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The semiclassical theory of discontinuous systems and ray-splitting billiards
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 137, Number 4, August 2015
- pp. 859-906
- 10.1353/ajm.2015.0027
- Article
- Additional Information
We analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities.
Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit
does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe
this system we introduce a dynamical system on the space of functions on phase space. To identify the quantum dynamics
in the semiclassical limit we compute the principal symbols of the Fourier integral operators associated to reflected
and refracted geodesic rays and identify the relation between classical and quantum dynamics. In particular we prove
a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity
for the ray-splitting dynamics.