Hyperbolic Low-Dimensional Invariant Tori¶and Summations of Divergent Series
Hyperbolic Low-Dimensional Invariant Tori¶and Summations of Divergent Series
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DOI：
10.1007/s002200200599
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发表时间：
2001-07
影响因子：
2.4
通讯作者：
G. Gallavotti;G. Gentile
中科院分区：
文献类型：
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作者：
G. Gallavotti;G. Gentile
考虑一类先验稳定的拟可积解析哈密顿系统，研究了低维双曲不变环面作为扰动参数的函数的正则性。我们证明，在自然非共振条件下，这样的环面是存在的，并且可以通过适当势的极大值或极小值来识别。它们是在圆盘内部解析的，以原点为中心，剥夺了原点具有二次尖点的正或负实轴周围的区域。不变环面允许在原点具有泰勒系数的渐近级数，该级数至多作为阶乘的幂增长，而对于任何阶数N的余数由自变量的第(N+1)次方乘以n的次方！遵循量子场论中微扰级数恢复的重整化群方法，证明了表示环面参数方程的(一般发散的)级数按微扰大小的幂的求和准则的存在性。
We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coefficients that grow at most as a power of a factorial and a remainder that to any orderNis bounded by the (N+1)-st power of the argument times a power ofN!. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory.