Lund, 22^{nd} - 24^{th} January 2004

Quantum Ergodicity - Basic Ideas and Recent Developments
We consider semiclassical pseudodifferential operators that are (Weyl-) quantisations of matrix valued symbols such as, e.g., Dirac operators. In this setting to each eigenvalue of the principal symbol of the Hamiltonian operator there corresponds an almost invariant subspace of the quantum Hilbert space as well as a classical dynamical system. The latter consists of a skew product flow built over the Hamiltonian flow generated by the eigenvalue of the principal symbol. Suitable invariant semiclassical observables are identified and an Egorov theorem is proven for them. Quantum ergodicity is then shown to hold for the projections of eigenvectors to an almost invariant subspace provided the corresponding skew product flow is ergodic.

Slides: 01 -- 02 -- 03 -- 04 -- 05 -- 06 -- 07 -- 08 -- 09 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 -- 16 -- 17 -- 18

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