Existence of Divergent Birkhoff Normal Forms of Hamiltonian Functions

where κ = 0, . . . , n, and λj is pure imaginary precisely when 1 ≤ j ≤ κ, and λ1, −λ1, . . . , λn, −λn are eigenvalues of Hzz(0)J with z = (x, y) and Jxj = yj = −J yj. One says that λ1, . . . , λn are non-resonant, if λ ·α ≡ λ1α1 + · · ·+λnαn 6= 0 for all multi-indices of integers α 6= 0. The Birkhoff normal form says that under the non-resonance condition on λ, there is a formal symplectic transformation of R sending h into ĥ that is a real formal power series in xj + y 2 j (1 ≤ j ≤ κ), (xk + ixl)(yk − iyl) (κ < k, l ≤ n). Notice that, up to the order of λ1, . . . , λn,−λ1, . . . ,−λn, the Birkhoff normal form ĥ is independent of the choice of the normalizing transformations. In [12], Siegel showed that the Birkhoff normal form cannot be realized by convergent symplectic transformations in general. In fact, Siegel [13] showed that when κ = n ≥ 2, for a real analytic function with any prescribed nonresonant λ1, . . . , λn and with generic higher order terms, there exists no convergent normalizing transformation. Despite Siegel’s divergence results and many other results, a basic question, which remains unsettled until now, is if there exists a divergent Birkhoff normal form arising from a real analytic function. This question was pointed out by Eliasson [2]. To the author’s knowledge, there seems no example of divergent normal form in other normal form problems in the literature. The divergence of Birkhoff normal form implies, of course, that of all normalizing transformations of the given function. The importance of such a divergent normal form was demonstrated by Pérez-Marco [9] very recently. In this paper we shall prove