Abstract
The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.
Highlights
The friction can be eliminated by setting x (t) = y(t) e−λ(t)/2, which yields a harmonic oscillator with no friction, but with a shifted frequency [29,30,31], ÿ + Ω2 (t)y = 0 where
The Junker–Inomata method of converting the time-dependent system into one with a constant frequency by switching from “real” to “fake time”, t → τ ( t ), ξ = τx induces a conformal transformation between the Bargmann metrics: dx2 + 2dtds − ω 2 (t) x2 dt2 frequency
K1 is in turn the Maslov-extended propagator of an oscillator with no friction and a constant frequency, as in (62)
Summary
Starting with a general quadratic Lagrangian in 1 + 1 spacetime dimensions with coordinates xand t, Junker and Inomata derived the equation of motion [12]:. The friction can be eliminated by setting x (t) = y(t) e−λ(t)/2 , which yields a harmonic oscillator with no friction, but with a shifted frequency [29,30,31],. For λ(t) = λ0 t and ω = ω0 = const., for example, we obtain the usual harmonic oscillator with a constant shifted frequency, Ω2 = ω02 − λ20 /4 = const. The frequency is in general time-dependent, though Ω = Ω(t); (14) is a. ≥ 0, Equation (12) describes a time-dependent oscillator with constant friction,. In addition, the frequency is constant ω (t) = ω0 = const., Equation (26) is solved algebraically by:.
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