Quantum Ergodicity and the Analysis of Semiclassical Pseudodifferential Operators

作者: Felix Wong

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摘要: This undergraduate thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand quantum ergodicity theorem Schnirelman (1974), Zelditch (1987), Colin de Verdi\`ere (1985) unique conjecture Rudnick Sarnak (1994). The former states that, on any Riemannian manifold negative curvature or ergodic geodesic flow, eigenfunctions Laplace-Beltrami operator equidistribute in phase space density 1. Under same assumptions, latter that induce a sequence Wigner probability measures fibers Hamiltonian space, these converge weak-* topology uniform Liouville measure. If true, implies such high-eigenvalue limit no exceptional "scarring" patterns. physically means finest details chaotic systems can never reflect their quantum-mechanical behaviors, even limit. The main contribution this contextualize question an elementary analytic geometric framework. In addition presenting summarizing numerous important proofs, as Verdi\`ere's proof theorem, we perform graphical simulations certain billiard flows expositorily discuss several themes study chaos.

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