Suresh Eswarathasan

STRONG SCARRING AND CLOSED HYPERBOLIC TRAJECTORIES

Abstract: 

There is a classical result in microlocal analysis which
states that given an elliptic periodic orbit ?, we can
construct quasimodes of order O(h^{?}) which
concentrate on ? (due to Colin de Verdiere, Ralston, and
others).  In the case of hyperbolic orbits, the well-known Gaussian
beam construction used to construct these quasimodes breaks down.

In joint with Stephane Nonnenmacher (U. Paris-Sud, Orsay), we
generalize a recent result of S. Brooks for the case of compact
hyperbolic surfaces.  Given a compact surface (M,g) and a hyperbolic
orbit ?', we construct logarithmic quasimodes (i.e. those of
order C h / | log h|) which are partially localized on ?'.
Furthermore, we give explicit relations between the order constant
C>0 for the quasimode and the corresponding localization properties.
Our construction uses a quantum normal form due to Sjoestrand and an
averaging procedure due to Vergini et. al.