Convolutions of semi-classical spectral distributions and periodic-orbit theory

We study the convolution of semi-classical spectral distributions associated to h-pseudodifferential operators on Rn. Under standard assumptions the micro-support of this object can be characterized via families of periodic orbits correlated simultaneously by energy and periods. When all the orbits are non-degenerate the convolution admits, as h tends to 0, an explicit asymptotic expansion in term of the respective dynamical systems. In this setting, this result validates the theory of orbits pairs used by physicists in quantum chaos. Some new contributions, related to the crossing of the period functions, are also analyzed.