= Bessel transition for an exponentially decaying oscillator
{c}
{title2=$T=\varepsilon t-\log(\varepsilon^{-1})$}
An oscillator with frequency $\omega(t)=e^{-\varepsilon t}$ leaves the <WKB approximation> when $|\omega_t|/\omega^2=\varepsilon e^{\varepsilon t}$ becomes order one. The shifted variable $T=\varepsilon t-\log(\varepsilon^{-1})$ changes its leading undamped equation to $Y_{TT}+e^{-2T}Y=0$. With $Z=e^{-T}$ this is the order-zero <Bessel differential equation>. Large-$Z$ <Bessel functions> match the earlier oscillations, while the small-$Z$ constant and logarithmic solutions become a constant and a linear function of late time.
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