title: Simple dynamical system with discrete bound states
abstract: We study numerically the dynamical system of a two-electron atom with the Darwin interaction as a model to investigate scale-dependent effects of the relativistic action-at-a-distance electrodynamics. This dynamical system consists 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 and (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 interaction), generates the phenomenon of scale-dependent stability: We find numerical evidence that stable nonionizing orbits can exist only for a discrete set of resonant energies. The Fourier transform of these nonionizing orbits is a set of sharp frequencies. The energies and sharp frequencies of the nonionizing orbits we study are in the quantum atomic range.
authors.affiliation: Departamento de Fisica, Universidade Federal de Sao Carlos, Rodovia Washington Luiz, km 235, Caixa Postal 676, Sao Carlos, Sao Paulo 13565-905, Brazil.
authors.lastname: De Luca J