|April 16, 2004: Alex Barnett, CIMS
Chaotic Billiards and Quantum Ergodicity
It is a long-standing question how quantum
eigenfunctions behave in the
semiclassical (short-wavelength) limit, when the corresponding classical
dynamics is chaotic. For instance, do they become ergodic
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 large-scale
numerical study on the rate of convergence to ergodicity, and
connections to 'arithmetic' billiards (number-theory 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.