  In this thesis, we investigate quantum ergodicity for two classes of
Hamiltonian systems satisfying intermediate dynamical hypotheses between the
well understood extremes of ergodic flow and quantum completely integrable
flow. These two classes are mixed Hamiltonian systems and KAM Hamiltonian
systems.
  Hamiltonian systems with mixed phase space decompose into finitely many
invariant subsets, only some of which are of ergodic character. It has been
conjectured by Percival that the eigenfunctions of the quantisation of this
system decompose into associated families of analogous character. The first
project in this thesis proves a weak form of this conjecture for a class of
dynamical billiards, namely the mushroom billiards of Bunimovich for a full
measure subset of a shape parameter $t\in (0,2]$.
  KAM Hamiltonian systems arise as perturbations of completely integrable
Hamiltonian systems. The dynamics of these systems are well understood and have
near-integrable character. The classical-quantum correspondence suggests that
the quantisation of KAM systems will not have quantum ergodic character. The
second project in this thesis proves an initial negative quantum ergodicity
result for a class of positive Gevrey perturbations of a Gevrey Hamiltonian
that satisfy a mild "slow torus" condition.
