Time operators of harmonic oscillators and their representations

A time operator T̂ϵ of the one-dimensional harmonic oscillator ĥϵ=12(p2+ϵq2) is rigorously constructed. It is formally expressed as T̂ϵ=121ϵ(arctan(ϵt̂0)+arctan(ϵt̂1)) with t̂0=p−1q and t̂1=qp−1. It is shown that the canonical commutation relation [hϵ,T̂ϵ]=−i1 holds true on a dense domain in the sense of sesqui-linear forms, and the limit of T̂ϵ as ϵ → 0 is shown. Finally a matrix representation of T̂ϵ and its analytic continuation are given.