Asymptotic analysis for Schrödinger Hamiltonians via Birkhoff–Gustavson normal form

Abstract

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.