A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators

TL;DR

This paper develops a semiclassical Egorov theorem and demonstrates quantum ergodicity for matrix-valued operators by decomposing the Hilbert space into almost invariant subspaces linked to classical systems.

Contribution

It introduces a decomposition of the Hilbert space into almost invariant subspaces and establishes quantum ergodicity for matrix-valued operators with classical ergodic systems.

Findings

Decomposition of Hilbert space into almost invariant subspaces.

Proof of Egorov theorem for certain matrix-valued observables.

Quantum ergodicity established for eigenvector projections under classical ergodicity.

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.