Arnold’s Web and Diffusion in the Stark–Quadratic–Zeeman Problem

The Arnold web and the Arnold diffusion arise when an integrable Hamiltonian system is slightly perturbed: the first concerns the peculiar topology characterizing the set of the resonance lines in phase space, the latter the extremaly slow motion (if any) along these lines. While Arnold has proved the possibility of diffusion, it is still unknown if the phenomenon is generic in realistic physical systems. The system we consider is the Hydrogen atom (or Kepler problem) subject to the combined action of a constant electric and magnetic field, which is known as Stark–Zeeman problem. We describe the results of numerical experiments: the Arnold web is clearly highlighted and, looking at the behaviour of the KAM frequencies on orbits of 10⁸ revolutions, evidence for the diffusion existence is reached.