Oscillator with Reflections and Superintegrability. Charles F. Dunkl. SIGMA 19 (2023), 025, 18 pages 1815-0659 2020 Mathematics Subject Classification: 81R12; 37J35; 33C45; 81Q05 arXiv:2210.14180 https://nasplib.isofts.kiev.ua/handle/123456789/211918 https://doi.org/10.3842/SIGMA.2023.025 The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group ₂, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated with finite reflection groups and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement. The author presented some of the material in a plenary lecture at the 34th International Colloquium on Group Theoretical Methods in Physics, Strasbourg, France, July 18–22, 2022. Also, the author thanks the referees for their careful reading and constructive suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The ₂ Harmonic Oscillator with Reflections and Superintegrability Article published earlier |
| spellingShingle | The ₂ Harmonic Oscillator with Reflections and Superintegrability Dunkl, Charles F. |
| title | The ₂ Harmonic Oscillator with Reflections and Superintegrability |
| title_full | The ₂ Harmonic Oscillator with Reflections and Superintegrability |
| title_fullStr | The ₂ Harmonic Oscillator with Reflections and Superintegrability |
| title_full_unstemmed | The ₂ Harmonic Oscillator with Reflections and Superintegrability |
| title_short | The ₂ Harmonic Oscillator with Reflections and Superintegrability |
| title_sort | ₂ harmonic oscillator with reflections and superintegrability |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211918 |
| work_keys_str_mv | AT dunklcharlesf the2harmonicoscillatorwithreflectionsandsuperintegrability AT dunklcharlesf 2harmonicoscillatorwithreflectionsandsuperintegrability |