On the limitations of the Birkhoff-Gustavson normal form approach

Abstract

Recently, the Birkhoff-Gustavson normal form (BGNF) approach has been used in both classical and semiclassical calculations. The authors consider the limitations associated with this approach using as an illustration the anharmonic oscillator. At the classical level, they emphasise the need to properly take into account the divergence properties of the BGNF series. By reviewing an earlier work they demonstrate that this divergence problem may, in some cases, be resolved. However, after considering several standard quantisation rules, they find that it is not possible to obtain the correct quantisation of a system via the BGNF. As a result, they maintain that care should be used when working with the BGNF.