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In quantum chaos, we make use of tools from microlocal analysis to examine how the dynamical features of a classical Hamiltonian system are reflected in the behaviour of the PDE that governs its quantisation.
The quantum ergodicity theorem of Shnirelman, Colin de Verdière, and Zelditch is a cornerstone theorem in this field, which states that if the geodesic flow on a closed Riemannian manifold is ergodic, then the Laplacian eigenfunctions satisfy an analogous equidistribution property in the high energy limit.
There is a longstanding conjecture by Percival on the analogous result for manifolds with geodesic flow of mixed character, with phase space that can be partitioned into multiple invariant subsets, only some of which are ergodic. In this talk, we shall discuss a partial result for the prototypical mixed system, the mushroom billiard.