Energy evolution in time-dependent harmonic oscillator

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional harmonic oscillator, whose Newton equation $\ddot{q}+{\omega }^2(t)q=0$ cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy $E_0$ at time $t=0$ and calculate rigorously the distribution of energy $E_1$ after time $t=T$, which is fully (all moments, including the variance ${\mu }^2$) determined by the first moment $\overline{E_1}$. For example, ${\mu }^2=E_0^2[(\overline{E_1}\mathrm{/}{E}_0{)}^2-(\omega (T)\mathrm{/}\omega (0){)}^2]\mathrm{/}2$, and all higher even moments are powers of ${\mu }^2$, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function $\omega (t)$ and is in this sense universal. In ideal adiabaticity $\overline{E_1}=\omega (T)E_0\mathrm{/}\omega (0)$, and the variance ${\mu }^2$ is zero, whilst for finite $T$ we calculate $\overline{E_1}$, and ${\mu }^2$ for the general case using exact WKB-theory to all orders. We prove that if $\omega (t)$isofclass${\cal C}^{m}$(allderivativesuptoandincludingtheorder$m$arecontinuous)$\mu \propto T^{-(m+1)}$,whilstforclass${\cal C}^{\infty}$itisknowntobeexponential$\mu \propto \exp (-\alpha T)$.