Comments on the Dynamics of the Pais-Uhlenbeck Oscillator
We discuss the quantum dynamics of the PU oscillator, i.e. the system with the Lagrangian 
 L = ½ [ ¨q² - (Ω₁² + Ω₂²) ·q² + Ω₁²Ω₂²q ] (+ nonlinear terms). 
 When Ω₁ ≠ Ω₂, the free PU oscillator has a pure point spectrum that is dense everywhere. When Ω₁ = Ω₂, the spectrum is contin...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149243 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator / A.V. Smilga // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728613314428928 |
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| author | Smilga, A.V. |
| author_facet | Smilga, A.V. |
| citation_txt | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator / A.V. Smilga // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We discuss the quantum dynamics of the PU oscillator, i.e. the system with the Lagrangian 
L = ½ [ ¨q² - (Ω₁² + Ω₂²) ·q² + Ω₁²Ω₂²q ] (+ nonlinear terms). 
When Ω₁ ≠ Ω₂, the free PU oscillator has a pure point spectrum that is dense everywhere. When Ω₁ = Ω₂, the spectrum is continuous, E ∊ {–∞, ∞}. The spectrum is not bounded from below, but that is not disastrous as the Hamiltonian is Hermitian and the evolution operator is unitary. Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved. We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates. The free nonstandard PU oscillator is unitary at Ω₁ ≠ Ω₂, but unitarity is broken in the equal frequencies limit.
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| first_indexed | 2025-12-07T19:09:49Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149243 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:09:49Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Smilga, A.V. 2019-02-19T19:16:58Z 2019-02-19T19:16:58Z 2009 Comments on the Dynamics of the Pais-Uhlenbeck Oscillator / A.V. Smilga // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 70H50; 70H14 https://nasplib.isofts.kiev.ua/handle/123456789/149243 We discuss the quantum dynamics of the PU oscillator, i.e. the system with the Lagrangian 
 L = ½ [ ¨q² - (Ω₁² + Ω₂²) ·q² + Ω₁²Ω₂²q ] (+ nonlinear terms). 
 When Ω₁ ≠ Ω₂, the free PU oscillator has a pure point spectrum that is dense everywhere. When Ω₁ = Ω₂, the spectrum is continuous, E ∊ {–∞, ∞}. The spectrum is not bounded from below, but that is not disastrous as the Hamiltonian is Hermitian and the evolution operator is unitary. Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved. We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates. The free nonstandard PU oscillator is unitary at Ω₁ ≠ Ω₂, but unitarity is broken in the equal frequencies limit. This paper is a contribution to the Proceedings of the VIIth Workshop “Quantum Physics with NonHermitian Operators” (June 29 – July 11, 2008, Benasque, Spain). I acknowledge warm hospitality at AEI in Golm, where this work was finished and thank P. Mannheim for useful correspondence. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Comments on the Dynamics of the Pais-Uhlenbeck Oscillator Article published earlier |
| spellingShingle | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator Smilga, A.V. |
| title | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator |
| title_full | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator |
| title_fullStr | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator |
| title_full_unstemmed | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator |
| title_short | Comments on the Dynamics of the Pais-Uhlenbeck Oscillator |
| title_sort | comments on the dynamics of the pais-uhlenbeck oscillator |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149243 |
| work_keys_str_mv | AT smilgaav commentsonthedynamicsofthepaisuhlenbeckoscillator |