Conjugate Operators of 1D-harmonic Oscillator
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909927198425088 |
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| author | Hiroshima, Fumio Teranishi, Noriaki |
| author_facet | Hiroshima, Fumio Teranishi, Noriaki |
| contents | A conjugate operator $T$ of one-dimensional harmonic oscillator $N$ is defined by an operator satisfying canonical commutation relation $[N,T]=-i\one$ on some domain but not necessarily a dense one. Examples of conjugate operators include the angle operator $\TA$ and the Galapon operator $\TG$. Let $\sT$ denote a set of conjugate operators of $N$ of the form $T_{ω,m}=\frac{i}{m}\log(ω\one-L^m)$ with $(ω, m)\in \overline{\DD}\times (\NN\setminus\{0\})$, where $L$ is a shift operator and $\DD$ denotes the open unit disc in the complex plane $\CC$. A classification of $\sT$ is given as $\sT=\sT_{\{0\}}\cup\sT_{\DD\setminus\{0\}}\cup \sT_{\partial \DD}$, where $\TA\in\sT_{\{0\}}$ and $\TG\in \sT_{\partial \DD}$. The classification is specified by a pair of parameters $(\om,m)\in\CC\times\NN$. Finally the time evolution $T_{\om,m}(t)=e^{itN} T_{\om,m}e^{-itN}$ for $T_{\om,m}\in\sT$ is investigated, and it is shown that $T_{\om,m}(t)$ is periodic with respect to~$t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conjugate Operators of 1D-harmonic Oscillator Hiroshima, Fumio Teranishi, Noriaki Mathematical Physics A conjugate operator $T$ of one-dimensional harmonic oscillator $N$ is defined by an operator satisfying canonical commutation relation $[N,T]=-i\one$ on some domain but not necessarily a dense one. Examples of conjugate operators include the angle operator $\TA$ and the Galapon operator $\TG$. Let $\sT$ denote a set of conjugate operators of $N$ of the form $T_{ω,m}=\frac{i}{m}\log(ω\one-L^m)$ with $(ω, m)\in \overline{\DD}\times (\NN\setminus\{0\})$, where $L$ is a shift operator and $\DD$ denotes the open unit disc in the complex plane $\CC$. A classification of $\sT$ is given as $\sT=\sT_{\{0\}}\cup\sT_{\DD\setminus\{0\}}\cup \sT_{\partial \DD}$, where $\TA\in\sT_{\{0\}}$ and $\TG\in \sT_{\partial \DD}$. The classification is specified by a pair of parameters $(\om,m)\in\CC\times\NN$. Finally the time evolution $T_{\om,m}(t)=e^{itN} T_{\om,m}e^{-itN}$ for $T_{\om,m}\in\sT$ is investigated, and it is shown that $T_{\om,m}(t)$ is periodic with respect to~$t$. |
| title | Conjugate Operators of 1D-harmonic Oscillator |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2404.12286 |