Bessel transition for an exponentially decaying oscillator ($T=\epsilon t-\log ({\epsilon }^{-1})$)

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.

For $y_{tt}+e^{-2\epsilon t}(\epsilon y_t+y)=0$, $y(0)=0$, $y_t(0)=1$, set $r=e^{-2\epsilon t}/2$. The exact equation becomes $r{y}_{rr}+(1-r)y_r+y/(2{\epsilon }^2)=0$. Let $m(t)=M(-1/(2{\epsilon }^2),1,r)$, using the Kummer function. It is regular at $r=0$ and tends to one. The Wronskian $W=m{y}_t-m_{t}y$ obeys $W_t=-\epsilon e^{-2\epsilon t}W$, with $W(0)=M(-1/(2{\epsilon }^2),1,1/2)$. Integration gives the displayed exact late slope. The other local solution grows at most logarithmically in $r$, so $m_{t}y\to 0$ and $W\to y_t$. This provides a check on a WKB approximation even near phases where its leading predicted slope vanishes.