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Suppose we are given an ODE problem $$ y''(x)+V(x)f(x)=E_{n} y(x) $$ with boundary conditions $ y(0)=y(\infty)=0$. Here $V(x)$ is a potential function.

Then is it always true that (for $n \gg 1$) $$ E_{n}^{\rm WKB} \sim E_{n} $$

where WKB means the eigenvalues evaluated via the WKB approach?

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And what is the question? Also, isn't the equality of energies suppose to be just approximative (i.e. $\sim$ instead of $=$)? – Marek Jul 31 '11 at 13:11
I think he is asking whether the WKB bound state energy for the nth bound state converges to the true energy in the limit $n\rightarrow\infty$. – BebopButUnsteady Jul 31 '11 at 22:00
yes, that is as Bebop said, i would like to know if for big big 'n' the approximate energies would converge to the exact ones – Jose Javier Garcia Aug 1 '11 at 8:44

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