  We examine the consequences of classical ergodicity for the localization
properties of individual quantum eigenstates in the classical limit. We note
that the well known Schnirelman result is a weaker form of quantum ergodicity
than the one implied by random matrix theory. This suggests the possibility of
systems with non-gaussian random eigenstates which are nonetheless ergodic in
the sense of Schnirelman and lead to ergodic transport in the classical limit.
These we call "weakly quantum ergodic.'' Indeed for a class of "slow ergodic"
classical systems, it is found that each eigenstate becomes localized to an
ever decreasing fraction of the available state space, in the semiclassical
limit. Nevertheless, each eigenstate in this limit covers phase space evenly on
any classical scale, and long-time transport properties betwen individual
quantum states remain ergodic due to the diffractive effects which dominate
quantum phase space exploration.
