On the Convergence of Formal Series Containing Small Divisors

Abstract

Poincare—Lindstedt series for the (formal) computation of quasi periodic solutions (in the context of real—analytic, nearly—integrable Hamiltonian dynamical systems) with fixed frequencies have been extensively studied, for over a century, from both the theoretical and applicative point of view. For applications, the Poincare-Lindstedt series provide a simple practical tool to explicitly compute the first few orders of perturbation theory; the problem of convergence has been instead much more controversial (famous is Poincare dubious statement in his Methodes nouvelles de la Mechanique Celeste). The matter was settled indirectly in the sixties thanks to KAM (Kolmogorov, Arnold, Moser) theory. “Indirectly” means that the convergence is obtained as a byproduct of estimates uniform in the smallness parameter rather than directly looking at the formal series and trying to check convergence by studying the rate of growth of coefficients (as in the classical Siegel’s approach to the small divisor problem arising in linearization of germs of complex analytic functions). As it is well known, the main problem with the “direct approach” is that the k th coefficient of the formal power series, if expanded in sums of monomials composed by Fourier coefficients of the Hamiltonian and of “small divisors” (appearing in the denominators of the monomials as linear integer combinations of the basic frequencies), contains, in general, monomials which diverge as k!. Hence, a “direct proof” has necessarily to deal with compensations, i.e., with the problem of grouping together all the “diverging” terms showing that they sum up to much smaller contributions, which can be bounded by a constant to the k th power. Direct proofs (in Hamiltonian setting) were given by H. Eliasson in 1988, by Gallavotti, Gentile and Mastropietro and, independently, by the authors in 1993 (for bibliography and more technical discussions see [1, 2, 3] and references therein).