= Solution
The transition equation reduces to the <Bessel differential equation> because $d/dT=-Z\,d/dZ$ and $d^2/dT^2=Z^2d^2/dZ^2+Z\,d/dZ$. Thus $Y_{TT}+e^{-2T}Y=0$ becomes $Z^2Y_{ZZ}+ZY_Z+Z^2Y=0$.
The <Bessel function of the first kind> has $J_0(Z)=1+O(Z^2)$ at zero, while the <Bessel function of the second kind> has $Y_0(Z)=(2/\pi)(\log(Z/2)+\gamma)+O(Z^2|\log Z|)$. Since $\log Z=-T$, these yield a constant and a linear function of $T$, explaining the late-time behavior. At large positive $Z$,
$$
J_0(Z)\sim\sqrt{\frac2{\pi Z}}\cos(Z-\pi/4),\qquad
Y_0(Z)\sim\sqrt{\frac2{\pi Z}}\sin(Z-\pi/4).
$$
Matching $A_0\cos(Z-\pi/4)+B_0\sin(Z-\pi/4)$ to $\sqrt{\pi/2}\sin(\varepsilon^{-1}-Z)$ fixes $A_0=\sqrt{\pi/2}\sin\theta$ and $B_0=-\sqrt{\pi/2}\cos\theta$. The large- and small-argument <asymptotic expansions> thereby connect the oscillatory <WKB approximation> to the nonoscillatory late-time solution through one <Bessel transition for an exponentially decaying oscillator>.
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