Mathematical Theory of Resonances (08w5092)
Organizers
Maciej Zworski (University of California at Berkeley)
Richard Froese (University of British Columbia)
Tanya Christiansen (University of Missouri)
Objectives
The last five years brought a string of new and surprising developments in the mathematical study of resonances:
Upper bound on the number of resonances for compactly supported perturbations were shown to be generically optimal by Christiansen and Hislop. The universal optimality is on the list of the 21st century Schr"odinger equation problems of Barry Simon. The difficulties are however highlighted by Tanya Christiansen's surprising example of families of complex potentials with {em no} resonances at all.
Fractal Weyl laws in chaotic scattering have been refined on the level of upper bounds by Sj"ostrand and Zworski. In the novel setting of open quantum maps, Nonnenmacher and Zworski showed that the bound is actually optimal. Numerical results of Lin, Lu, Sridhar, and Strain suggest optimality in a great range of settings. Heuristic arguments of Schomerus-Tworzyd{l}o provide conjectures related to random matrix theory, while the microwave experiments of Kuhl-St"ockmann are equally promising.
The work of Burq on the minimal distance of resonances to the real axis have been greatly refined by Vodev and his collaborators. In chaotic setting, the work of Nalini Anantharaman and Nonnenmacher led to developments in estimating quantum decay rates in terms of topological pressure. Resonances are gaining prominence in the study of nonlinear evolution problems, for instance in the work of Krieger and Schlag. Roughly speaking, the linearization of an equation leads to a Schr"odinger operator with a zero resonance. The study of resonances is also close to the study of spectra of non-self-adjoint operators, and these
arise as linearization of many equations, including Navier-Stokes (the Orr-Sommerfeld equation).
In geometric scattering Guillarmou has resolved many subtle problems surrounding meromorphic continuation of the resolvent on conformally compact spaces (which appear, for instance in AdS/CFT correspondence of string theory), and continued that work alone, and with Naud, and Sa Barreto. One example is a functional equation for the Selberg zeta function of convex co-compact quotient.
The goal of this workshop is to bring together experts working these and other aspects of the mathematical theory of resonances and their applications.
