S.Zelditch
Quantum ergodicity of C* dynamical systems
(68K, AmsLatex (amsart))
ABSTRACT. The purpose of this paper is to generalize some basic notions and results
on quantum
ergodicity ( [Sn], [CV], [Su], [Z.1], [Z.2]) to a wider class of $C^*$
dynamical
systems $(\ A, G, \alpha)$ which we call {\it quantized Gelfand-Segal
systems} (Definition 1.1). The key feature of such a system is an invariant
state $\omega$ which
in a certain sense
is the barycenter of the normal invariant states.
By the Gelfand-Segal construction, it induces a new system $(\A_{\omega},
G, \alpha_{\omega}),$ which will play the role of the classical limit.
Our main abstract result
(Theorem 1 ) shows that if $(\A, G, \alpha)$ is a quantized GS system, if
the classial limit
is abelian (or if $(\A, \omega)$ is a ``G-abelian" pair), and if
$\omega $ is an ergodic state, then ``almost all" the ergodic normal
invariant states
$\rho_j$ of the system tend to $\omega $ as the ``energy"
$E(\rho_j)\rightarrow
\infty$. This leads to an intrinsic notion of the quantum ergodicity of
a quantized GS system
in terms of operator time and space averages
(Definition 0.1), and to the result that a quantized GS system is
quantum
ergodic if its classical limit is an ergodic abelian system
(or if $(\A, \omega)$ is an ergodic G-abelian pair) (Theorem 2).
Concrete applications include a simplified proof
of quantum ergodicity of the wave group of a compact Riemannian
manifold with ergodic geodesic flow, as well as extensions to manifolds
with concave boundary and ergodic billiards, to quotient Hamiltonian
systems on
symplectic quotients and to ergodic Hamiltonian subsystems on sympletic
subcones.