Periodic-orbit spectra of hydrogen and helium

The semiclassical expression for the spectral density (Gutzwiller’s trace formula) can be viewed as a Fourier expansion in terms of classical (quasi)periodic orbits. The inverse Fourier transform of the full quantum spectral density therefore not only gives the periodic orbit spectrum in the semiclassical limit, but may also provide a tool to characterize the quantum spectrum. We investigate the scaled energy Fourier transform spectrum of the hydrogen atom and doubly excited states of helium. For exactly solvable hydrogenic Rydberg spectra we study the ‘‘window’’ effect due to the finite length of the spectrum, the influence of resonance width, and of quantum defects. In helium a large number of periodic orbits including previously unknown orbits can be identified. © 1996 The American Physical Society.