Action integral for a linear-to-square-root potential
= Action integral for a linear-to-square-root potential
For the even potential
$$
V(x)=\begin{cases}|x|/4,&|x|\leq4,\\
\sqrt{|x|}-1,&|x|\geq4,
\end{cases}
$$
the turning points are $\pm4E$ for $0\leq E\leq1$ and $\pm(E+1)^2$ for $E\geq1$. Its action is
$$
f(E)=\begin{cases}
\dfrac{16}{3}E^{3/2},&0\leq E\leq1,\\[4pt]
\dfrac{16}{3}E^{3/2}+\dfrac{16}{15}(E-1)^{5/2},&E\geq1.
\end{cases}
$$