WKB quantization condition

For an oscillatory interval terminated by a hard boundary and a simple turning point, matching the boundary condition to the decaying solution gives an action integral equal to an integer multiple of $\pi$ with the turning-point phase correction.

For two simple turning points $a(E)<b(E)$ surrounding one classically allowed well,

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