Conjugate Operators of 1D-harmonic Oscillator

Conjugate operator T of 1D-harmonic oscillator N=1/2 (p²+q²-1) is defined by an operator satisfying canonical commutation relation $[N,T]=-i$ on some domain but not necessarily dense. The angle operator TA=1/2 ( q⁻¹ p+ pq⁻¹ ) and Galapon operator TG=i ∑ₙ₌₀^∞(∑m≠ n(vₘ,·) /n-mvₙ) are examples of conjugate operators, where vₙ denotes the set of normalized eigenvectors of N. Let T be a subset of conjugate operators of N. A classification of T are given as T=T_\0\∪ T_D \0\∪ T∂ D, and TA∈ T_\0\ and TG∈ T∂ D are shown. Here the classification is specified by a pair of parameters (ω,m)∈ C× N. Finally the time evolution Tω,m(t)=eitN Tω,me-itN for Tω,m∈ T is investigated, and show that Tω,m(t) is periodic.