Differential equations to compute $\hbar$ corrections of the trace formula

In this paper a new method for computation of higher order corrections to the saddle point approximation of the Feynman path integral is discussed. The saddle point approximation leads to local Schr\"odinger problems around classical orbits. Especially, the saddle point approximation leads to Schr\"odinger problems around classical periodic orbits when it is applied to the trace of Green's function. These local Schr\"odinger problems, in semiclassical approximation, can be solved exactly on the basis of local analytic functions. Then the corrections of the semiclassical result can be treated perturbatively. The strength of the perturbation is proportional to $\hbar$. The perturbation problem leads to ordinary differential equations. We propose these equations for numerical calculation of corrections, since they can easily be solved by computers. We give quantum mechanical generalizations of the semiclassical zeta functions. Two simple examples are included in order to demonstrate the effectiveness of the method.