phase approximation). New divergences might pop up that need to be cured using the same method as step 3 Use (tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states . Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i.e. periodic orbits. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books. Closed orbit theory Closed-orbit theory was developed by J.B. Delos, M.L. Du, J. Gao, and J. Shaw. It is similar to periodic-orbit theory, except that closed-orbit theory is applicable only to atomic and molecular spectra and yields the oscillator strength density (observable photo-absorption spectrum) from a specified initial state whereas periodic-orbit theory yields the density of states. Only orbits that begin and end at the nucleus are important in closed-orbit theory. Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state. For Rydberg atoms and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time. According to closed-orbit theory, the average oscillator strength density at constant is given by a smooth background plus an oscillatory sum of the form is a phase that depends on the Maslov index and other details of the orbits. is the recurrence amplitude of a closed orbit for a given initial state (labeled ). It contains information about the stability of the orbit, its initial and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy Coulomb wave. For scaling systems such as Rydberg atoms in strong fields, the Fourier transform of an oscillator strength spectrum computed at fixed as a function of is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to . Closed-orbit theory has found broad agreement with a number of chaotic systems, including diamagnetic hydrogen, hydrogen in parallel electric and magnetic fields, diamagnetic lithium, lithium in an electric field, the ion in crossed and parallel