understanding the oscillating part of the Gutzwiller trace

Question

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 ??

Answer 1

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...