= Solution
For the <WKB approximation>, put $Q(x)=|x|-E$. Away from the <turning points> $x=\pm E$, the forbidden-region solutions have the form $Q^{-1/4}\exp(\pm\int\sqrt Q\,dx)$; in the allowed region put $k(x)=\sqrt{E-|x|}$ and use $k^{-1/2}\exp(\pm i\int k\,dx)$. The decaying tails selected by the <boundary conditions> are
$$
y\sim C_+(x-E)^{-1/4}e^{-\frac23(x-E)^{3/2}}\ (x>E),\qquad
y\sim C_-(-x-E)^{-1/4}e^{-\frac23(-x-E)^{3/2}}\ (x<-E).
$$
The simple-turning-point Airy connection formula matches these to
$$
2C_+k(x)^{-1/2}\sin\left(\int_x^E k(s)ds+\frac\pi4\right),\qquad
2C_-k(x)^{-1/2}\sin\left(\int_{-E}^x k(s)ds+\frac\pi4\right).
$$
Compatibility requires the <Bohr-Sommerfeld quantization> condition
$$
\int_{-E}^E\sqrt{E-|x|}\,dx=\frac43E^{3/2}\sim\pi(n+1/2),
$$
with levels indexed from $n=0$. Therefore
$$
\boxed{E_n\sim\left[\frac{3\pi}{4}(n+1/2)\right]^{2/3},\qquad E_n=O(n^{2/3}).}
$$
The cusp at zero does not alter the leading high-energy count. An exact matching check is available: on $x\ge0$ the decaying solution is $\operatorname{Ai}(x-E)$, and parity at zero imposes $\operatorname{Ai}'(-E)=0$ for even states or $\operatorname{Ai}(-E)=0$ for odd states. Their interlacing large negative zeros give the same combined leading quantization. Reindexing from one changes only the constant phase shift, not the requested growth law.
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