  We have developed a Green's function formalism based on the use of an
overcomplete semicoherent basis of vortex states, specially devoted to the
study of the Hamiltonian quantum dynamics of electrons at high magnetic fields
and in an arbitrary potential landscape smooth on the scale of the magnetic
length. This formalism is used here to derive the exact Green's function for an
arbitrary quadratic potential in the special limit where Landau level mixing
becomes negligible. This solution remarkably embraces under a unified form the
cases of confining and unconfining quadratic potentials. This property results
from the fact that the overcomplete vortex representation provides a more
general type of spectral decomposition of the Hamiltonian operator than usually
considered. Whereas confining potentials are naturally characterized by
quantization effects, lifetime effects emerge instead in the case of
saddle-point potentials. Our derivation proves that the appearance of lifetimes
has for origin the instability of the dynamics due to quantum tunneling at
saddle points of the potential landscape. In fact, the overcompleteness of the
vortex representation reveals an intrinsic microscopic irreversibility of the
states synonymous with a spontaneous breaking of the time symmetry exhibited by
the Hamiltonian dynamics.
