By the work of Siegel it is well known that as a rule the Birkhoff normal form of a real analytic Hamiltonian system whose eigenvalues satisfies suitable non-resonance condition cannot be realized by convergent symplectic transformations. We show the existence of divergent Birkhoff normal forms for suitable Hamiltonian systems. Our calculation shows how the small divisors appear in the normal forms, from which the divergence is derived by using Siegel’s methods of small divisors.
"Existence of divergent Birkhoff normal forms of Hamiltonian functions." Illinois J. Math. 56 (1) 85 - 94, Spring 2012. https://doi.org/10.1215/ijm/1380287461