On the summation of the Birkhoff–Gustavson normal form of an anharmonic oscillator

The classical Birkhoff–Gustavson normal form (BGNF) has played an important role in finding approximate constants of motion, and semiclassical energies. In this paper, this role is examined in detail for the well-known anharmonic oscillator H=1/2(p2+x2+gx4). It is shown that, with appropriate restrictions, this is the only perturbation series that preserves the period of this system. This series has a nonzero radius of convergence in contrast to the zero radius of convergence of its quantum analog, the Rayleigh–Schrödinger perturbation series. In addition, the BGNF is generated to high order, and a technique is given based on Padé approximants for summing this series. The summation of this series makes possible an accurate comparison of torus quantization energies with the known quantum energies over the entire range of quantum numbers. This example also demonstrates that divergence of the BGNF series of a Hamiltonian is not sufficient to refute its global integrability.