The time-dependent harmonic oscillator revisited

Abstract

We re-examine the time-dependent harmonic oscillator q = −ω 2 q under various regularity assumptions. Where ω(t) is continuously differentiable we reduce its integration to that of a single first order equation, i.e. the Hamilton equation for the angle variable ψ alone (the action variable I does not appear). Reformulating the generic Cauchy problem for ψ(t) as a Volterra-type integral equation and applying the fixed point theorem we obtain a sequence {ψ (h) } h∈N 0 converging uniformly to ψ in every compact time interval; if ω varies slowly or little, already ψ (0) approximates ψ well for rather long time lapses. The discontinuities of ω, if any, determine those of ψ, I. The zeroes of q, q are investigated with the help of two Riccati equations. As a demo of the potential of our approach, we briefly argue that it yields good approximations of the solutions (as well as exact bounds on them) and thereby may simplify the study of: the stability of the trivial one; the occurrence of parametric resonance when ω(t) is periodic; the adiabatic invariance of I; the asymptotic expansions in a slow time parameter ε; etc.

On the impact of short laser pulses on cold diluted plasmas

“…The proof is in appendix 9.3. For small ξ ( Ĵa , σa ) is a good approximation of ( Ĵ, σ) expressed in closed form in terms of the family of solutions of (29-30); actually, it is the first element of a sequence of approximations of ( Ĵ, σ) that can be determined using the method of [43].…”

“…We briefly report here some properties of the 'time-dependent' harmonic oscillator studied in [43]. By Proposition 1 in [43], every nontrivial solution of the equation…”

“…We apply the results [43] recalled in section 9.2 to the solution D(y,y 0 ) ≡ q(y 0 ) of ( 121), with y 0 playing the role of 'time' x. We denote by {y n (y)} n∈Z the sequence of zeroes associated via Proposition 1 of [43], and by Ǐ, ψ the corresponding action, angle variables.…”

On the breakdown of the perturbative interaction picture in Big Crunch/Big Bang or the true reason why perturbative string amplitudes on temporal orbifolds diverge

“…We will follows essentially Tseytlin et at [37] which refers to [38] but we will be careful in distinguish between Heisenberg and Schrodinger representation and this should make things more clear. For a newer point of view on the problem see also [39].…”

On the breakdown of the perturbative interaction picture in Big Crunch/Big Bang or the true reason why perturbative string amplitudes on temporal orbifolds diverge

“…We will follows essentially Tseytlin et at [45] which refers to [46] but we will be careful in distinguish between Heisenberg and Schrodinger representation and this should make things more clear. For a newer point of view on the problem see also [47].…”