 Bessel transition for an exponentially decaying oscillator

An oscillator with frequency $\omega (t)=e^{-\epsilon t}$ leaves the WKB approximation when $|{\omega }_t|/{\omega }^2=\epsilon e^{\epsilon t}$ becomes order one. The shifted variable $T=\epsilon t-\log ({\epsilon }^{-1})$ 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.