Differential equations to compute h̄ corrections of the trace formula Gábor

In this paper a new method for computation of higher order corrections to the saddle point approximation of the Feynman path integral is introduced. The saddle point approximation leads to local Schrödinger problems around classical orbits. Especially, the saddle point approximation leads to Schrödinger problems around classical periodic orbits when it is applied to the trace of Green’s function. These local Schrödinger 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 h̄. 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, spectral determinant and trace formula. Feynman’s path integral in quantum mechanics[1] and similar path integrals for stochastic systems are the most intuitive tools of modern theoretical physics. But calculations with path integral are difficult[2] in general. One can often find numerical solution of the underlying partial differential equation easier. The most convenient asymptotic method to evaluate the path