FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
September 20, 2012
at 4:00 p.m.
Fields Insititue, Room 230
Commissariat à l'énergie atomique, Saclay
Counting stationary modes: a discrete view of geometry and
Co-sponsored by the Fields Institute and Department of Mathematics,
University of Toronto
In this lecture I first plan to present the historical context leading
to Hermann Weyl's first result on the high frequency eigenvalue
counting for the Laplacian on a planar domain. I will then sketch
the mathematical developments on that question, and various extensions
of this result, including a semiclassical version useful in quantum
mechanics, as well as the case of "fractal domains". Such
spectral asymptotics can often reveal a lot of information on the
geometry of the domain (or manifold) and the associated geodesic
(or Hamiltonian) dynamics.
I will then switch to the study of (quantum) scattering systems.
Such systems admit a discrete set of complex-valued "generalized
eigenvalues", called resonances. Counting such resonances has
proved a difficult task, mainly due to the nonselfadjoint nature
of the problem.
Yet, I will present some resonance counting estimates, which may
also reflect some dynamical features of the corresponding classical
dynamics; this is the case, for instance, of the "fractal Weyl's
law" expected to hold for chaotic scattering systems.