Jens Marklof, Zeev Rudnick
Quantum unique ergodicity for parabolic maps
(59K, LaTeX 2e)
ABSTRACT. We study the ergodic properties of quantized ergodic maps of the torus. It
is known that these satisfy quantum ergodicity: For almost all
eigenstates, the expectation values of quantum observables converge to the
classical phase-space average with respect to Liouville measure of the
corresponding classical observable.
The possible existence of any exceptional subsequences of eigenstates is
an important issue, which until now was unresolved in any example. The
absence of exceptional subsequences is referred to as quantum unique
ergodicity (QUE). We present the first examples of maps which satisfy
QUE: Irrational skew translations of the two-torus, the parabolic
analogues of Arnold's cat maps. These maps are classically uniquely
ergodic and not mixing. A crucial step is to find a quantization recipe
which respects the quantum-classical correspondence principle.
In addition to proving QUE for these maps, we also give results on the
rate of convergence to the phase-space average. We give upper bounds which
we show are optimal. We construct special examples of these maps for which
the rate of convergence is arbitrarily slow.