The classical and quantum-mechanical answers to this question are totally different.
Classically you have the standard 1/r^2 potential well and can solve the two-body problem analytically to find the conditions for capture and escape (which turn out to be elliptical and parabolic orbits), given a known starting position and velocity. However, whether you count an electron orbiting a proton with an eccentricity of 0.99 as an "atom" might depend on whether you call Pluto a planet! You also need to disregard radiation which will lead to fast decaying orbits.
Quantum mechanically I know a lot less about. You can probably find solutions to the static Schroedinger equation in a coulomb potential and argue they represent an atom, but a free electron appears in the maths as a wave with a time-dependant element so I don't know how it would be approached.