Symmetry reduction and semiclassical analysis of axially symmetric systems

Abstract

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 $(\ensuremath{\rho}{,z,p}_{\ensuremath{\rho}}{,p}_{z})$ and have ${p}_{\ensuremath{\varphi}}=m\ensuremath{\Elzxh}$, where $m$ is the azimuthal quantum number. For $m\ensuremath{\ne}0$, 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 $m>0$, however, is not as dramatic as our numerical results indicate, suggesting that contributing orbits occur in topologically equivalent families within which ${p}_{\ensuremath{\varphi}}$ varies smoothly.