Symmetry Reduction and Semiclassical Analysis of Axially Symmetric Systems

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We derive a semiclassical trace formula for a symmetry reduced part of the spectrum in axially symmetric systems. The classical orbits that contribute are closed in (ρ,z,pρ,pz) and have pϕ= mℏ where m is the azimuthal quantum number. For m > 0, these orbits vary with energy and almost never lie on periodic trajectories in the full phase space in contrast to the case of discrete symmetries. The transition from m=0 to m > 0 is however not dramatic as our numerical results indicate, suggesting that contributing orbits occur in topologically equivalent families within which pϕvaries smoothly.