95-264 S.Zelditch

Quantum ergodicity of C* dynamical systems (68K, AmsLatex (amsart)) Jun 12, 95

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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.

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