Hamilton Jacobi equation and divergent series summations

Giovanni Gallavotti University of Rome La Sapienza, Italy

The Hamilton Jacobi equation for resonant motions in quasi integrable Hamiltonian systems (ie analytic integrable systems with an analytic perturbation proportional to a small parameter ε) admits a formal power series solution. It is a conjecture that the series is not convergent: I shall present a resummation method which leads to a convergent series solution of the equation for values of the perturbation parameter for which a solution to the equation can be expected to exist and which form a set with 0 as a density point but with open dense complement. The issue of possible lack of uniqueness is stressed and briefly discussed.