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Existence of divergent Birkhoff normal forms of Hamiltonian functions

Xianghong Gong
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Abstract

We show that for $n \geq 2$ there exist real analytic Hamiltonian systems on $\mathbf{R}^{2n}$ with non-resonant eigenvalues at a singular point, of which the Birkhoff normal form itself is divergent. The proof of the result is achieved by analyzing how the small-divisors enter the normal form.

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Xianghong Gong. 2003-07-29. Existence of divergent Birkhoff normal forms of Hamiltonian functions. https://arxiv.org/abs/math/0307379

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