April 16, 2004: Alex Barnett, CIMS
Chaotic Billiards and Quantum Ergodicity
It is a longstanding question how quantum
eigenfunctions behave in the
semiclassical (shortwavelength) limit, when the corresponding classical
dynamics is chaotic. For instance, do they become ergodic
(equidistributed
across space) or are remnants (scars) of classical periodic orbits
important? I will introduce `billiards' (where `quantum' simply means:
Laplacian operator with Dirichlet boundary conditions), one of the
paradigm quantum chaotic systems. I will present my recent largescale
numerical study on the rate of convergence to ergodicity, and
connections to 'arithmetic' billiards (numbertheory territory).
To pique the interest of numerical analysts in the audience, I will
outline the extremely fast 'scaling method' for solving the Laplace
eigenproblem that I have helped develop.
