Simple dynamical system with discrete bound states

We study numerically the dynamical system of a two-electron atom with the D arwin interaction as a model to investigate scale-dependent effects of the relativistic action-at-a-distance electrodynamics. This dynamical system co nsists of a small perturbation of the Coulomb dynamics for energies in the atomic range. The key properties of the Coulomb dynamics are (i) a peculiar mixed-type phase space with sparse families of stable nonionizing orbits a nd (ii) scale-invariance symmetry, with all orbits defined by an arbitrary scale parameter. The combination of this peculiar chaotic dynamics [(i) and (ii)], with the scale-dependent relativistic corrections (Darwin interacti on), generates the phenomenon of scale-dependent stability: We find numeric al evidence that stable nonionizing orbits can exist only for a discrete se t of resonant energies. The Fourier transform of these nonionizing orbits i s a set of sharp frequencies. The energies and sharp frequencies of the non ionizing orbits we study are in the quantum atomic range.