Birkhoff-Gustavson normal form in classical and quantum mechanics

A quantum analogue of the classical transformations to the Birkhoff-Gustavson normal form is derived. With unitary transformations taking over the role of canonical transformations, the author finds a striking similarity to Lie transformations in the classical case. The author suggests that the quantum normal form is identical to the Rayleigh-Schrodinger perturbation series and thus usually only asymptotic to the eigenvalues. The convergence question of both classical and quantum normal forms is discussed in some detail.