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 $[N,T]=-i\text{one}$ on some domain but not necessarily a dense one. Examples of conjugate operators include the angle operator $\text{TA}$ and the Galapon operator $\text{TG}$. Let $\text{sT}$ denote a set of conjugate operators of $N$ of the form $T_{\omega ,m}=\frac{i}{m}\log (\omega \text{one}-L^m)$ with $(\omega ,m)\in \overline{\text{DD}}\times (\text{NN}\setminus \{0\})$, where $L$ is a shift operator and $\text{DD}$ denotes the open unit disc in the complex plane $\text{CC}$. A classification of $\text{sT}$ is given as $\text{sT}={\text{sT}}_{\{0\}}\cup {\text{sT}}_{\text{DD}\setminus \{0\}}\cup {\text{sT}}_{\mathrm{\partial }\text{DD}}$, where $\text{TA}\in {\text{sT}}_{\{0\}}$ and $\text{TG}\in {\text{sT}}_{\mathrm{\partial }\text{DD}}$. The classification is specified by a pair of parameters $(\text{om},m)\in \text{CC}\times \text{NN}$. Finally the time evolution $T_{\text{om},m}(t)=e^{itN}{T}_{\text{om},m}{e}^{-itN}$ for $T_{\text{om},m}\in \text{sT}$ is investigated, and it is shown that $T_{\text{om},m}(t)$ is periodic with respect to~$t$.