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shouldn't be the quantization formula (in one dimension) equal to

$ N_{smooth}(E)+N_{osc}(E) = \oint_{C}p.dq $ ??

where the Oscillating term is just the correction from Gutzwiller trace formula or a sum over Orbits

why is just the oscillating term ignored ..

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can you justify? The traditional higher order corrections are simply fractions of integers on the left hand side which correspond to reflection phases for reflecting orbits. –  Ron Maimon Apr 8 '12 at 20:27
    
i was more interested in the inverse problem ¨can we fromthe smooth and oscillating part of the eigenvalue staircase obtaine the potential $ N(_{osc}(E)+N_{smooth}(E)=A \int_{0}^{a(E)} dx (E-V(x)^{1/2} $ in this case the inverse of the potential$ V^{-1}(x)$ should depend ¿even in the semiclassical approximation on the SMOOTH and OSCILLATING part of the eingenvalue counting function –  Jose Javier Garcia Apr 8 '12 at 21:17
    
: Then you are asking about the problem of reconstructing the potential from the energy levels, which is impossible without symmetry conditions. This was asked here before... –  Ron Maimon Apr 8 '12 at 22:43
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