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to solve and obtain the potential of a 1-D Hamiltonian we must solve an integral equation

$$ N(E)= A \int_{0}^{E}\frac{V^{-1}(x)}{\sqrt{E-x}}$$

fro a some constant 'A' , then my question is since the approximated espectral fucntion

$$ N(E)= <N(E)>+ \sum_{p.p.o}A_{p}e^{\frac{iS(E)}{\hbar}}+c.c $$

with $ <N(E)>$ meaning the WKB approximation (smooth) to the spectra staircase and p.p.o means summation over the periodic orbits with $ S(E)= \sqrt{2m} l_{p.o}$ the action over the closed orbits , then can i from the Gutzwiller trace formula recover the potential ? this is my ansatz which i think is valid at least for one dimension

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