Quantum Birkhoff–Gustavson normal form in some completely resonant cases

The goal of this paper is to present an explicit computation of the Birkhoff–Gustavson normal form in the 1:1, 1:2 and 1:3 resonances, by using the concept of Weyl quantization, especially when it is about the first terms. We use in particular the Weyl symbolic calculus and the abstract theorem of the quantum Birkhoff–Gustavson normal form in infinite dimension.