Existence of divergent Birkhoff normal forms of Hamiltonian functions

We show that for $n\ge 2$ 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.