标题： 在PCR3体问题中扰动函数的解析性质 显示英文标题

标题： On the analytic properties of the perturbing function in the PCR3Body Problem

摘要： 我们提供了一种新的展开方式，将PCR3体问题的扰动函数的傅里叶系数表示为Hansen系数的形式。 这为我们提供了在KAM理论应用区域（即偏心率和半长轴取小值的情况，参见例如\cite{Celletti-Chierchia}）中的系数的精确渐近公式。 此外，在上述区域内，我们研究了互质模态$(m,k) \in \Z^2$的傅里叶系数的零点存在性，以及模态$(m,k)$、$(2m,2k)$和$(m,k)$、$(2m,2k)$、$(3m,3k)$对应的系数之间的公共零点的存在性。 得益于之前的展开，此数值分析已进行到偏心率和半长轴的$60$阶。 这是可能将\cite{Singular KAM, BBCZ}应用于 PCR3Body 问题的第一步，这将意味着相空间中所谓的“非环面”集合的测度从$O(1-\sqrt{\e})$（由标准 KAM 理论所暗示）减少到$O(1-\e |\log\e|^c )$，对于某些$c>0$而言。 显示英文摘要

摘要： 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 semi-major axis see e.g. \cite{Celletti-Chierchia}). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \Z^2$ and the presence of common zeros 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 a first step for a possible application of \cite{Singular KAM, BBCZ} to PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so called "non--torus" set from $O(1-\sqrt{\e})$ (implied by standard KAM theory) to $O(1-\e |\log\e|^c )$ for some $c>0$.