Bohr sommerfeld quantiztion rule and Gutzwiller trace

assuming we can evaluate the eigenvalue staircase $N(E)$ in both manners with the Bohr-Sommerfeld quantization rule

$N(E)2\pi \hbar = \oint _{C}p.dq$

and using the Gutzwiller trace $N(E)= N_{smooth} (E) + \delta N_{osc} (E)$

here the oscillating part is given by a sum over orbits in the phase space

are the two formulations equivalent ??

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 Question(v1) seems to be essentially a duplicate of OP's previous questions here and here. – Qmechanic♦ May 26 '12 at 13:04