Eigenfunctions of Bunimovich billiard in coherent state representation
by Fernando P. Simonotti
Abstract
We study the semiclassical limit of eigenfunctions of classicaly chaotic
systems.
Particularly, we focus our analysis on the scarring phenomenon, in
which the
probability density shows spectacular enhancements along periodic orbits
of the
underlying classical system. We develop to this effect a method for
detecting
scars in the quantum spectrum. Using this construction on the stadium
billiard,
we are able, by means of symbolic dynamics, to identify scars of single
periodic
orbits and of families of them in the spectrum.
Moreover, we are able to obtain a semiclassical expression for the
projector
onto eigenfunctions bt means of the Fredholm theory. We express the
projector
in the coherent state basis, thus obtaining the semiclassical Husimi
representation of the stadium eigenfunctions, which is written in terms
of
classical invariants: periodic points, their monodromy matrices and
Maslov
indices.
Keywords: quantum chaos, semiclassical limit, Fredholm methods,
scars, Bunimovich stadium.
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