 02174 Jens Bolte and Rainer Glaser
 A semiclassical Egorov theorem and quantum ergodicity for matrix
valued operators
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Apr 9, 02

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Abstract. We study the semiclassical time evolution of observables given by matrix
valued pseudodifferential operators and construct a decomposition of the
Hilbert space $L^2(\rz^d)\otimes\kz^n$ into a finite number of almost
invariant subspaces. For a certain class of observables, that is preserved
by the time evolution, we prove an Egorov theorem. We then associate with
each almost invariant subspace of $L^2(\rz^d)\otimes\kz^n$ a classical
system on a product phase space $\TRd\times\cO$, where $\cO$ is a compact
symplectic manifold on which the classical counterpart of the matrix
degrees of freedom is represented. For the projections of eigenvectors
of the quantum Hamiltonian to the almost invariant subspaces we finally
prove quantum ergodicity to hold, if the associated classical systems are
ergodic.
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