Asymptotic analysis for Schrödinger Hamiltonians via Birkhoff–Gustavson normal form

This article reviews the Birkhoff–Gustavson normal form theorem (BGNF) near an equilibrium point of a quantum Hamiltonian. The BGNF process is thereafter used to investigate the spectrum of Schrödinger operators in the 1:1, 1:2 and 1:3 resonances. A computer program is proposed to compute the coefficients of the BGNF up to any order.