= Solution
Between the <classical turning points>, $q<0$, while boundedness selects exponentially decaying <WKB approximation> solutions in the forbidden regions $x<a$ and $x>b$. The <Airy turning-point connection formula> at the left turning point matches the left decaying solution to
$$
\psi_L(x)\sim C_L|q(x)|^{-1/4}
\sin\left(\frac1\epsilon\int_a^x|q(\xi)|^{1/2}\,d\xi+\frac\pi4\right)
$$
in the allowed region. Matching the right decaying solution across $b$ gives
$$
\psi_R(x)\sim C_R|q(x)|^{-1/4}
\sin\left(\frac1\epsilon\int_x^b|q(\xi)|^{1/2}\,d\xi+\frac\pi4\right).
$$
For these expressions to be proportional throughout $(a,b)$, their phases must be compatible. Each simple turning point contributes a phase shift $\pi/4$, giving the <two-turning-point WKB quantization condition>
$$
\boxed{
\frac1\epsilon\int_a^b|q(\xi)|^{1/2}\,d\xi
=\left(n+\frac12\right)\pi,
\qquad n=0,1,2,\ldots.}
$$
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