Conjugate Operators of 1D-harmonic Oscillator

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 $T_{\omega,m}=\frac{i}{m}\log(\omega\one-L^m)$ with $(\omega, m)\in \overline{\DD}\times (\NN\setminus\{0\})$, 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~$t$.