Conjugate Operators of 1D-harmonic Oscillator
Fumio Hiroshima, Noriaki Teranishi
TL;DR
The paper develops a rigorous operator framework for conjugate operators of the 1D harmonic oscillator, defining a broad family T_{ω,m} via logarithmic representations and classifying them into three regimes. It connects angle and Galapon operators to shift operators and logarithms, and extends the construction to generalized logarithmic forms using ad_N eigenvectors to enforce canonical commutation on CCR-domains. A comprehensive classification is presented, with T_A and T_G recovered as special cases, and a detailed examination of time evolution showing periodic behavior with period 2π/m. The work advances the mathematical understanding of time observables through ultra-weak canonical relations, offering structured methods to generate and analyze non densely defined conjugate operators and their discrete time evolution.
Abstract
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~$t$.