A trace formula for the semiclassical limit of some Hermitian operators

The aim of this paper is to provide an algorithm for calculating the leading order contributions for the semiclassical series of a functional of trace-class Hermitian operators. We base our study on the linear space structure of the space of bounded Hermitian operators on quantum Hilbert space. We work with a basis of operators with natural leading order Weyl symbols, performing stationary phase calculations with no assumption on their smoothness. The stationary points are tilted orbits near the periodic ones in phase space. We manage to get control of the error in the technique, which scales like some power a of 1/N, with 1/2 < a ≤ 1. The calculations are directly applicable to quantum maps, and we provide some examples in finite dimension with quantum perturbed cat maps