ħ corrections in semiclassical formulas for smooth chaotic dynamics

The validity of semiclassical expansions in the power of $ħ$ 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 formulas are existing, a quantitative comparison is still missing. In this paper, extending the theory developed by Gaspard et al. [Adv. Chem. Phys. 90, 105 (1995)], based on the classical Green’s functions, we present an efficient method allowing the calculation of $ħ$ corrections for the propagator, the quantum Green’s function, and their traces. In particular, we show that the previously published expressions for $ħ$ corrections to the traces are incomplete.