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Action integral for a linear-to-square-root potential

For the even potential

V (x) = {∣ x ∣/4, ∣ x ∣ - 1, ∣ x ∣ \leq 4, ∣ x ∣ \geq 4,

the turning points are

\pm 4 E

for

0 \leq E \leq 1

and

\pm (E + 1)^{2}

for

E \geq 1

. Its action is

f (E) = ⎩ ⎨ ⎧ \frac{16}{3} E^{3/2}, \frac{16}{3} E^{3/2} + \frac{16}{15} (E - 1)^{5/2}, 0 \leq E \leq 1, E \geq 1.

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