# understanding the oscillating part of the Gutzwiller trace

given the density of states according to Gutzwiller's trace formula

$g(E)= g_{smooth}(E)+ g_{osc}(E)$

i know that the 'smooth' part comes from $g_{smooth}(E)= \iint dxdp \delta(E-p^{2}-V(x))$ for one dimensional system

however how it is the oscillating part of the trace obtained ?? :D i mean the sum over lenghts of the orbit (in the phase space)

also how does the condition for WKB energies appear ?? $\oint _{C} p.dq= 2\pi \hbar (n+ \alpha)$ from the Gutzwiller trace ??

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The oscillatory part is nothing but Thomas-Fermi approximation or more riguresly, this is a version (someone should correct me if I am wrong) Weyl's formula

Regrading on how to obtain the WKB from the trace formula: You can read the 2 papers by Berry and Tabor on how they derived a trace formula (like that of Gutzwiller) but to the case of integrable systems. From the derivation there you can see how the EBK pop up...

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