Conjugate Operators of 1D-harmonic Oscillator

2024/04/18 by Fumio Hiroshima, Hiroshima, Fumio, Noriaki Teranishi +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Elasticity and Wave Propagation #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2404.12286

openalex publication_date 2024/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=(i)/(m)log(ω\one-L^m) with (ω, m)∈ \DD× (\NN∖\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\∪\sT_\DD∖\0\∪ \sT_{∂ \DD}, where \TA∈\sT_\0\ and \TG∈ \sT_{∂ \DD}. The classification is specified by a pair of parameters (\om,m)∈\CC×\NN. Finally the time evolution T_\om,m(t)=e^itN T_\om,me^-itN for T_\om,m∈\sT is investigated, and it is shown that T_\om,m(t) is periodic with respect to~t.

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