Existence of divergent Birkhoff normal forms of Hamiltonian functions

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.