The semiclassical theory of discontinuous systems and ray-splitting billiards
Dmitry Jakobson
Yuri Safarov
Alexander Strohmaier
Yves C. de Verdiere
2134/20023
https://repository.lboro.ac.uk/articles/journal_contribution/The_semiclassical_theory_of_discontinuous_systems_and_ray-splitting_billiards/9388199
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 (raysplitting)
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.
2016-01-14 11:32:04
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Mathematical Sciences not elsewhere classified