A Galton-Watson estimate for Dyson series

We consider the question of convergence of particular series of integrals, which are labeled by rooted trees.

Necessary and sufficient criteria for convergence are obtained, together with an explicit expression for the sum.

The technique used is strongly reminiscent of the generating function approach of Galton and Watson to branching processes.

The interest in these series derives from the Dyson series expansion for the perturbation of a free quantum dynamics by a local potential: the convergence of the series imlies that the perturbed dynamics exists and is unitarily equivalent with the free one.