Semiclassical action integral ($f(E)$)

The semiclassical action integralis strictly increasing for a confining one-dimensional potential. Consequently the two-turning-point WKB quantization condition has at most one energy at each quantum number.

For the even potentialthe turning points are $\pm 4E$ for $0\le E\le 1$ and $\pm (E+1{)}^2$ for $E\ge 1$. Its action is