Quantum Ergodicity

The Quantum Ergodicity Theorem of Shnirelman, Colin de Verdière and Zelditch is a result that combines in a nice way several different concepts. Namely, suppose {(M,g)} is a compact Riemannian manifold with ergodic geodesic flow. Let {\{\phi_k\}} be the appropriately normalized eigenfunctions of the Laplacian {\Delta}. Then the measures with density {\frac 1 N\sum_{k=1}^N |\phi_k|^2 \cdot d{\mathrm{Vol}}} tend weakly to the usual volume {d{\mathrm{Vol}}}.

I find the result interesting because it makes a dynamical assumption on the metric and says something about the eigenfunctions of the Laplacian. The tool connecting these two aspects is microlocal analysis. It appears in the proof, but not in the statement of the result.

Below the fold is a brief discussion of the main ideas that go into the proof. Weyl’s law for the density of eigenvalues appears as a rather simple result of the general formalism of microlocal analysis.

I am following a series of lectures notes due to Faure and Anantharaman available here.

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