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 continuous, E ∊ {–∞, ∞...
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| Дата: | 2009 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2009
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149243 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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| id |
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irk-123456789-1492432019-02-20T01:27:59Z Comments on the Dynamics of the Pais-Uhlenbeck Oscillator Smilga, A.V. 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. 2009 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149243 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Smilga, A.V. |
| spellingShingle |
Smilga, A.V. Comments on the Dynamics of the Pais-Uhlenbeck Oscillator Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Smilga, A.V. |
| author_sort |
Smilga, A.V. |
| title |
Comments on the Dynamics of the Pais-Uhlenbeck Oscillator |
| title_short |
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_sort |
comments on the dynamics of the pais-uhlenbeck oscillator |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149243 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:32:32Z |
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2023-05-20T17:32:32Z |
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