Marko Robnik, Luca Salasnich
WKB to all orders and the accuracy of the semiclassical quantization
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ABSTRACT. We perform a systematic WKB expansion to all orders
for a one--dimensional system with potential $V(x)=U_0/\cos^2{(\alpha x)}$.
We are able to sum the series to the exact energy spectrum.
Then we show that at any finite order the error of the WKB approximation
measured in {\em the natural units of the mean energy level spacing}
does not go to zero when the quantum number goes to infinity.
Therefore we make the general conclusion that the semiclassical
approximations fail to predict the individual energy levels within a
vanishing fraction of the mean energy level spacing.
(J. Phys. A: Math. Gen. 30, 1711-1718 (1997))