CRM-McGill Applied Math Seminar
Tuesday Mar 11, 2008, 3:35pm
McGill, Burnside Hall Room 1205

Speaker: Alex Barnett, Dartmouth

Title: "Eigenmodes and quantum chaos: Lost on the frequency axis? Check your
Dirichlet-to-Neumann map!"

Abstract:
The Dirichlet eigenmode (or `drum') problem describes vibrations of an
elastic membrane, acoustic cavity, or quantum particle, and is a
paradigm for more complicated applications to electromagnetic and
optical resonators.  When the wavelength is much shorter than the
cavity size this becomes a challenging multiscale problem, and
boundary methods are essential.  I will explain an accelerated cousin
of the method of particular solutions (MPS, a global basis
approximation method) which allows O(k) modes to be calculated in the
effort usually required for a single mode, k being the wavenumber.  It
removes the need for expensive root-searches along the wavenumber (ie
frequency) axis.  At very high frequencies and many cavity shapes, it
is the fastest method known, 10^3 times faster than either MPS or
boundary integral methods.

This has enabled large-scale numerical study of the asymptotic
properties of planar eigenmodes. I will present recent data on
`dynamical tunneling' in Bunimovich's mushroom cavity, which has both
chaotic and integrable motion.  If time I will mention recent work
with T. Betcke on the Helmholtz BVP using fundamental solutions bases,
which in analytic shapes give spectral accuracy approaching only 2
degrees of freedom per wavelength on the boundary