$\hbar$ corrections in semi-classical formula for smooth chaotic dynamics

The validity of semiclassical expansions in the power of $\hbar$ for the quantum Green's function have been extensively tested for billiards systems, but in the case of chaotic dynamics with smooth potential, even if formula are existing, a quantitative comparison is still missing. In this paper, extending the theory developed by Gaspard et al., Adv. Chem. Phys. XC 105 (1995), based on the classical Green's functions, we present an efficient method allowing the calculation of $\hbar$ corrections for the propagator, the quantum Green's function, and their traces. Especially, we show that the previously published expressions for $\hbar$ corrections to the traces are incomplete.