Davide Zaccaria

In this note, we briefly discuss how the singular KAM theory of [7] — which was worked out for the mechanical case $\frac12 |y|^2+\varepsilon f(x)$ — can be extended to convex real-analytic nearly integrable Hamiltonian systems with Hamiltonian in action-angle variables given by $h(y)+\varepsilon f(x)$ with $h$ convex and $f$ generic.

We provide a new expansion of the Fourier coefficient of the perturbing function of the PCR3Body problem in terms of Hansen coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i.e., small value of eccentricity and semimajor axis. See, e.g., [17]). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \mathbb{Z}^2$ and the presence of common zeros as functions of actions between coefficients relative to modes $(m,k)$,$(2m,2k)$ and $(m,k)$,$(2m,2k)$,$(3m,3k)$. Thanks to the previous expansion, this numerical analysis is done up to order $60$ in the power of eccentricity and semimajor axis. This is the first step for a possible application of [4, 9] to the PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so-called ``non-torus'' set from $O(1-\sqrt{\varepsilon})$ (implied by standard KAM theory) to $O(1-\varepsilon |\log\varepsilon|^c )$ for some $c>0$.