Dynamical, Spectral, and Unique Continuation Properties on Waveguides

Organizers of this minisymposium are

Mathematical models of waveguides have a plethora of applications in Mathematical Physics, Photonics, Materials Science, and Engineering. A thorough analysis of the governing operator often enables a precise prediction of the resulting dynamics and associated a-priori bounds. The study of systems of waveguides and their 1-dimensional versions involving metric graphs is particularly interesting due to nontrivial topological features of these structures, that spill over into analytical properties as well. The same remains true also for simplified discrete graph models. This is related to such intriguing effects as chaotic scattering and Anderson (de)localization in the context of random Schrödinger operators. A prominent fact is that, for evolution or time-harmonic PDEs of Schrödinger-type in the continuum space, the unique continuation property holds under very mild assumptions on the coefficients, even in strong quantitative ways. However, the unique continuation property no longer holds true for the corresponding differential and difference equations on metric and discrete graphs, as well as for Maxwell equations with non-Lipschitz anisotropic material parameters in spatial domains. This makes certain analytical features dramatically different. The study of dynamical systems on metric graphs provides important insights into the effects connected with non-unique continuation.
This minisymposium is designed to facilitate the communication between specialists having complementary and partially overlapping areas of expertise in the fields of evolution PDEs on waveguides, quantum graphs, and time-harmonic problems involving discrete and continuous Schrödinger operators.