On the Analytic Properties of the Perturbing Function in the PCR3Body Problem

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 $$c>0$$ .