J.~Asch and A.~Knauf, Motion in periodic potenials.
Nonlinearity {\bf 11} (1998), 175--200.
Abstract: We consider motion in a periodic potential in a classical,
quantum, and semiclassical context. Various results on the distribution
of asymptotic velocities are proven.
J.~Asch and A.~Knauf, Quantum Transport on KAM Tori.
Comm. Math. Phys.\ {\bf 205} (1999), 113--128.
Abstract: Although quantum tunneling between phase space tori occurs,
it is suppressed in the semiclassical limit $\hbar\searrow 0$ for the
Schr\"{o}dinger equation of a particle in $\bR^d$ under the influence
of a smooth periodic potential. In particular this implies that the
distribution of quantum group velocities near energy $E$ converges to
the distribution of the classical asymptotic velocities near $E$, up to
a term of the order $\cO(1/\sqrt{E}.